Small Signal Stability and Dynamics

Chapter 1

System Stability

loads are connected by a power system.

1.1 Overview of power system structure

cables. Due to excessive power loss (I

3

Kenneth Eloghene Okedu

1. Technical Background

Introductory Chapter: Power

Among the various available energy systems, electrical energy is the most popu-

Earlier electric network stations supplied DC (direct current) power for lightning. The power was generated by DC generators and distributed by underground

such system could only travel short distances from their stations. When electrical transformers were invented, it created room for the prevalence of the AC (alternating current) system over the DC system. This is because the electrical transformers were able to raise the level of AC voltage for transmission and distribution. The AC system was further boosted with the invention of induction motors to replace DC motors. In addition, the merits of the AC system became apparent due to the fact that more power can be produced at higher voltages at convenience

As a result of the apparent advantages of the AC system, the single-phase and three-phase AC systems emerged. Many electric companies and independent power producers were operating at different frequencies. However, as the need for interconnection and parallel operation became imperative, a standard frequency of either 50 or 60Hz was adopted. Consequently, transmission voltages rose steadily and gave birth to extra high voltages (EHVs) mostly used for commercial purposes. It may be more economical to convert EHV based on AC to EHV based on DC, when considering power transmission over long distances. This would involve transmission of the power via a two-line system and its inversion from DC back to AC at the other terminal. From the literature, it was reported that it is of more benefit to consider DC lines when the transmission distance is 500 km or more. It should be noted that DC lines possesses no reactance and they have the ability of transferring more power considering the same conductor size than AC lines. The

R), at low voltage, the energy delivered from

2

because of the lack of commutators in the AC generators [1].

lar form, because it can be transported easily at high efficiency and reasonable cost from one place to the other. Electrical machine is a device that converts mechanical energy to electrical energy or vice versa. In the earlier case, the machine is known as a generator, while in the latter case, it is called a motor. The action of magnetic field is used in both machines for the conversion of energy from one form to the other. A power system is a network of components that is well designed and structured to efficiently transmit and distribute electrical energy produced by generators to locations where they are utilized. Generators, motors and other utility

## Chapter 1

## Introductory Chapter: Power System Stability

Kenneth Eloghene Okedu

## 1. Technical Background

Among the various available energy systems, electrical energy is the most popular form, because it can be transported easily at high efficiency and reasonable cost from one place to the other. Electrical machine is a device that converts mechanical energy to electrical energy or vice versa. In the earlier case, the machine is known as a generator, while in the latter case, it is called a motor. The action of magnetic field is used in both machines for the conversion of energy from one form to the other. A power system is a network of components that is well designed and structured to efficiently transmit and distribute electrical energy produced by generators to locations where they are utilized. Generators, motors and other utility loads are connected by a power system.

#### 1.1 Overview of power system structure

Earlier electric network stations supplied DC (direct current) power for lightning. The power was generated by DC generators and distributed by underground cables. Due to excessive power loss (I 2 R), at low voltage, the energy delivered from such system could only travel short distances from their stations. When electrical transformers were invented, it created room for the prevalence of the AC (alternating current) system over the DC system. This is because the electrical transformers were able to raise the level of AC voltage for transmission and distribution. The AC system was further boosted with the invention of induction motors to replace DC motors. In addition, the merits of the AC system became apparent due to the fact that more power can be produced at higher voltages at convenience because of the lack of commutators in the AC generators [1].

As a result of the apparent advantages of the AC system, the single-phase and three-phase AC systems emerged. Many electric companies and independent power producers were operating at different frequencies. However, as the need for interconnection and parallel operation became imperative, a standard frequency of either 50 or 60Hz was adopted. Consequently, transmission voltages rose steadily and gave birth to extra high voltages (EHVs) mostly used for commercial purposes.

It may be more economical to convert EHV based on AC to EHV based on DC, when considering power transmission over long distances. This would involve transmission of the power via a two-line system and its inversion from DC back to AC at the other terminal. From the literature, it was reported that it is of more benefit to consider DC lines when the transmission distance is 500 km or more. It should be noted that DC lines possesses no reactance and they have the ability of transferring more power considering the same conductor size than AC lines. The

main advantage of DC transmission is in the scenario where two remotely located large power systems are to be connected via a tie line. In this case, the DC tie line transmission system acts as a synchronous link between the two rigid power systems eliminating the instability problem that is common with the AC links. However, the production of harmonics that requires filtering in addition to the large amount of reactive power compensation required at both ends of the line is a major setback of the DC link system [1, 2].

1.2.2 Transformers

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

1.2.4 Loads

5

Transformers are another major component of a power system that allows power to be transmitted with minimal loss over a long distance. Power is transferred with very high efficiency from one level of voltage to another level by the use of the transformers. In a transformer, the power transferred to the secondary side is almost the same as the primary side except for losses in the transformer. The product VI on the secondary side of the transformer is approximately same as that of the primary side. Hence, a step-up transformer of turns ratio a will reduce the secondary current by a ratio of 1=a. Consequently, this will reduce losses in the line, which makes the transmission of power over long distance possible. Due to the insulation requirements and practical design problems, the generated voltage is limited to low values. Therefore, the step-up transformers are used for transmission of power, while at the receiving end of the transmission line, the step-down transformers are used to reduce the voltage to the required values for distribution and utilization. The transmitted power might undergo several transformations between generator and end users. Recent generators usually generate electrical power at voltages of 13.8–24 kV and transmission lines operate at very high voltages in order to reduce transmission losses. Electrical loads consume power at various voltage levels of 110 and 220 V,

for residential and up to 4160 V for industrial applications [1, 2, 4].

The transmission and distribution lines are also known as power lines. They connect generators to loads, and transmit electrical power from one place to the other at minimal loss. Transmission lines also interconnect neighboring utilities, which permits not only economic dispatch of power within regions during steadystate or normal working conditions but also transfer of power between regions during emergencies. Thus, transmission lines are designed to efficiently transfer

lines, they run at very high voltages [1, 2]. Upon receiving the power at the area of the end user, the transmission voltage is stepped down and the power is supplied through distribution lines to the final customer. Much less power is carried by the distribution lines and they operate for shorter distances at lower voltages without prohibitive losses compared to the transmission lines. The distribution system could be either overhead or underground. In recent times, the growth of the underground

The loads on a power system are of different types. These loads could be electric motors, electric lighting, and others. However, a broad division of loads in a power system could be: industrial, commercial, and residential. The transmission system could serve very large industrial loads directly, while small industrial loads are served by the primary distribution network. The industrial loads are mainly composite loads and induction motors. The composite loads depend on voltage and frequency and they form bulk of the system load. Commercial and residential loads are made of lighting, heating, and cooling loads and they are independent of frequency with small or negligible reactive power consumption. Kilowatts or megawatts are used to define and express the real power of loads. The real power should be available to the end users and the magnitude of the load varies throughout the day. A composite of the demands made by various classes of utility end users gives

2

R) in the

electrical power over long distances. In order to reduce resistive losses (I

distribution has been rapid in modern residential constructions.

1.2.3 Transmission and distribution (power lines)

The interconnection of the entire or overall network system is known as the power grid. When the system is divided into several geographical regions, they are called power pools. In an interconnected system or grid network, there exist fewer generators that are required as reserve for peak load and spinning reserve. The power grid allows energy penetration and transmission in a more reliable and economical way due to the fact that power can readily be transferred from one area to another. Most times, it may be cheaper for a power-producing company to purchase bulk power from the interconnected system instead of generating its own power.

#### 1.2 Power system components

The major components of modern power systems are as follows.

### 1.2.1 Generators

Generators are one of the essential components of a power system. They produce electrical energy distributed by a power system. Most generators produce electrical energy by converting mechanical energy to electrical energy through the action of a magnetic field. The converted mechanical energy comes from a prime mover, which is a device that spins the generator. Steam and water turbines are some usual forms of prime movers, but in remote locations diesel engines have been used. Prime movers can operate based on many energy sources like water, coal, natural gas, oil, and nuclear energy. The prime mover based on water appears to be one of the best because it is non-polluting and requires no fuel cost. Nuclear power plants are expensive to construct and elaborate safety measures are required. Although, the cost of fuel is low and they are non-polluting energy sources in nature. Therefore, a combination of hydroelectric and nuclear power generators to power a given system would result in low fuel cost and the system can effectively run for long at full power rating. Coal plants are the most common source of electrical power generation because coal is a relatively cheap fuel. But it is unfortunate that coal is one of the most polluting fuel sources. Antipollution features are required in coal-fired plants in order to control pollution. Natural gas is a much cleaner and better energy source compared to coal. Its burning process emits little pollution and it is relatively cheaper. The main drawback of natural gas is that it is difficult to transport over long distances due to the fact that it is flammable in nature. Oil is much easier to transport; however, it is more polluting and more expensive than natural gas. Generally, coal, oil, and natural gas are the traditional non-renewable energy sources and this is a demerit of these energy sources. Wind, solar, biomass, and geothermal resources are other sources of energy that are renewable in nature; however, they are not yet economical when compared to the traditional sources of energy [1–3]. It should be noted that no source of electrical energy has all it takes to be perfect in producing electricity. All the available sources have their pros and cons and sometimes, a combination of two or more energy mix used in hybrid energy system is encouraged.

#### 1.2.2 Transformers

main advantage of DC transmission is in the scenario where two remotely located large power systems are to be connected via a tie line. In this case, the DC tie line transmission system acts as a synchronous link between the two rigid power systems eliminating the instability problem that is common with the AC links. However, the production of harmonics that requires filtering in addition to the large amount of reactive power compensation required at both ends of the line is a major

The interconnection of the entire or overall network system is known as the power grid. When the system is divided into several geographical regions, they are called power pools. In an interconnected system or grid network, there exist fewer generators that are required as reserve for peak load and spinning reserve. The power grid allows energy penetration and transmission in a more reliable and economical way due to the fact that power can readily be transferred from one area to another. Most times, it may be cheaper for a power-producing company to purchase bulk power from the interconnected system instead of generating its own

The major components of modern power systems are as follows.

Generators are one of the essential components of a power system. They produce electrical energy distributed by a power system. Most generators produce electrical energy by converting mechanical energy to electrical energy through the action of a magnetic field. The converted mechanical energy comes from a prime mover, which is a device that spins the generator. Steam and water turbines are some usual forms of prime movers, but in remote locations diesel engines have been used. Prime movers can operate based on many energy sources like water, coal, natural gas, oil, and nuclear energy. The prime mover based on water appears to be one of the best because it is non-polluting and requires no fuel cost. Nuclear power plants are expensive to construct and elaborate safety measures are required. Although, the cost of fuel is low and they are non-polluting energy sources in nature. Therefore, a combination of hydroelectric and nuclear power generators to power a given system would result in low fuel cost and the system can effectively run for long at full power rating. Coal plants are the most common source of electrical power generation because coal is a relatively cheap fuel. But it is unfortunate that coal is one of the most polluting fuel sources. Antipollution features are required in coal-fired plants in order to control pollution. Natural gas is a much cleaner and better energy source compared to coal. Its burning process emits little pollution and it is relatively cheaper. The main drawback of natural gas is that it is difficult to transport over long distances due to the fact that it is flammable in nature. Oil is much easier to transport; however, it is more polluting and more expensive than natural gas. Generally, coal, oil, and natural gas are the traditional non-renewable energy sources and this is a demerit of these energy sources. Wind, solar, biomass, and geothermal resources are other sources of energy that are renewable in nature; however, they are not yet economical when compared to the traditional sources of energy [1–3]. It should be noted that no source of electrical energy has all it takes to be perfect in producing electricity. All the available sources have their pros and cons and sometimes, a combination of two or more energy mix

setback of the DC link system [1, 2].

Power System Stability

1.2 Power system components

used in hybrid energy system is encouraged.

4

power.

1.2.1 Generators

Transformers are another major component of a power system that allows power to be transmitted with minimal loss over a long distance. Power is transferred with very high efficiency from one level of voltage to another level by the use of the transformers. In a transformer, the power transferred to the secondary side is almost the same as the primary side except for losses in the transformer. The product VI on the secondary side of the transformer is approximately same as that of the primary side. Hence, a step-up transformer of turns ratio a will reduce the secondary current by a ratio of 1=a. Consequently, this will reduce losses in the line, which makes the transmission of power over long distance possible. Due to the insulation requirements and practical design problems, the generated voltage is limited to low values. Therefore, the step-up transformers are used for transmission of power, while at the receiving end of the transmission line, the step-down transformers are used to reduce the voltage to the required values for distribution and utilization. The transmitted power might undergo several transformations between generator and end users. Recent generators usually generate electrical power at voltages of 13.8–24 kV and transmission lines operate at very high voltages in order to reduce transmission losses. Electrical loads consume power at various voltage levels of 110 and 220 V, for residential and up to 4160 V for industrial applications [1, 2, 4].

## 1.2.3 Transmission and distribution (power lines)

The transmission and distribution lines are also known as power lines. They connect generators to loads, and transmit electrical power from one place to the other at minimal loss. Transmission lines also interconnect neighboring utilities, which permits not only economic dispatch of power within regions during steadystate or normal working conditions but also transfer of power between regions during emergencies. Thus, transmission lines are designed to efficiently transfer electrical power over long distances. In order to reduce resistive losses (I 2 R) in the lines, they run at very high voltages [1, 2]. Upon receiving the power at the area of the end user, the transmission voltage is stepped down and the power is supplied through distribution lines to the final customer. Much less power is carried by the distribution lines and they operate for shorter distances at lower voltages without prohibitive losses compared to the transmission lines. The distribution system could be either overhead or underground. In recent times, the growth of the underground distribution has been rapid in modern residential constructions.

#### 1.2.4 Loads

The loads on a power system are of different types. These loads could be electric motors, electric lighting, and others. However, a broad division of loads in a power system could be: industrial, commercial, and residential. The transmission system could serve very large industrial loads directly, while small industrial loads are served by the primary distribution network. The industrial loads are mainly composite loads and induction motors. The composite loads depend on voltage and frequency and they form bulk of the system load. Commercial and residential loads are made of lighting, heating, and cooling loads and they are independent of frequency with small or negligible reactive power consumption. Kilowatts or megawatts are used to define and express the real power of loads. The real power should be available to the end users and the magnitude of the load varies throughout the day. A composite of the demands made by various classes of utility end users gives

the daily load curve, and the greatest value of load during a period of 24 h is known as maximum or peak demand. Some key factors like the load factor (ratio of average load over a designated period of time to the peak load occurring in that period), utilization factor (ratio of maximum demand to the installed capacity), and plant factor (product of 8760 h and the ratio of annual energy generation to the plant capacity) help judge the performance of the system. In order for the a power system plant to operate economically, the load factor must be high, while the utilization and plant factors indicate how well the system capacity is usually operated and utilized [1, 5, 6].

called the power angle or torque angle. During disturbances, the rotor accelerates/ decelerates with respect to the synchronously rotating air gap, thus a relative motion begins. The equation describing this relative motion is known as the swing

where δ is the electrical radian; H is the per unit inertia constant; Pm∧Pe are the per unit mechanical and electrical power, respectively; and f <sup>0</sup> is the frequency of

Consider a generator connected to a major substation of a very large system via a

The substation bus voltage and frequency are assumed to remain constant (infi-

terminal voltage Vg can be eliminated by converting the Y connected impedances to

dt<sup>2</sup> <sup>¼</sup> Pm � Pe (1)

dt<sup>2</sup> <sup>¼</sup> Pm � Pe (2)

<sup>d</sup>. The node representing the generator

H πf <sup>0</sup>

H 180f <sup>0</sup>

1.5 Stability studies for synchronous generator models

transmission line as shown below (Figures 1 and 2).

behind the direct axis transient reactance X<sup>0</sup>

Δ with admittances as [1]

One machine connected to an infinite bus.

Equivalent circuit of one machine connected to an infinite bus.

Figure 1.

Figure 2.

7

d2 δ

nite bus). This is because its characteristics do no change regardless of power supplied or consumed by it. The generator is represented by a constant voltage

d2 δ

equation given below [1, 2].

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

the system. With δ in degrees, then

#### 1.2.5 Protection system

The protection system for a power system involves a variety of protective devices like current, voltage, power sensors, relays, fuses, and circuit breakers. The protective devices that are connected directly to the circuits are known as switchgears (e.g., instrument transformers, circuit breakers, disconnect switches, fuses, and lighting arresters). The presence of these devices is required in order to de-energize the power system either in scenarios of normal operation or in the occurrence of faults [1, 2]. The control house contains the associated control equipment and protective relays. There are basically two types of failures in a power system: overloads and faults. Overload conditions occur when the components in the power system are supplying more power than they were designed to carry safely. This scenario usually occurs when the total demand on the power system surpasses the capability of the system to supply power. Overloads often occur in new residential or industrial construction areas of the power system due to expansion. There are measures in place for the power system operator to immediately correct and control overload conditions due to the robustness of the system in order to avoid damage to the power network. On the other hand, fault conditions occur when one or more of the phases in a power system are shorted to ground or to each other (i.e., single phase to ground, three phases to ground, line to line, etc.). When a phase is open circuited, faults also occur in such situation. During periods of short circuit, very large currents flow and damage the entire power system if no measures are in place to quickly stop it. Faults must be cleared as quickly as possible in a power system when they occur, unlike overloads. For this reason, relays are employed to automatically open circuit breakers and isolate faulty areas; then, they are sensed in a power system [7, 8].

#### 1.3 Power system stability

The tendency of a power system to develop restoring forces equal to or greater than the disturbing forces to maintain the state of equilibrium is known as stability. Power system stability problems are usually divided into two parts: steady state and transient. Steady-state stability refers to the ability of the power system to regain synchronism after small or slow disturbances like gradual power change. An extension of steady-state stability is dynamic stability [1]. Dynamic stability is concerned with small disturbances lasting for a long time with inclusion of automatic control devices. Transient stability deals with effects of large, sudden disturbances like fault occurrence, sudden outage of a line, and sudden application or removal of loads.

#### 1.4 The swing equation

The position of the rotor axis and the resultant magnetic field axis is fixed under normal working conditions based on their relations. The angle between the two is

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

the daily load curve, and the greatest value of load during a period of 24 h is known as maximum or peak demand. Some key factors like the load factor (ratio of average load over a designated period of time to the peak load occurring in that period), utilization factor (ratio of maximum demand to the installed capacity), and plant factor (product of 8760 h and the ratio of annual energy generation to the plant capacity) help judge the performance of the system. In order for the a power system plant to operate economically, the load factor must be high, while the utilization and plant factors indicate how well the system capacity is usually operated and

The protection system for a power system involves a variety of protective devices like current, voltage, power sensors, relays, fuses, and circuit breakers. The

switchgears (e.g., instrument transformers, circuit breakers, disconnect switches, fuses, and lighting arresters). The presence of these devices is required in order to de-energize the power system either in scenarios of normal operation or in the occurrence of faults [1, 2]. The control house contains the associated control equipment and protective relays. There are basically two types of failures in a power system: overloads and faults. Overload conditions occur when the components in the power system are supplying more power than they were designed to carry safely. This scenario usually occurs when the total demand on the power system surpasses the capability of the system to supply power. Overloads often occur in new residential or industrial construction areas of the power system due to expansion. There are measures in place for the power system operator to immediately correct and control overload conditions due to the robustness of the system in order to avoid damage to the power network. On the other hand, fault conditions occur when one or more of the phases in a power system are shorted to ground or to each other (i.e., single phase to ground, three phases to ground, line to line, etc.). When a phase is open circuited, faults also occur in such situation. During periods of short circuit, very large currents flow and damage the entire power system if no measures are in place to quickly stop it. Faults must be cleared as quickly as possible in a power system when they occur, unlike overloads. For this reason, relays are

employed to automatically open circuit breakers and isolate faulty areas; then, they

The tendency of a power system to develop restoring forces equal to or greater than the disturbing forces to maintain the state of equilibrium is known as stability. Power system stability problems are usually divided into two parts: steady state and transient. Steady-state stability refers to the ability of the power system to regain synchronism after small or slow disturbances like gradual power change. An extension of steady-state stability is dynamic stability [1]. Dynamic stability is concerned with small disturbances lasting for a long time with inclusion of automatic control devices. Transient stability deals with effects of large, sudden disturbances like fault occurrence, sudden outage of a line, and sudden application or removal of loads.

The position of the rotor axis and the resultant magnetic field axis is fixed under normal working conditions based on their relations. The angle between the two is

protective devices that are connected directly to the circuits are known as

utilized [1, 5, 6].

Power System Stability

1.2.5 Protection system

are sensed in a power system [7, 8].

1.3 Power system stability

1.4 The swing equation

6

called the power angle or torque angle. During disturbances, the rotor accelerates/ decelerates with respect to the synchronously rotating air gap, thus a relative motion begins. The equation describing this relative motion is known as the swing equation given below [1, 2].

$$\frac{H}{\pi \mathcal{f}\_0} \frac{d^2 \delta}{dt^2} = P\_m - P\_\epsilon \tag{1}$$

where δ is the electrical radian; H is the per unit inertia constant; Pm∧Pe are the per unit mechanical and electrical power, respectively; and f <sup>0</sup> is the frequency of the system. With δ in degrees, then

$$\frac{H}{180f\_0} \frac{d^2 \delta}{dt^2} = P\_m - P\_e \tag{2}$$

#### 1.5 Stability studies for synchronous generator models

Consider a generator connected to a major substation of a very large system via a transmission line as shown below (Figures 1 and 2).

The substation bus voltage and frequency are assumed to remain constant (infinite bus). This is because its characteristics do no change regardless of power supplied or consumed by it. The generator is represented by a constant voltage behind the direct axis transient reactance X<sup>0</sup> <sup>d</sup>. The node representing the generator terminal voltage Vg can be eliminated by converting the Y connected impedances to Δ with admittances as [1]

Figure 1. One machine connected to an infinite bus.

Figure 2. Equivalent circuit of one machine connected to an infinite bus.

$$\begin{aligned} y\_{10} &= \frac{Z\_L}{jX\_d'Z\_s + jX\_d'Z\_L + Z\_LZ\_s} \\ y\_{20} &= \frac{jX\_d'}{jX\_d'Z\_s + jX\_d'Z\_L + Z\_LZ\_s} \\ y\_{12} &= \frac{Z\_s}{jX\_d'Z\_s + jX\_d'Z\_L + Z\_LZ\_s} \end{aligned} \tag{3}$$

Writing the node equations for the above diagram gives

$$\begin{aligned} I\_1 &= (y\_{10} + y\_{12})E' - y\_{12}V \\ I\_2 &= -y\_{12}E' + (y\_{20} + y\_{12})V \end{aligned} \tag{4}$$

The above equations can be written in terms of the bus admittance matrix

$$
\begin{bmatrix} I\_1 \\ I\_2 \end{bmatrix} = \begin{bmatrix} Y\_{11} & Y\_{12} \\ Y\_{21} & Y\_{22} \end{bmatrix} \begin{bmatrix} E' \\ V \end{bmatrix} \tag{5}
$$

1.6 Small disturbances' steady-state stability

power in Eq. (9) into Eq. (1) gives [1]

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

for the system are given respectively by

9

stability and is given by:

Figure 3. Power angle curve.

The steady-state stability refers to the ability of the power system to remain in

ð10Þ

ð11Þ

ð12Þ

ð13Þ

ð14Þ

synchronism when subjected to small disturbances. Substituting the electrical

Solving the above differential equation results in synchronizing coefficient denoted by PS: This coefficient plays an important part in determining the system

The damping power and dimensionless damping ratio are respectively defined as

where δ is the damping coefficient. The response time constant and settling time

The natural frequency of the marginally stable oscillation is

The diagonal elements of the bus admittance are Y<sup>11</sup> ¼ y<sup>10</sup> þ y<sup>12</sup> and Y<sup>22</sup> ¼ y<sup>20</sup> þ y12. The off-diagonal elements are Y<sup>12</sup> ¼ Y<sup>21</sup> ¼ �y12: Expressing the voltages and admittances in polar form, the real power at node 1 is given by the following expression [1, 6].

$$\begin{split} P\_e &= \Re[E'I\_1^\*] \\ &= \Re[|E'|\angle\delta(|Y\_{11}|\angle-\theta\_{11})|E'|\angle-\delta+|Y\_{12}|\angle-\theta\_{12}|\dot{V}(\angle0)] \end{split} \tag{6}$$

In most systems, ZL∧ZS are predominantly inductive. If all resistances are neglected, <sup>θ</sup><sup>11</sup> <sup>¼</sup> <sup>θ</sup><sup>12</sup> <sup>¼</sup> <sup>90</sup>o, then <sup>Y</sup><sup>12</sup> <sup>¼</sup> <sup>B</sup><sup>12</sup> <sup>¼</sup> <sup>1</sup>=X12. The simplified expression for power is

$$\begin{aligned} \vert P\_e \rangle &= \vert E' \vert \vert V \rangle \vert \vert B\_{12} \vert \cos(\delta - 90^\circ) \rangle \\\\ \vert P\_e \rangle &= \frac{\vert E' \vert \vert V \rangle}{\vert X\_{12} \vert} \sin(\delta) \end{aligned} \tag{7}$$

The above equation is the simplified form of the power equation and basic to the understanding of all stability problems. The equation shows that the power transmitted depends upon the transfer reactance and the angle between the two voltages. The curve Pe versus δ is known as the power angle curve shown below (Figure 3).

Maximum power is transferred at a displacement of 90° . The maximum power is called the steady-state stability limit and is given by:

$$P\_{max} = \frac{|E'||V|}{|X\_{12}|} \tag{8}$$

A further increase of the electrical power causes loss of synchronism, thus,

$$P\_e = P\_{\text{max}} \sin \delta \tag{9}$$

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

Figure 3. Power angle curve.

ð3Þ

ð4Þ

ð5Þ

ð6Þ

ð7Þ

ð8Þ

. The maximum power is

Pe ¼ Pmaxsinδ (9)

Writing the node equations for the above diagram gives

following expression [1, 6].

Power System Stability

power is

8

The above equations can be written in terms of the bus admittance matrix

The diagonal elements of the bus admittance are Y<sup>11</sup> ¼ y<sup>10</sup> þ y<sup>12</sup> and Y<sup>22</sup> ¼ y<sup>20</sup> þ y12. The off-diagonal elements are Y<sup>12</sup> ¼ Y<sup>21</sup> ¼ �y12: Expressing the voltages and admittances in polar form, the real power at node 1 is given by the

In most systems, ZL∧ZS are predominantly inductive. If all resistances are neglected, <sup>θ</sup><sup>11</sup> <sup>¼</sup> <sup>θ</sup><sup>12</sup> <sup>¼</sup> <sup>90</sup>o, then <sup>Y</sup><sup>12</sup> <sup>¼</sup> <sup>B</sup><sup>12</sup> <sup>¼</sup> <sup>1</sup>=X12. The simplified expression for

The above equation is the simplified form of the power equation and basic to the understanding of all stability problems. The equation shows that the power transmitted depends upon the transfer reactance and the angle between the two voltages. The curve Pe versus δ is known as the power angle curve shown below (Figure 3).

A further increase of the electrical power causes loss of synchronism, thus,

Maximum power is transferred at a displacement of 90°

called the steady-state stability limit and is given by:

#### 1.6 Small disturbances' steady-state stability

The steady-state stability refers to the ability of the power system to remain in synchronism when subjected to small disturbances. Substituting the electrical power in Eq. (9) into Eq. (1) gives [1]

$$\frac{H}{\pi f\_0} \frac{d^2 \delta}{dt^2} = \dot{P}\_m - \dot{P}\_{max} \sin \delta \tag{10}$$

Solving the above differential equation results in synchronizing coefficient denoted by PS: This coefficient plays an important part in determining the system stability and is given by:

$$P\_s = \left. \frac{dP}{d\delta} \right|\_{\delta\_0} = P\_{max} \cos \delta\_0 \tag{11}$$

The natural frequency of the marginally stable oscillation is

$$
\omega\_n = \sqrt{\frac{\pi f\_0}{H} P\_s} \tag{12}
$$

The damping power and dimensionless damping ratio are respectively defined as

$$P\_d = D \frac{d\delta}{dt} \tag{13}$$

$$
\zeta = \frac{D}{2} \sqrt{\frac{\pi f\_0}{HP\_\theta}} \tag{14}
$$

where δ is the damping coefficient. The response time constant and settling time for the system are given respectively by

$$\eta = \frac{1}{\zeta \omega\_n} = \frac{2H}{\pi f\_0 D} \tag{15}$$

$$t\_s \cong 4\tau\tag{16}$$

#### 1.7 Transient stability

Transient stability studies involve the determination of whether or not synchronism is maintained after the machine has been subjected to severe disturbances. A method known as the equal area criterion can be used for a quick prediction of stability. Consider a synchronous machine connected to an infinite bus bar. The swing equation with damping neglected is given by

$$\frac{H}{\pi f\_0} \frac{d^2 \delta}{dt^2} = P\_m - P\_e = P\_a \tag{17}$$

Figure 5.

Figure 6.

Figure 7.

Figure 8.

11

Equal area criterion—maximum power limit.

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

One machine system connected to infinite bus, three-phase fault at F, at the sending end.

One machine system connected to infinite bus, three-phase fault at F, away from the sending end.

Equal area criterion for a three-phase fault at the sending end.

where Pa is the accelerating power. Scenarios for the equal area criterion are described below (Figure 4).

For a sudden step increase in input power, this is represented by the horizontal line Pm1. Since Pm1>Pe0,the accelerating power on the rotor is positive and the power angle δ increases. The excess energy stored in the rotor during the initial acceleration is [1]

$$\int\_{\delta\_0}^{\delta\_1} (P\_{m1} - P\_e) d\delta = \text{area } abc = \text{area } A\_1 \tag{18}$$

With increase in δ, the electrical power increases, and when δ ¼ δ1, the electrical power matches the new input power Pm1. For a situation where Pm < Pe, the rotor decelerates toward synchronous speed until δ ¼ δmax: The energy given up by the rotor as it decelerates back to synchronous speed is

$$\int\_{\delta\_0}^{\delta\_1} (P\_{m1} - P\_e) d\delta = \text{area } abc = \text{area } A\_1 \tag{19}$$

Figure 4. Equal area criterion—sudden change of load.

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

Figure 5.

ð15Þ

ð16Þ

ð17Þ

ð18Þ

ð19Þ

1.7 Transient stability

Power System Stability

described below (Figure 4).

acceleration is [1]

Figure 4.

10

Equal area criterion—sudden change of load.

swing equation with damping neglected is given by

rotor as it decelerates back to synchronous speed is

Transient stability studies involve the determination of whether or not synchronism is maintained after the machine has been subjected to severe disturbances. A method known as the equal area criterion can be used for a quick prediction of stability. Consider a synchronous machine connected to an infinite bus bar. The

where Pa is the accelerating power. Scenarios for the equal area criterion are

line Pm1. Since Pm1>Pe0,the accelerating power on the rotor is positive and the power angle δ increases. The excess energy stored in the rotor during the initial

For a sudden step increase in input power, this is represented by the horizontal

With increase in δ, the electrical power increases, and when δ ¼ δ1, the electrical power matches the new input power Pm1. For a situation where Pm < Pe, the rotor decelerates toward synchronous speed until δ ¼ δmax: The energy given up by the

Equal area criterion—maximum power limit.

#### Figure 6.

One machine system connected to infinite bus, three-phase fault at F, at the sending end.

Figure 7. Equal area criterion for a three-phase fault at the sending end.

Figure 8. One machine system connected to infinite bus, three-phase fault at F, away from the sending end.

References

McGraw-Hill; 2005

McGraw-Hill; 2005

Hall of India; 2003

Global Media; 2010

& Sons; 2013

13

[1] Saadat H. Power System Analysis. International Student Edition. New York, United States of America:

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

[2] Chapman SJ. Electrical Machinery & Power System Fundamentals. New York, United States of America:

[4] Narendra K, Sanjiv K. Power System Analysis. West Sussex, United Kingdom:

[5] Pdiyar KR. Power Systems Dynamics: Stability and Control. West Sussex, United Kingdom: Global Media; 2008

[6] JanBialek M, Janusz Bumby J. Power Systems Dynamics: Stability and Control. 2nd ed. West Sussex, United Kingdom: John Wiley & Sons; 2009

[7] Balakrishnan BM, Hewitson R. Leslie, Practical Power System Protection. United States: Newnes; 2005

[8] Horowitz S, Phadke AG, Niemira JK. Power System Relaying. 4th ed. West Sussex, United Kingdom: John Wiley

[3] Painthankar YG, Bhide SR. Fundamentals of Power System Protection. New Delhi, India: Prentice

Figure 9. Equal area criterion for a three-phase fault away from the sending end.

The equal area criterion is used to determine the maximum additional power Pm which can be applied for stability to be maintained. This could be termed as application to sudden increase in power input as shown in Figure 5. Figures 6–9 show the application to three-phase fault considering the equal area criterion [1].

## Author details

Kenneth Eloghene Okedu Department of Electrical and Electronic Engineering, National University of Science and Technology (Glasgow Caledonian University), Muscat, Oman

\*Address all correspondence to: kenokedu@yahoo.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Introductory Chapter: Power System Stability DOI: http://dx.doi.org/10.5772/intechopen.84497

## References

The equal area criterion is used to determine the maximum additional power Pm which can be applied for stability to be maintained. This could be termed as application to sudden increase in power input as shown in Figure 5. Figures 6–9 show the application to three-phase fault considering the equal area criterion [1].

Equal area criterion for a three-phase fault away from the sending end.

Department of Electrical and Electronic Engineering, National University of Science and Technology (Glasgow Caledonian University), Muscat, Oman

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: kenokedu@yahoo.com

provided the original work is properly cited.

Author details

12

Figure 9.

Power System Stability

Kenneth Eloghene Okedu

[1] Saadat H. Power System Analysis. International Student Edition. New York, United States of America: McGraw-Hill; 2005

[2] Chapman SJ. Electrical Machinery & Power System Fundamentals. New York, United States of America: McGraw-Hill; 2005

[3] Painthankar YG, Bhide SR. Fundamentals of Power System Protection. New Delhi, India: Prentice Hall of India; 2003

[4] Narendra K, Sanjiv K. Power System Analysis. West Sussex, United Kingdom: Global Media; 2010

[5] Pdiyar KR. Power Systems Dynamics: Stability and Control. West Sussex, United Kingdom: Global Media; 2008

[6] JanBialek M, Janusz Bumby J. Power Systems Dynamics: Stability and Control. 2nd ed. West Sussex, United Kingdom: John Wiley & Sons; 2009

[7] Balakrishnan BM, Hewitson R. Leslie, Practical Power System Protection. United States: Newnes; 2005

[8] Horowitz S, Phadke AG, Niemira JK. Power System Relaying. 4th ed. West Sussex, United Kingdom: John Wiley & Sons; 2013

Chapter 2

Abstract

Application of the Trajectory

Stability Analysis

Alejandro Pizano Martínez

bifurcation, parameter sensitivities

1. Introduction

15

Enrique Arnoldo Zamora Cárdenas,

and Claudio Rubén Fuerte Esquivel

Sensitivity Theory to Small Signal

The security assessment of power systems represents one of the principal studies

that must be carried out in energy control centers. In this context, small-signal stability analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. This chapter details a practical approach for assessing the stability of power system's equilibrium points in real time based on the concept of trajectory sensitivity theory. This approach provides complementary information to that given by selective modal analysis: it determines how the state variables linked with the critical eigenvalues are affected by the system's parameters and also determines the way of judging how the system's parameters affect the oscillatory behavior of a power system. The WSCC 9 bus and a 190-buses equivalent system of the Mexican power system are used to demonstrate the generality of the approach as well as how its application in energy

management systems is suitable for power system operation and control.

Keywords: small-signal stability, equilibrium points, selective modal analysis, Hopf

Small-signal stability (SSS) is the ability of a power system to maintain synchronism when subjected to small disturbances such as small load and/or generation changes [1]. The analysis of SSS consists of assessing the stability of an equilibrium point (EP), as well as determining the most influential state variables in the stability of the operating point. For small enough disturbances, the system behavior can be studied via the theory of linear systems around an equilibrium point [2]. The stability of an EP is assessed by eigenvalue analysis (eigenanalysis) according to the Lyapunov criterion [3], which states that an EP will be stable in the small-signal sense, if all system eigenvalues of the system matrix are located on the left side of the complex plane. On the contrary, the EP will be unstable if at least one eigenvalue is located on the right side of the imaginary axis. In this context, the resulting dominant eigenvalue from the eigen-analysis is called the critical eigenvalue, and its association to the state variables is investigated by selective modal analysis (SMA)

Chapter 2

## Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

Enrique Arnoldo Zamora Cárdenas, Alejandro Pizano Martínez and Claudio Rubén Fuerte Esquivel

## Abstract

The security assessment of power systems represents one of the principal studies that must be carried out in energy control centers. In this context, small-signal stability analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. This chapter details a practical approach for assessing the stability of power system's equilibrium points in real time based on the concept of trajectory sensitivity theory. This approach provides complementary information to that given by selective modal analysis: it determines how the state variables linked with the critical eigenvalues are affected by the system's parameters and also determines the way of judging how the system's parameters affect the oscillatory behavior of a power system. The WSCC 9 bus and a 190-buses equivalent system of the Mexican power system are used to demonstrate the generality of the approach as well as how its application in energy management systems is suitable for power system operation and control.

Keywords: small-signal stability, equilibrium points, selective modal analysis, Hopf bifurcation, parameter sensitivities

## 1. Introduction

Small-signal stability (SSS) is the ability of a power system to maintain synchronism when subjected to small disturbances such as small load and/or generation changes [1]. The analysis of SSS consists of assessing the stability of an equilibrium point (EP), as well as determining the most influential state variables in the stability of the operating point. For small enough disturbances, the system behavior can be studied via the theory of linear systems around an equilibrium point [2]. The stability of an EP is assessed by eigenvalue analysis (eigenanalysis) according to the Lyapunov criterion [3], which states that an EP will be stable in the small-signal sense, if all system eigenvalues of the system matrix are located on the left side of the complex plane. On the contrary, the EP will be unstable if at least one eigenvalue is located on the right side of the imaginary axis. In this context, the resulting dominant eigenvalue from the eigen-analysis is called the critical eigenvalue, and its association to the state variables is investigated by selective modal analysis (SMA)

[4, 5]. Based on the participation factors analysis (PFA), the SMA provides those state variables having the highest influence in the EP stability by means of their coupling to the critical eigenvalue [6, 7]. Since the system eigenvalues are directly related to its dynamic performance, different forms of instability in a power system can be studied by means of well-defined structures of eigenvalues which are called bifurcations.

SMA-TS approach uses an index of sensitivity quantification, which facilitates identifying the influence of the system loads around the HB points. The index allows to rank the power system loads in order to predict the most critical loading directions toward a B point. Its application is suitable to monitoring in real time the power system operation and improve the security. The results of the study cases of two real systems, i.e., WSCC 9-bus and an equivalent of the Mexican power system of 190-buses, are

An electric power system can be represented analytically by a set of differentialalgebraic equations (DAEs), as given by (1), where x is a n-dimensional vector of dynamic state variables with initial conditions x tð Þ¼ <sup>0</sup> x0, y is a m-dimensional vector of instantaneous state (algebraic) variables, (usually the real and imaginary parts or the magnitudes and phase angles of the complex node voltages) with initial conditions y tð Þ¼ <sup>0</sup> y0, and β is a set of time-invariant parameters of the system. The dynamics of the equipment, e.g., generators and controls, is explicitly modeled by the set of differential equations through the function fð Þ� . The set of algebraic equations 0 ¼ gð Þ� represents the stator algebraic equations and mismatch power

> <sup>x</sup>\_ <sup>¼</sup> f xð Þ ; <sup>y</sup>; <sup>β</sup> <sup>f</sup> : <sup>R</sup><sup>n</sup>þmþ<sup>p</sup> ! <sup>R</sup><sup>n</sup> <sup>0</sup> <sup>¼</sup> g xð Þ ; <sup>y</sup>; <sup>β</sup> <sup>g</sup> : <sup>R</sup><sup>n</sup>þmþ<sup>p</sup> ! <sup>R</sup><sup>m</sup> x∈X⊂R<sup>n</sup> y∈Y⊂R<sup>m</sup> β∈β⊂R<sup>p</sup>:

Let β<sup>0</sup> be the nominal values of β, and assume that the nominal set of DAEs x\_ ¼ f x; y; β<sup>0</sup> ð Þ, 0 ¼ g x; y; β<sup>0</sup> ð Þ has a unique nominal trajectory solution x t; x0; y0; β<sup>0</sup>

y t; <sup>x</sup>0; <sup>y</sup>0; <sup>β</sup> � � over <sup>t</sup><sup>∈</sup> <sup>t</sup>0; <sup>t</sup> ½ � end that is close to the nominal trajectory solution. This

ðtend t0

The sensitivities of the dynamic and algebraic state vectors with respect to a chosen system's parameter, <sup>x</sup><sup>β</sup> <sup>¼</sup> <sup>∂</sup>xð Þ� <sup>=</sup>∂<sup>β</sup> and <sup>y</sup><sup>β</sup> <sup>¼</sup> <sup>∂</sup>yð Þ� <sup>=</sup>∂β, at a time <sup>t</sup> along the trajectory are obtained from (4) and (5), which in turn are obtained from the partial

> ∂x ∂β þ

> > <sup>∂</sup>gð Þ� ∂y

<sup>∂</sup>fð Þ� ∂y

> ∂y ∂β þ

∂y ∂β þ

<sup>∂</sup>gð Þ�

of DAEs (1) has a unique perturbed trajectory solution x t; <sup>x</sup>0; <sup>y</sup>0; <sup>β</sup> � � and

xðÞ¼ � x<sup>0</sup> þ

ðtend t0

<sup>0</sup> <sup>¼</sup> <sup>∂</sup>gð Þ� ∂x

<sup>∂</sup>fð Þ� ∂x

> ∂x ∂β þ

perturbed solution is given by (2) and (3) [19, 20]:

derivative of (2) and (3) with respect to β:

<sup>∂</sup>xð Þ� ∂β ¼

� � over <sup>t</sup><sup>∈</sup> <sup>t</sup>0; <sup>t</sup> ½ � end , where <sup>t</sup><sup>0</sup> and tend are the initial and final times, respectively, of the study time period. Thus, for all β sufficiently close to β0, the set

(1)

� �

f xð Þ ð Þ� ; yð Þ� ; β ds (2)

0 ¼ g xð Þ ð Þ� ; yð Þ� ; β : (3)

<sup>∂</sup>fð Þ� ∂β � �ds (4)

<sup>∂</sup><sup>β</sup> : (5)

used to show the performance and applicability of the proposed approach.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

2. Trajectory sensitivity theory

DOI: http://dx.doi.org/10.5772/intechopen.81490

flow equations at each node:

2.1 Analytical formulation

and y t; x0; y0; β<sup>0</sup>

17

The theory of bifurcations is a powerful mathematical tool based on eigenanalysis to assess the stability of EPs in nonlinear systems [8]. This theory consists of searching for specific eigenvalue structures associated to different instability forms that appear on power systems [9]. One of the most common local bifurcations that can appear in the power system operation is the Hopf bifurcation (HB), which occurs when the system matrix contains a pair of purely imaginary eigenvalues causing undamped oscillatory behavior [10–13]. Any parameter variation in the system may result in complicated behavior until the system stability changes. This point, where the stability changes, is defined as a bifurcation point. In this chapter the system loads are changed in order to analyze the stability of the EPs. The maximum load in a specific direction that a power system can provide before the appearance of a bifurcation point establishes the loading limit in that direction. Thus, the loading limit is directly associated with the stability margin of the system. The critical eigenvalue of an EP is used as an index of the stability margin. After a load change, an eigen-analysis permits us to assess the stability margin closeness. A small margin indicates closeness to a bifurcation point (instability).

SSS analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. Control strategies employed in electric power systems are usually tested by means of an assessment of the stability improvement. Thus, the influence of parameters and components in the EP stability provides an insight to achieve the best control. In this context, the participation factors let us know the highest association between state variables and the critical eigenvalue dominating the EP stability [6, 7]. In this way, PFA allows the selection via the associated states to the critical eigenvalue of those components that will provide the best control in EP stability. Although the PFA selects the most sensitive states in the EP stability, it is not possible to identify in a direct form the most influential parameters, e.g., those most sensitive loads influencing the stability. In order to achieve this, the PFA must be combined with other methods providing parameter sensitivity features. In [14, 15] the authors combined the PFA and modal controllability of the weak damping oscillatory modes to obtain an optimal location of static VAR compensators (SVCs). In [16] the authors presented an approach to examine the effect of loads in the system stability by using participation factors and mode shape analysis. In Ref. [7], both a voltage stability analysis and a low-frequency oscillation analysis were performed by using the SSS analysis in the EPs. As the system faces increased loading conditions, bifurcation points appear, and the participating generators are identified by means of the most associated states obtained from the SMA.

This chapter presents an alternative method based on the trajectory sensitivity (TS) theory [17, 18] to investigate the stability of the EPs by using a time-domain simulation. In this approach which will be referred to as selective modal analysis and trajectory sensitivities (SMA-TS), the TS were computed with respect to the load parameters; therefore, besides the stability assessment and the participating states, this approach also has the ability to identify those most sensitive load parameters influencing the critical states. The stability assessment consists of just examining the TS oscillations. However, the stability analysis of the EP is never perturbed. The TS oscillations required for the stability assessment are produced by means of the initial condition values selected for the sensitivity variables. The

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

SMA-TS approach uses an index of sensitivity quantification, which facilitates identifying the influence of the system loads around the HB points. The index allows to rank the power system loads in order to predict the most critical loading directions toward a B point. Its application is suitable to monitoring in real time the power system operation and improve the security. The results of the study cases of two real systems, i.e., WSCC 9-bus and an equivalent of the Mexican power system of 190-buses, are used to show the performance and applicability of the proposed approach.

## 2. Trajectory sensitivity theory

[4, 5]. Based on the participation factors analysis (PFA), the SMA provides those state variables having the highest influence in the EP stability by means of their coupling to the critical eigenvalue [6, 7]. Since the system eigenvalues are directly related to its dynamic performance, different forms of instability in a power system can be studied by means of well-defined structures of eigenvalues which are called

The theory of bifurcations is a powerful mathematical tool based on eigenanalysis to assess the stability of EPs in nonlinear systems [8]. This theory consists of searching for specific eigenvalue structures associated to different instability forms that appear on power systems [9]. One of the most common local bifurcations that can appear in the power system operation is the Hopf bifurcation (HB), which occurs when the system matrix contains a pair of purely imaginary eigenvalues causing undamped oscillatory behavior [10–13]. Any parameter variation in the system may result in complicated behavior until the system stability changes. This point, where the stability changes, is defined as a bifurcation point. In this chapter the system loads are changed in order to analyze the stability of the EPs. The maximum load in a specific direction that a power system can provide before the appearance of a bifurcation point establishes the loading limit in that direction. Thus, the loading limit is directly associated with the stability margin of the system. The critical eigenvalue of an EP is used as an index of the stability margin. After a load change, an eigen-analysis permits us to assess the stability margin closeness. A small margin indicates closeness to a bifurcation point (instability). SSS analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. Control strategies employed in electric power systems are usually tested by means of an assessment of the stability improvement. Thus, the influence of parameters and components in the EP stability provides an insight to achieve the best control. In this context, the participation factors let us know the highest association between state variables and the critical eigenvalue dominating the EP stability [6, 7]. In this way, PFA allows the selection via the associated states to the critical eigenvalue of those components that will provide the best control in EP stability. Although the PFA selects the most sensitive states in the EP stability, it is not possible to identify in a direct form the most influential parameters, e.g., those most sensitive loads influencing the stability. In order to achieve this, the PFA must be combined with other methods providing parameter sensitivity features. In [14, 15] the authors combined the PFA and modal controllability of the weak damping oscillatory modes to obtain an optimal location of static VAR compensators (SVCs). In [16] the authors presented an approach to examine the effect of loads in the system stability by using participation factors and mode shape analysis. In Ref. [7], both a voltage stability analysis and a low-frequency oscillation analysis were performed by using the SSS analysis in the EPs. As the system faces increased loading conditions, bifurcation points appear, and the participating generators are identified by means

of the most associated states obtained from the SMA.

16

This chapter presents an alternative method based on the trajectory sensitivity (TS) theory [17, 18] to investigate the stability of the EPs by using a time-domain simulation. In this approach which will be referred to as selective modal analysis and trajectory sensitivities (SMA-TS), the TS were computed with respect to the load parameters; therefore, besides the stability assessment and the participating states, this approach also has the ability to identify those most sensitive load parameters influencing the critical states. The stability assessment consists of just examining the TS oscillations. However, the stability analysis of the EP is never perturbed. The TS oscillations required for the stability assessment are produced by means of the initial condition values selected for the sensitivity variables. The

bifurcations.

Power System Stability

An electric power system can be represented analytically by a set of differentialalgebraic equations (DAEs), as given by (1), where x is a n-dimensional vector of dynamic state variables with initial conditions x tð Þ¼ <sup>0</sup> x0, y is a m-dimensional vector of instantaneous state (algebraic) variables, (usually the real and imaginary parts or the magnitudes and phase angles of the complex node voltages) with initial conditions y tð Þ¼ <sup>0</sup> y0, and β is a set of time-invariant parameters of the system. The dynamics of the equipment, e.g., generators and controls, is explicitly modeled by the set of differential equations through the function fð Þ� . The set of algebraic equations 0 ¼ gð Þ� represents the stator algebraic equations and mismatch power flow equations at each node:

$$\begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}, \boldsymbol{y}, \boldsymbol{\beta}) & \quad f: \mathfrak{R}^{n+m+p} &\to \mathfrak{R}^{n} \\ \mathbf{0} &= \mathbf{g}(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\beta}) & \quad \mathbf{g}: \mathfrak{R}^{n+m+p} &\to \mathfrak{R}^{m} \\ \boldsymbol{\alpha} &\in \mathbf{X} \subset \mathfrak{R}^{n} & \boldsymbol{y} \in \mathbf{Y} \subset \mathfrak{R}^{m} & \boldsymbol{\beta} \in \mathfrak{R}^{p} \mathbf{C}^{p} \end{aligned} \tag{1}$$

#### 2.1 Analytical formulation

Let β<sup>0</sup> be the nominal values of β, and assume that the nominal set of DAEs x\_ ¼ f x; y; β<sup>0</sup> ð Þ, 0 ¼ g x; y; β<sup>0</sup> ð Þ has a unique nominal trajectory solution x t; x0; y0; β<sup>0</sup> � � and y t; x0; y0; β<sup>0</sup> � � over <sup>t</sup><sup>∈</sup> <sup>t</sup>0; <sup>t</sup> ½ � end , where <sup>t</sup><sup>0</sup> and tend are the initial and final times, respectively, of the study time period. Thus, for all β sufficiently close to β0, the set of DAEs (1) has a unique perturbed trajectory solution x t; <sup>x</sup>0; <sup>y</sup>0; <sup>β</sup> � � and y t; <sup>x</sup>0; <sup>y</sup>0; <sup>β</sup> � � over <sup>t</sup><sup>∈</sup> <sup>t</sup>0; <sup>t</sup> ½ � end that is close to the nominal trajectory solution. This perturbed solution is given by (2) and (3) [19, 20]:

$$\mathfrak{x}(\cdot) = \mathfrak{x}\_0 + \int\_{t\_0}^{t\_{end}} f(\mathfrak{x}(\cdot), \mathfrak{y}(\cdot), \beta) \, ds \tag{2}$$

$$\mathbf{0} = \mathbf{g}(\mathbf{x}(\cdot), \mathbf{y}(\cdot), \boldsymbol{\beta}). \tag{3}$$

The sensitivities of the dynamic and algebraic state vectors with respect to a chosen system's parameter, <sup>x</sup><sup>β</sup> <sup>¼</sup> <sup>∂</sup>xð Þ� <sup>=</sup>∂<sup>β</sup> and <sup>y</sup><sup>β</sup> <sup>¼</sup> <sup>∂</sup>yð Þ� <sup>=</sup>∂β, at a time <sup>t</sup> along the trajectory are obtained from (4) and (5), which in turn are obtained from the partial derivative of (2) and (3) with respect to β:

$$\frac{\partial \mathfrak{x}(\cdot)}{\partial \beta} = \int\_{t\_0}^{t\_{end}} \left( \frac{\partial f(\cdot)}{\partial \mathbf{x}} \frac{\partial \mathfrak{x}}{\partial \beta} + \frac{\partial f(\cdot)}{\partial \mathbf{y}} \frac{\partial \mathfrak{y}}{\partial \beta} + \frac{\partial f(\cdot)}{\partial \beta} \right) ds \tag{4}$$

$$\mathbf{O} = \frac{\partial \mathbf{g}(\cdot)}{\partial \mathbf{x}} \frac{\partial \mathbf{x}}{\partial \beta} + \frac{\partial \mathbf{g}(\cdot)}{\partial \mathbf{y}} \frac{\partial \mathbf{y}}{\partial \beta} + \frac{\partial \mathbf{g}(\cdot)}{\partial \beta} . \tag{5}$$

Lastly, the smooth evolution of the sensitivities along the trajectory (6) and (7) is obtained by differentiating (4) and (5) with respect to t:

$$\dot{\boldsymbol{x}}\_{\boldsymbol{\beta}} = \frac{\partial \boldsymbol{f}(\cdot)}{\partial \mathbf{x}} \frac{\partial \mathbf{x}}{\partial \boldsymbol{\beta}} + \frac{\partial \boldsymbol{f}(\cdot)}{\partial \mathbf{y}} \frac{\partial \mathbf{y}}{\partial \boldsymbol{\beta}} + \frac{\partial \boldsymbol{f}(\cdot)}{\partial \boldsymbol{\beta}} \equiv \boldsymbol{f}\_{\mathbf{x}} \mathbf{x}\_{\boldsymbol{\beta}} + \boldsymbol{f}\_{\mathbf{y}} \mathbf{y}\_{\boldsymbol{\beta}} + \boldsymbol{f}\_{\boldsymbol{\beta}} \mathbf{y} \quad \mathbf{x}\_{\boldsymbol{\beta}}(\mathbf{t}\_{0}) = \mathbf{0} \tag{6}$$

$$\mathbf{0} = \frac{\partial \mathbf{g}(\cdot)}{\partial \mathbf{x}} \frac{\partial \mathbf{x}}{\partial \boldsymbol{\beta}} + \frac{\partial \mathbf{g}(\cdot)}{\partial \mathbf{y}} \frac{\partial \mathbf{y}}{\partial \boldsymbol{\beta}} + \frac{\partial \mathbf{g}(\cdot)}{\partial \boldsymbol{\beta}} \equiv \mathbf{g}\_x \mathbf{x}\_{\boldsymbol{\beta}} + \mathbf{g}\_y \mathbf{y}\_{\boldsymbol{\beta}} + \mathbf{g}\_{\boldsymbol{\beta}} \mathbf{y} \quad \mathbf{y}\_{\boldsymbol{\beta}}(\mathbf{t} \mathbf{o}) = \mathbf{0} \tag{7}$$

where f <sup>x</sup>, f <sup>y</sup>, f <sup>β</sup>, gx, gy, and g<sup>β</sup> are time-varying matrices computed along the system trajectories.

## 3. Trajectory sensitivity analysis

#### 3.1 Sensitivity discretization

TS computation is obtained by means of the sequential solution of the nonlinear DAE system (1) and the linear time-varying DAE system (6) and (7). By applying the trapezoidal rule to algebraize the differential equations, both DAE systems are converted into the following systems of algebraic difference Eqs. (8)–(11).

$$F\_1(\cdot) = \mathbf{x}^{k+1} - \mathbf{x}^k - \frac{\Delta t}{2} \left( f^{k+1} + f^k \right) = \mathbf{0} \tag{8}$$

$$F\_2(\cdot) = \mathbf{g}^{k+1} = \mathbf{0} \tag{9}$$

Once the states have been computed for a new time step, the TS can be calculated. To do this, the linear time-varying systems (10) and (11) rearranged as indicated in (13) should be evaluated with the recently computed states and solved

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

<sup>¼</sup> xk <sup>β</sup> þ Δt <sup>2</sup> <sup>f</sup> k xxk <sup>β</sup> þ f k y yk <sup>β</sup> þ f k <sup>β</sup> þ f kþ1 β

which represents a very small computational burden [17]. This is because the coefficient matrix on the left side of (13) corresponds to the Jacobian matrix already

It is very important to note that at each time step Δt, the solution of (13) for the

It is clear that the solution of the linear time-varying system (13) uses the

matrix whose Np columns are the ð Þ n þ m -dimension TS vectors with respect to

As numerically demonstrated in [19–23], when the system approaches an unsta-

Sβ<sup>1</sup> Sβ<sup>2</sup> ⋯ SβNp � � |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} S

ble operation condition, the sensitivities of state trajectories have more rapid changes in magnitudes and larger excursions than the state trajectories. Hence, as the system approaches its stability boundary, the TS approach infinity [19]. In this sense, it is possible to associate the sensitivity information with the stability level of the system for a particular system parameter. For this purpose, an Euclidian norm of the trajectory sensitivity vector, referred to as a sensitivity norm, is proposed in [23] as a measure of proximity to instability. This sensitivity norm also permits the computation of the critical parameters whose variations steer the system much faster to an unstable operation condition. In this chapter, such a sensitivity norm is used to provide a time-varying index of proximity to oscillatory instability for a nggenerator system defined at each integration time step Δt, as shown in (15), which by computing sensitivities of rotor angle and speed trajectories with respect to active power loads measures the load power's effect on the system's small-signal

same Jacobian matrix already factored for solving the TS calculation with respect to any β parameter. Taking advantage of this observation, the solution approach described in Section 3.2 can be directly extended to provide an efficient linear computation of multiparameter trajectory sensitivities, as shown in (14). In this case, it is only necessary to carry out Np forward/backward substitutions to calculate all TS vectors with respect to Np parameters of the

2 4 � �

3 5

: (13)

: (14)

�gkþ<sup>1</sup> β


<sup>β</sup> uniquely requires a forward/backward substitution,

<sup>β</sup><sup>i</sup> ∀i ¼ 1, …, Np, where S is a ð Þ� n þ m NP sensitivity

<sup>¼</sup> <sup>B</sup>β<sup>1</sup> <sup>B</sup>β<sup>2</sup> <sup>⋯</sup> <sup>B</sup>βNp � � |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} B

for the TS xkþ<sup>1</sup>

TS calculation xkþ<sup>1</sup>

system, i.e., xkþ<sup>1</sup>

Np parameters:

<sup>I</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 <sup>x</sup> � <sup>Δ</sup><sup>t</sup>

stability:

19

2 4 <sup>I</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 <sup>x</sup> � <sup>Δ</sup><sup>t</sup>

2 4

<sup>β</sup> ykþ<sup>1</sup> β h i<sup>T</sup>

gkþ<sup>1</sup> <sup>x</sup> gkþ<sup>1</sup>


3.3 Multiparameter sensitivity

2 f kþ1 y

DOI: http://dx.doi.org/10.5772/intechopen.81490

y

<sup>β</sup> and ykþ<sup>1</sup>

<sup>β</sup><sup>i</sup> and y<sup>k</sup>þ<sup>1</sup>

2 f kþ1 y

3 5

y

g<sup>k</sup>þ<sup>1</sup> <sup>x</sup> g<sup>k</sup>þ<sup>1</sup>


3.4 Sensitivity quantification

:

3 5

factored used in the final NR iteration to solve (12).

xkþ<sup>1</sup> β ykþ<sup>1</sup> β


" #

$$F\_3(\cdot) = \mathbf{x}\_{\beta}^{k+1} - \mathbf{x}\_{\beta}^k - \frac{\Delta t}{2} \left( f\_{\mathbf{x}}^{k+1} \mathbf{x}\_{\beta}^{k+1} + f\_{\mathbf{y}}^{k+1} \mathbf{y}\_{\beta}^{k+1} + f\_{\beta}^{k+1} + f\_{\mathbf{z}}^k \mathbf{x}\_{\beta}^k + f\_{\mathbf{y}}^k \mathbf{y}\_{\beta}^k + f\_{\beta}^k \right) = \mathbf{0} \tag{10}$$

$$F\_4(\cdot) = \mathbf{g}\_{\ge}^{k+1} \mathbf{x}\_{\boldsymbol{\beta}}^{k+1} + \mathbf{g}\_{\ge}^{k+1} \mathbf{y}\_{\boldsymbol{\beta}}^{k+1} + \mathbf{g}\_{\boldsymbol{\beta}}^{k+1} = \mathbf{0} \tag{11}$$

where Δt is the integration time step and the superscript k is an index for the time instant tk at which variables and functions are evaluated, e.g., <sup>x</sup><sup>k</sup> <sup>¼</sup> x tð Þ<sup>k</sup> and f <sup>k</sup> <sup>¼</sup> f xk; yk � �.

#### 3.2 Linear sensitivity computation

Once algebraized both DAE systems (8)–(11), the Newton-Raphson (NR) algorithm is used to provide an approximate solution of the algebraized nonlinear systems (8) and (9). In this case, the resulting linearized system J i <sup>Δ</sup>X<sup>i</sup> ¼ �F Xð Þ<sup>i</sup> , whose representation in expanded form is given in (12), provides the approximate solution ΔX<sup>i</sup> = Δxk Δy<sup>k</sup> � �<sup>T</sup> , which is updated at each NR ith iteration, i.e., xkþ<sup>1</sup> <sup>¼</sup> xk <sup>þ</sup> <sup>Δ</sup>xk <sup>y</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>y</sup><sup>k</sup> <sup>þ</sup> <sup>Δ</sup>y<sup>k</sup> � �<sup>T</sup> , until a selected convergence criterion is satisfied. Note also that J is the Jacobian matrix resulting of the linearization around an EP.

The initial guess x<sup>k</sup>þ<sup>1</sup> <sup>0</sup> <sup>¼</sup> xk <sup>y</sup><sup>k</sup>þ<sup>1</sup> <sup>0</sup> <sup>¼</sup> <sup>y</sup><sup>k</sup> � �<sup>T</sup> is used to start the NR algorithm from given values x<sup>k</sup> y<sup>k</sup> � �<sup>T</sup> :

$$\underbrace{\begin{bmatrix} I - \frac{\Delta t}{2} f\_{\mathbf{x}}^{k+1} & -\frac{\Delta t}{2} f\_{\mathbf{y}}^{k+1} \\ \mathbf{g}\_{\mathbf{x}}^{k+1} & \mathbf{g}\_{\mathbf{y}}^{k+1} \end{bmatrix}}\_{\mathbf{j}^{i}} \underbrace{\begin{bmatrix} \Delta \mathbf{x}^{k} \\ \Delta \mathbf{y}^{k} \end{bmatrix}}\_{\Delta \mathbf{X}^{i}} = -\underbrace{\begin{bmatrix} F\_{1}(\cdot) \\ F\_{2}(\cdot) \end{bmatrix}}\_{F(\cdot)^{i}}.\tag{12}$$

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

Once the states have been computed for a new time step, the TS can be calculated. To do this, the linear time-varying systems (10) and (11) rearranged as indicated in (13) should be evaluated with the recently computed states and solved for the TS xkþ<sup>1</sup> <sup>β</sup> ykþ<sup>1</sup> β h i<sup>T</sup> : <sup>I</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 <sup>x</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 y 2 3 xkþ<sup>1</sup> β " # <sup>β</sup> þ Δt <sup>2</sup> <sup>f</sup> k xxk <sup>β</sup> þ f k y yk <sup>β</sup> þ f k <sup>β</sup> þ f kþ1 β � � 2 3

$$\underbrace{\begin{bmatrix} I - \frac{\omega}{2} f\_x^{k+1} & -\frac{\omega}{2} f\_y^{k+1} \\ \mathbf{g}\_x^{k+1} & \mathbf{g}\_y^{k+1} \end{bmatrix}}\_I \underbrace{\begin{bmatrix} \mathbf{x}\_{\beta}^{\kappa+1} \\ \mathbf{y}\_{\beta}^{k+1} \end{bmatrix}}\_S = \underbrace{\begin{bmatrix} \mathbf{x}\_{\beta}^k + \frac{\omega}{2} \left( f\_x^k \mathbf{x}\_{\beta}^k + f\_y^k \mathbf{y}\_{\beta}^k + f\_{\beta}^k + f\_{\beta}^{\kappa+1} \right) \\ -\mathbf{g}\_{\beta}^{k+1} \end{bmatrix}}\_B. \tag{13}$$

It is very important to note that at each time step Δt, the solution of (13) for the TS calculation xkþ<sup>1</sup> <sup>β</sup> and ykþ<sup>1</sup> <sup>β</sup> uniquely requires a forward/backward substitution, which represents a very small computational burden [17]. This is because the coefficient matrix on the left side of (13) corresponds to the Jacobian matrix already factored used in the final NR iteration to solve (12).

#### 3.3 Multiparameter sensitivity

Lastly, the smooth evolution of the sensitivities along the trajectory (6) and (7)

<sup>∂</sup><sup>β</sup> � <sup>f</sup> <sup>x</sup>x<sup>β</sup> <sup>þ</sup> <sup>f</sup> <sup>y</sup>y<sup>β</sup> <sup>þ</sup> <sup>f</sup> <sup>β</sup>; xβð Þ¼ <sup>t</sup><sup>0</sup> 0 (6)

<sup>∂</sup><sup>β</sup> � gxx<sup>β</sup> <sup>þ</sup> gyy<sup>β</sup> <sup>þ</sup> <sup>g</sup>β; yβð Þ¼ <sup>t</sup><sup>0</sup> 0 (7)

is obtained by differentiating (4) and (5) with respect to t:

∂y ∂β þ

<sup>∂</sup>gð Þ� ∂y

∂y ∂β þ

<sup>F</sup>1ðÞ¼ � xkþ<sup>1</sup> � xk � <sup>Δ</sup><sup>t</sup>

<sup>F</sup>4ðÞ¼ � <sup>g</sup><sup>k</sup>þ<sup>1</sup>

kþ1 <sup>y</sup> y<sup>k</sup>þ<sup>1</sup> <sup>β</sup> þ f

<sup>β</sup> <sup>þ</sup> <sup>g</sup><sup>k</sup>þ<sup>1</sup>

Once algebraized both DAE systems (8)–(11), the Newton-Raphson (NR) algorithm is used to provide an approximate solution of the algebraized nonlinear

whose representation in expanded form is given in (12), provides the approximate

satisfied. Note also that J is the Jacobian matrix resulting of the linearization around

where Δt is the integration time step and the superscript k is an index for the time instant tk at which variables and functions are evaluated, e.g., <sup>x</sup><sup>k</sup> <sup>¼</sup> x tð Þ<sup>k</sup> and

<sup>x</sup> x<sup>k</sup>þ<sup>1</sup>

systems (8) and (9). In this case, the resulting linearized system J

<sup>0</sup> <sup>¼</sup> xk <sup>y</sup><sup>k</sup>þ<sup>1</sup>

g<sup>k</sup>þ<sup>1</sup> <sup>x</sup> g<sup>k</sup>þ<sup>1</sup>


" #<sup>i</sup>

y

:

<sup>I</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 <sup>x</sup> � <sup>Δ</sup><sup>t</sup> 2 f kþ1 y

<sup>∂</sup>fð Þ�

<sup>∂</sup>gð Þ�

where f <sup>x</sup>, f <sup>y</sup>, f <sup>β</sup>, gx, gy, and g<sup>β</sup> are time-varying matrices computed along the

TS computation is obtained by means of the sequential solution of the nonlinear DAE system (1) and the linear time-varying DAE system (6) and (7). By applying the trapezoidal rule to algebraize the differential equations, both DAE systems are converted into the following systems of algebraic difference Eqs. (8)–(11).

<sup>2</sup> <sup>f</sup>

<sup>k</sup>þ<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>k</sup> � �

> kþ1 <sup>β</sup> þ f k xxk <sup>β</sup> þ f k y yk <sup>β</sup> þ f k β

<sup>β</sup> <sup>þ</sup> <sup>g</sup><sup>k</sup>þ<sup>1</sup>

, which is updated at each NR ith iteration, i.e.,

<sup>0</sup> <sup>¼</sup> <sup>y</sup><sup>k</sup> � �<sup>T</sup> is used to start the NR algorithm from

Δx<sup>k</sup> Δy<sup>k</sup> " #<sup>i</sup>


, until a selected convergence criterion is

¼ � <sup>F</sup>1ð Þ� F2ð Þ� � �<sup>i</sup>

> |fflfflfflfflffl{zfflfflfflfflffl} <sup>F</sup>ð Þ� <sup>i</sup>

� �

<sup>y</sup> y<sup>k</sup>þ<sup>1</sup>

<sup>F</sup>2ðÞ¼ � <sup>g</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>0</sup> (9)

¼ 0 (8)

<sup>β</sup> ¼ 0 (11)

i

<sup>Δ</sup>X<sup>i</sup> ¼ �F Xð Þ<sup>i</sup>

: (12)

,

¼ 0 (10)

<sup>∂</sup>fð Þ� ∂y

<sup>x</sup>\_ <sup>β</sup> <sup>¼</sup> <sup>∂</sup>fð Þ� ∂x

Power System Stability

system trajectories.

<sup>F</sup>3ðÞ¼ � xkþ<sup>1</sup>

<sup>k</sup> <sup>¼</sup> f xk; yk � �.

f

an EP.

18

<sup>β</sup> � <sup>x</sup><sup>k</sup>

<sup>β</sup> � <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>f</sup> kþ1 <sup>x</sup> xkþ<sup>1</sup> <sup>β</sup> þ f

3.2 Linear sensitivity computation

xkþ<sup>1</sup> <sup>¼</sup> xk <sup>þ</sup> <sup>Δ</sup>xk <sup>y</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>y</sup><sup>k</sup> <sup>þ</sup> <sup>Δ</sup>y<sup>k</sup> � �<sup>T</sup>

solution ΔX<sup>i</sup> = Δxk Δy<sup>k</sup> � �<sup>T</sup>

The initial guess x<sup>k</sup>þ<sup>1</sup>

given values x<sup>k</sup> y<sup>k</sup> � �<sup>T</sup>

∂x ∂β þ

3. Trajectory sensitivity analysis

3.1 Sensitivity discretization

∂x ∂β þ

<sup>0</sup> <sup>¼</sup> <sup>∂</sup>gð Þ� ∂x

> It is clear that the solution of the linear time-varying system (13) uses the same Jacobian matrix already factored for solving the TS calculation with respect to any β parameter. Taking advantage of this observation, the solution approach described in Section 3.2 can be directly extended to provide an efficient linear computation of multiparameter trajectory sensitivities, as shown in (14). In this case, it is only necessary to carry out Np forward/backward substitutions to calculate all TS vectors with respect to Np parameters of the system, i.e., xkþ<sup>1</sup> <sup>β</sup><sup>i</sup> and y<sup>k</sup>þ<sup>1</sup> <sup>β</sup><sup>i</sup> ∀i ¼ 1, …, Np, where S is a ð Þ� n þ m NP sensitivity matrix whose Np columns are the ð Þ n þ m -dimension TS vectors with respect to Np parameters:

$$\underbrace{\begin{bmatrix} I - \frac{\Delta t}{2} f\_x^{k+1} & -\frac{\Delta t}{2} f\_y^{k+1} \\ \hline & g\_x^{k+1} & g\_y^{k+1} \end{bmatrix}}\_{I} \underbrace{\begin{bmatrix} \mathcal{S}\_{\beta 1} & \mathcal{S}\_{\beta 2} & \cdots & \mathcal{S}\_{\beta Np} \end{bmatrix}}\_{S} = \underbrace{\begin{bmatrix} \mathcal{B}\_{\beta 1} & \mathcal{B}\_{\beta 2} & \cdots & \mathcal{B}\_{\beta Np} \end{bmatrix}}\_{B}. \tag{14}$$

#### 3.4 Sensitivity quantification

As numerically demonstrated in [19–23], when the system approaches an unstable operation condition, the sensitivities of state trajectories have more rapid changes in magnitudes and larger excursions than the state trajectories. Hence, as the system approaches its stability boundary, the TS approach infinity [19]. In this sense, it is possible to associate the sensitivity information with the stability level of the system for a particular system parameter. For this purpose, an Euclidian norm of the trajectory sensitivity vector, referred to as a sensitivity norm, is proposed in [23] as a measure of proximity to instability. This sensitivity norm also permits the computation of the critical parameters whose variations steer the system much faster to an unstable operation condition. In this chapter, such a sensitivity norm is used to provide a time-varying index of proximity to oscillatory instability for a nggenerator system defined at each integration time step Δt, as shown in (15), which by computing sensitivities of rotor angle and speed trajectories with respect to active power loads measures the load power's effect on the system's small-signal stability:

$$\text{SNV}\_{\rho}(t\_k) = \sqrt{\sum\_{m=1}^{\frac{n}{\rho}} \left( \left( \frac{\partial \delta\_m(t\_k)}{\partial \beta\_\rho} - \frac{\partial \delta\_j(t\_k)}{\partial \beta\_\rho} \right)^2 + \left( \frac{\partial \alpha\_m(t\_k)}{\partial \beta\_\rho} \right)^2 \right)} \quad \forall \rho = 1, \cdots, Np, \quad \text{(15)}$$

Step 3. Determine the critical eigenvalues of JR, and perform a SMA to identify

Step 4. Compute the sensitivities of associated state variables with respect to the selected system's parameters (load powers) at the equilibrium point, xt!<sup>∞</sup> <sup>β</sup> and yt!<sup>∞</sup> <sup>β</sup> ,

> <sup>¼</sup> xk <sup>β</sup> þ Δt <sup>2</sup> <sup>f</sup> <sup>x</sup>x<sup>k</sup>

Step 5. Quantify the interaction between the system parameters and the associated state variables by using the sensitivity index (15). Since this index is a function of the sensitivities of those state variables directly associated with the oscillatory modes and the critical eigenvalues, it can be used to quantify the effect of the ith parameter on these variables. In this case, the highest values of the sensitivity norms indicate the most sensitive parameters. Furthermore, the sensitivity index value

This section presents the analysis of the TS applied on assessing the SSS analysis. The effectiveness of the SMA-TS approach is numerically tested by analyzing the WSCC 9-buses and 3-generator systems [7] and a reduced equivalent system corresponding to the Mexican power system consisting of 190-buses with 46 generators. For the purpose of the studies presented in this section, the system generators are represented by means of the two-axis model with a simple fast exciter loop containing max/min ceiling limits. For this case the system loads are represented by means of the constant power load model; however, the generality of the method

In this subsection the conventional modal analysis is employed in order to investigate the small-signal stability of the WSCC power system, whose diagram is given in Figure 1. Table 1 presents the modal analysis for different EPs as the active

SSS and SMA were performed to investigate the different operating points corresponding to the different levels of loading. Eigen-analysis revealed the SSS of EPs, whereas participation factors were used to identify the most associated states to the critical eigenvalue at each EP, as given in Table 1. The first column represents the load changes at bus 5. The second column presents the critical eigenvalue for the EP. The third column presents the most associated states to the critical mode

2 4

<sup>β</sup> <sup>þ</sup> <sup>f</sup> <sup>y</sup>y<sup>k</sup>

�g<sup>β</sup>


� �

<sup>β</sup> þ 2f <sup>β</sup>

3 5

(16)

by solving (16). The integration process is started with initial conditions

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

xkþ<sup>1</sup> β y<sup>k</sup>þ<sup>1</sup> β


increases as the system is approaching an oscillatory stability problem.

" #

relate our proposal to the theory of trajectory sensitivities:

xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 for the parameter sensitivities, while the state and algebraic variables are set at their equilibrium values during the solution process. The time evolution of sensitivities is computed under the assumption that a very small perturbation is carried out in the system such that the state and algebraic variables are infinitesimally perturbed; their values, therefore, can be considered constant during the computation of the sensitivity index. These assumptions permit us to directly

the associate state variables.

DOI: http://dx.doi.org/10.5772/intechopen.81490

<sup>I</sup> � <sup>Δ</sup><sup>t</sup>

<sup>2</sup> <sup>f</sup> <sup>x</sup> � <sup>Δ</sup><sup>t</sup>

gx gy " #�

<sup>2</sup> f <sup>y</sup>


4. Small-signal stability analysis

allows to consider any load model.

4.1 Modal analysis WSCC system

power at bus 5 is increasing.

21

� � � � xe;ye ð Þ;<sup>β</sup>

where j denotes the reference generator.

In this case, the time evolution of sensitivities is necessary to quantify the loads' influence on the possible occurrence of an oscillatory instability, where the highest values of the sensitivity norms SN<sup>ρ</sup> indicate the most sensitive loads for the EP's stability. Thus, it should be noted that the critical load powers are those with the largest values of sensitivity index within the integration period, not the largest final value.

Note that this sensitivity norm has been successfully applied before for developing suitable approaches to improve the transient stability of power systems, e.g., for the estimation of the critical clearing time of a faulted system [23], the best possible location of FACTS controllers for transient stability enhancement [24–26], and the thyristor-controlled series compensator (TCSC) control design to enhance transient stability [27].

### 3.5 Sensitivity initial conditions

The analysis of a stationary operating point based on this approach is performed by keeping the corresponding EP constant during the whole simulation. Thus, during this simulation period with no perturbation, which is considered from the instant of time t<sup>0</sup> and the infinity t∞, the TS also remain into the same stationary behavior established at nonzero constant values xβð Þ t<sup>∞</sup> 6¼ 0 and yβð Þ t<sup>∞</sup> 6¼ 0. However, as our approach is based on catching the oscillatory behavior of trajectory sensitivities, the values xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 are arbitrarily used as the initial conditions of the sensitivity variables. Then, such an initial condition perturbation starts an oscillatory behavior on the TS, whose transient period evolves from the perturbed initial condition xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 to the stationary final condition xβð Þ t<sup>∞</sup> ; yβð Þ t<sup>∞</sup> � �. Such a transient behavior provides the necessary TS information to assess the influence of the states and parameters (load powers) in the stability of the analyzed EP. It is important to note that because of the EP stay undisturbed during the TS transient simulation, f <sup>x</sup>, f <sup>y</sup>, f <sup>β</sup> and gx, gy, g<sup>β</sup> are considered time-invariant matrices in the linear sensitivity model (13), where only the TS xkþ<sup>1</sup> <sup>β</sup> ykþ<sup>1</sup> β h i<sup>T</sup> are time-varying. Thus, a reduced computational burden is required for the TS simulation.

### 3.6 Approach for small-signal stability with trajectory sensitivities

The application of the SMA-TS approach to assess the effect of a set of system's parameters on the stability of the equilibrium points is summarized as follows:

Step 1. For an arbitrary set of fixed parameters β, the system's equilibrium is computed by solving the set of nonlinear algebraic equations (1) for x and y considering x\_ ¼ 0. The NR algorithm is applied to obtain this solution given by the values xe and ye that satisfy 0 <sup>¼</sup> f xe; ye; <sup>β</sup> � � and 0 <sup>¼</sup> g xe; ye; <sup>β</sup> � �.

Step 2. Based on the Schur and the implicit function theorems [28, 29], compute the reduced Jacobian matrix JR <sup>¼</sup> <sup>f</sup> <sup>x</sup> � <sup>f</sup> <sup>y</sup>g�<sup>1</sup> <sup>y</sup> g<sup>x</sup> � �� � � xe;ye ð Þ;<sup>β</sup> that has the same dynamic and algebraic properties of the system's Jacobian matrix.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

Step 3. Determine the critical eigenvalues of JR, and perform a SMA to identify the associate state variables.

Step 4. Compute the sensitivities of associated state variables with respect to the selected system's parameters (load powers) at the equilibrium point, xt!<sup>∞</sup> <sup>β</sup> and yt!<sup>∞</sup> <sup>β</sup> , by solving (16). The integration process is started with initial conditions xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 for the parameter sensitivities, while the state and algebraic variables are set at their equilibrium values during the solution process. The time evolution of sensitivities is computed under the assumption that a very small perturbation is carried out in the system such that the state and algebraic variables are infinitesimally perturbed; their values, therefore, can be considered constant during the computation of the sensitivity index. These assumptions permit us to directly relate our proposal to the theory of trajectory sensitivities:

$$\underbrace{\begin{bmatrix} I - \frac{\Delta t}{2} f\_{\times} & -\frac{\Delta t}{2} f\_{\times} \\ \mathbf{g}\_{\times} & \mathbf{g}\_{\times} \end{bmatrix} \Bigg|\_{\{\mathbf{x}, y, \boldsymbol{\rho}\}} \underbrace{\begin{bmatrix} \mathbf{x}\_{\boldsymbol{\rho}}^{k+1} \\ \mathbf{y}\_{\boldsymbol{\rho}}^{k+1} \end{bmatrix}}\_{\mathbf{S}} = \underbrace{\begin{bmatrix} \mathbf{x}\_{\boldsymbol{\rho}}^{k} + \frac{\Delta t}{2} \left( f\_{\times} \mathbf{x}\_{\boldsymbol{\rho}}^{k} + f\_{\times} \mathbf{y}\_{\boldsymbol{\rho}}^{k} + 2 \mathbf{f}\_{\boldsymbol{\rho}} \right) \\ -\mathbf{g}\_{\boldsymbol{\rho}} \end{bmatrix}}\_{\mathbf{B}} \tag{16}$$

Step 5. Quantify the interaction between the system parameters and the associated state variables by using the sensitivity index (15). Since this index is a function of the sensitivities of those state variables directly associated with the oscillatory modes and the critical eigenvalues, it can be used to quantify the effect of the ith parameter on these variables. In this case, the highest values of the sensitivity norms indicate the most sensitive parameters. Furthermore, the sensitivity index value increases as the system is approaching an oscillatory stability problem.

### 4. Small-signal stability analysis

This section presents the analysis of the TS applied on assessing the SSS analysis. The effectiveness of the SMA-TS approach is numerically tested by analyzing the WSCC 9-buses and 3-generator systems [7] and a reduced equivalent system corresponding to the Mexican power system consisting of 190-buses with 46 generators. For the purpose of the studies presented in this section, the system generators are represented by means of the two-axis model with a simple fast exciter loop containing max/min ceiling limits. For this case the system loads are represented by means of the constant power load model; however, the generality of the method allows to consider any load model.

#### 4.1 Modal analysis WSCC system

In this subsection the conventional modal analysis is employed in order to investigate the small-signal stability of the WSCC power system, whose diagram is given in Figure 1. Table 1 presents the modal analysis for different EPs as the active power at bus 5 is increasing.

SSS and SMA were performed to investigate the different operating points corresponding to the different levels of loading. Eigen-analysis revealed the SSS of EPs, whereas participation factors were used to identify the most associated states to the critical eigenvalue at each EP, as given in Table 1. The first column represents the load changes at bus 5. The second column presents the critical eigenvalue for the EP. The third column presents the most associated states to the critical mode

SNρð Þ¼ tk

Power System Stability

value.

transient stability [27].

3.5 Sensitivity initial conditions

∑ ng

@

<sup>∂</sup>δmð Þ tk ∂βρ

where j denotes the reference generator.

m¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!<sup>2</sup> 0

þ

In this case, the time evolution of sensitivities is necessary to quantify the loads' influence on the possible occurrence of an oscillatory instability, where the highest values of the sensitivity norms SN<sup>ρ</sup> indicate the most sensitive loads for the EP's stability. Thus, it should be noted that the critical load powers are those with the largest values of sensitivity index within the integration period, not the largest final

Note that this sensitivity norm has been successfully applied before for developing suitable approaches to improve the transient stability of power systems, e.g., for the estimation of the critical clearing time of a faulted system [23], the best possible location of FACTS controllers for transient stability enhancement [24–26], and the thyristor-controlled series compensator (TCSC) control design to enhance

The analysis of a stationary operating point based on this approach is performed by keeping the corresponding EP constant during the whole simulation. Thus, during this simulation period with no perturbation, which is considered from the instant of time t<sup>0</sup> and the infinity t∞, the TS also remain into the same stationary behavior established at nonzero constant values xβð Þ t<sup>∞</sup> 6¼ 0 and yβð Þ t<sup>∞</sup> 6¼ 0. However, as our approach is based on catching the oscillatory behavior of trajectory sensitivities, the values xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 are arbitrarily used as the initial conditions of the sensitivity variables. Then, such an initial condition perturbation starts an oscillatory behavior on the TS, whose transient period evolves from the perturbed initial con-

dition xβð Þ¼ t<sup>0</sup> yβð Þ¼ t<sup>0</sup> 0 to the stationary final condition xβð Þ t<sup>∞</sup> ; yβð Þ t<sup>∞</sup>

Thus, a reduced computational burden is required for the TS simulation.

3.6 Approach for small-signal stability with trajectory sensitivities

values xe and ye that satisfy 0 <sup>¼</sup> f xe; ye; <sup>β</sup> � � and 0 <sup>¼</sup> g xe; ye; <sup>β</sup> � �.

and algebraic properties of the system's Jacobian matrix.

the reduced Jacobian matrix JR <sup>¼</sup> <sup>f</sup> <sup>x</sup> � <sup>f</sup> <sup>y</sup>g�<sup>1</sup>

20

linear sensitivity model (13), where only the TS xkþ<sup>1</sup>

transient behavior provides the necessary TS information to assess the influence of the states and parameters (load powers) in the stability of the analyzed EP. It is important to note that because of the EP stay undisturbed during the TS transient simulation, f <sup>x</sup>, f <sup>y</sup>, f <sup>β</sup> and gx, gy, g<sup>β</sup> are considered time-invariant matrices in the

The application of the SMA-TS approach to assess the effect of a set of system's

Step 1. For an arbitrary set of fixed parameters β, the system's equilibrium is computed by solving the set of nonlinear algebraic equations (1) for x and y considering x\_ ¼ 0. The NR algorithm is applied to obtain this solution given by the

Step 2. Based on the Schur and the implicit function theorems [28, 29], compute

<sup>y</sup> g<sup>x</sup> � ��

�

parameters on the stability of the equilibrium points is summarized as follows:

<sup>∂</sup>ωmð Þ tk ∂βρ

vuuut <sup>∀</sup><sup>ρ</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Np, (15)

1 A

� �

� xe;ye ð Þ;<sup>β</sup> that has the same dynamic

are time-varying.

<sup>β</sup> ykþ<sup>1</sup> β h i<sup>T</sup> . Such a

� <sup>∂</sup>δjð Þ tk ∂βρ

!<sup>2</sup>

and the critical eigenvalue of the EP is complex. The proximity to the HB point is qualitatively assessed by observing the damping of the TS oscillation, i.e., if the TS oscillation is positively damped, the system is operating before the HB (stable EP); however, if the TS oscillation is undamped, the system operates after the HB

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

Multiparameter sensitivity is used to assess the load influence around the HB. Such an assessment requires only one simulation to identify the critical loading direction on approaching HB. The TS computation in all cases started from second

In order to qualitatively analyze the oscillatory behavior of the EPs around the HB, Figures 2–4 show the TS of the dynamic variables (generator states) with respect to the load embedded at bus 5. The oscillation waveforms and their peak

Parameter sensitivities with respect to PL<sup>5</sup> ¼ 4:3 pu, λcrit ¼ �0:5395 � 6:8512i, [30].

Parameter sensitivities with respect to PL<sup>5</sup> ¼ 4:4 pu, λcrit ¼ �0:0305 � 6:1462i, [30].

(unstable EP).

one onward.

Figure 2.

Figure 3.

23

5.1 Stability around the Hopf bifurcation

DOI: http://dx.doi.org/10.5772/intechopen.81490

Figure 1. WSCC 9-buses, 3-generators.


#### Table 1. Selective modal analysis.

(eigenvalue), obtained by selecting the highest magnitudes of participation factors, which are given in the fourth column in the table.

As the load embedded at bus 5 increased, the stability of the new EP decreased with respect to the previous one. The power system oscillatory instability called the HB was detected when the load changed from 4.4 to 4.5 p.u. The SMA around the HB revealed generator 2 as the most participative in the unstable EP. It is important to observe in Table 1 that generator 2 is the most associated with the critical eigenvalues for all analyzed EPs.

## 5. Trajectory sensitivity analysis: WSCC system

In order to test the proposed method based on TS to assess the EP stability, operating points before and after the HB point were investigated [30]. As the active power increases at bus 5, as was performed in [7], the system proximity to the bifurcation points is assessed by using the analysis of TS. The rotor angle and speed sensitivities with respect to the load active powers were traced and observed through the time. The TS oscillations provide qualitative information used to investigate the proximity to bifurcation points. Such sensitivity oscillations agree with the critical eigenvalues obtained by the SMA at each EP, as reported in Table 1. Thus, the oscillatory behavior in the TS indicates that the EP is around the HB point Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

and the critical eigenvalue of the EP is complex. The proximity to the HB point is qualitatively assessed by observing the damping of the TS oscillation, i.e., if the TS oscillation is positively damped, the system is operating before the HB (stable EP); however, if the TS oscillation is undamped, the system operates after the HB (unstable EP).

Multiparameter sensitivity is used to assess the load influence around the HB. Such an assessment requires only one simulation to identify the critical loading direction on approaching HB. The TS computation in all cases started from second one onward.

#### 5.1 Stability around the Hopf bifurcation

In order to qualitatively analyze the oscillatory behavior of the EPs around the HB, Figures 2–4 show the TS of the dynamic variables (generator states) with respect to the load embedded at bus 5. The oscillation waveforms and their peak

Figure 2. Parameter sensitivities with respect to PL<sup>5</sup> ¼ 4:3 pu, λcrit ¼ �0:5395 � 6:8512i, [30].

Figure 3. Parameter sensitivities with respect to PL<sup>5</sup> ¼ 4:4 pu, λcrit ¼ �0:0305 � 6:1462i, [30].

(eigenvalue), obtained by selecting the highest magnitudes of participation factors,

PL5ðpuÞ λcrit Associated states PF

d2, E<sup>0</sup>

d2, E<sup>0</sup>

d2, E<sup>0</sup>

<sup>q</sup>1, δ2,ω2, Efd1, E<sup>0</sup>

<sup>q</sup>1, δ2,ω2, Efd1, E<sup>0</sup>

<sup>q</sup>1, δ2,ω2, Efd1, E<sup>0</sup>

<sup>q</sup>1, δ2,ω2, Efd1, E<sup>0</sup>

<sup>q</sup>1, Efd<sup>1</sup> 1.0, 0.99, 0.18, 0.18, 0.16

<sup>q</sup>1, Efd<sup>1</sup> 1.0, 0.99, 0.28, 0.39, 0.35

<sup>q</sup>1, Efd<sup>1</sup> 1.0, 0.99, 0.46, 0.99, 0.83

<sup>d</sup><sup>2</sup> 1.0, 0.85, 0.85, 0.79, 0.38

<sup>d</sup><sup>2</sup> 1.0, 0.81, 0.80, 0.76, 0.32

<sup>d</sup><sup>2</sup> 1.0, 0.79, 0.78, 0.74, 0.28

<sup>d</sup><sup>2</sup> 1.0, 0.77, 0.75, 0.76, 0.26

As the load embedded at bus 5 increased, the stability of the new EP decreased with respect to the previous one. The power system oscillatory instability called the HB was detected when the load changed from 4.4 to 4.5 p.u. The SMA around the HB revealed generator 2 as the most participative in the unstable EP. It is important to observe in Table 1 that generator 2 is the most associated with the critical

In order to test the proposed method based on TS to assess the EP stability, operating points before and after the HB point were investigated [30]. As the active power increases at bus 5, as was performed in [7], the system proximity to the bifurcation points is assessed by using the analysis of TS. The rotor angle and speed sensitivities with respect to the load active powers were traced and observed through the time. The TS oscillations provide qualitative information used to investigate the proximity to bifurcation points. Such sensitivity oscillations agree with the critical eigenvalues obtained by the SMA at each EP, as reported in Table 1. Thus, the oscillatory behavior in the TS indicates that the EP is around the HB point

which are given in the fourth column in the table.

4.2 �0.5085 � 7.3001i ω2, δ2, E<sup>0</sup>

4.3 �0.5395 � 6.8512i ω2, δ2, E<sup>0</sup>

4.4 �0.0305 � 6.1462i δ2,ω2, E<sup>0</sup>

4.5 0.7064 � 5.8935i E<sup>0</sup>

4.6 1.5118 � 5.7190i E<sup>0</sup>

4.7 2.6677 � 5.5020i E<sup>0</sup>

4.8 5.3798 � 4.5835i E<sup>0</sup>

5. Trajectory sensitivity analysis: WSCC system

eigenvalues for all analyzed EPs.

Table 1.

22

Figure 1.

WSCC 9-buses, 3-generators.

Power System Stability

Selective modal analysis.

values allow to assess and to determine the EP stability as well as its most associated states because of the loading increase. Figure 2 shows the TS with respect to P<sup>5</sup> ¼ 4:3 pu where the sensitivity oscillations are damped. This agrees with the corresponding critical eigenvalue λcrit ¼ �0:5395 � 6:8512i which indicates that the system is not too close to the HB point. However, Figure 3 shows sensitivity oscillations with a very small damping when P<sup>5</sup> ¼ 4:4 pu, where the critical eigenvalue λcrit ¼ �0:0305 � 6:1462i is very close to the imaginary axis in the complex plane and hence close to a HB point. This means that a very small variation in the load parameter could steer the system to operate in an unstable EP. Figure 4 shows the TS behavior when P<sup>5</sup> ¼ 4:41 pu, which corresponds to an unstable EP after a HB point. From Figures 2–4, it is clear that the highest peaks of the TS oscillations in all cases correspond to generator 2. Therefore, such a generator is the most influential in the EP stability according to the most associated states reported in Table 1. However, this correspondence between associated states and TS is not always kept as will be shown in the upcoming sections.

For the two cases, the TS calculation required to compute 14000 and 1400 forward/ backward substitutions, respectively. Such a difference is equivalent to reduce in 90% the number of sensitivity solutions. For the case with Δt ¼ 0:1 sec, 140 forward/backward substitutions were required, which represents a reduction of 99% of such solutions required to assess the transient. Although for this case, a small difference between trajectory sensitivities results, this is still negligible. This TS-based method can assess at the same time the effect of NP parameters in the EP stability, whereas in the method of eigenvalues and modal analysis, it is not

Effect of the integration time step on the evolution of parameter sensitivities, [30].

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

DOI: http://dx.doi.org/10.5772/intechopen.81490

The participation factors provide the change of critical eigenvalues to the change of the states of critical machines ð Þ <sup>∂</sup>λ=∂<sup>x</sup> , which establishes the SMA. However, this analysis does not provide information about the critical parameter (critical load in this case) influencing the EP stability but only the critical generator and its most participative states. Thus, the modal analysis does not allow the direct identification of the most sensitive loading directions in the stability of the EPs around a HB point. In practice, load increments not have a unique loading direction as in the previous study, which results in a valid consideration only for academic interest. All loads are then constantly varying in NP directions, so that it is very important to have a general tool to assess the stability of the EPs, especially operating under stressed conditions of loading. In this context, besides identifying the critical generators (states), the SMA-TS approach based on TS identifies the loading directions which

Multiparameter analysis of TS allows computing trajectory sensitivities with respect to NP parameters in a power system [17] at the same time, as explained in Section 3.3, and can be used to find out the influence of the different loading directions on the SSS around a HB point. For this purpose, Figure 6 shows the sensitivity norm SN through the time with respect to the three embedded loads in the system, where the highest peaks indicate the maximum influence of the

corresponding load in the oscillatory behavior. The load demand corresponds to the

possible.

Figure 5.

5.3 Most sensitive loads to Hopf bifurcation

are most sensitive to oscillatory instabilities.

base case as provided in [7].

25

It should be noted from the figures that the load demand at bus 5 increases as the peaks of TS are higher. This qualitative information indicates that such a load variation has a major impact in the load angle and speed of generator 2 (red lines) than the state variables associated with generator 3 (blue lines). Note that this line of reasoning also applies for unstable EPs, i.e., after passing the HB point, as can be observed in Figure 4. Thus, a short simulation in time is enough for this approach to determine the EP stability and their most influencing generators.

#### 5.2 Simulation efficiency

In order to observe the effect of the time-step integration Δt in the computational burden, Figure 5 shows the rotor angle sensitivity of generator 2 with respect to the active power embedded at bus 5. Such a trajectory was computed using three different time-step values. The figure shows sensitivity oscillations considering a period of 14 seconds.

It is important to observe that the big difference between Δt ¼ 0:001 sec and Δt ¼ 0:01 sec results in negligible differences in the resulting trajectory sensitivities. Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

Figure 5. Effect of the integration time step on the evolution of parameter sensitivities, [30].

For the two cases, the TS calculation required to compute 14000 and 1400 forward/ backward substitutions, respectively. Such a difference is equivalent to reduce in 90% the number of sensitivity solutions. For the case with Δt ¼ 0:1 sec, 140 forward/backward substitutions were required, which represents a reduction of 99% of such solutions required to assess the transient. Although for this case, a small difference between trajectory sensitivities results, this is still negligible. This TS-based method can assess at the same time the effect of NP parameters in the EP stability, whereas in the method of eigenvalues and modal analysis, it is not possible.

#### 5.3 Most sensitive loads to Hopf bifurcation

The participation factors provide the change of critical eigenvalues to the change of the states of critical machines ð Þ <sup>∂</sup>λ=∂<sup>x</sup> , which establishes the SMA. However, this analysis does not provide information about the critical parameter (critical load in this case) influencing the EP stability but only the critical generator and its most participative states. Thus, the modal analysis does not allow the direct identification of the most sensitive loading directions in the stability of the EPs around a HB point. In practice, load increments not have a unique loading direction as in the previous study, which results in a valid consideration only for academic interest. All loads are then constantly varying in NP directions, so that it is very important to have a general tool to assess the stability of the EPs, especially operating under stressed conditions of loading. In this context, besides identifying the critical generators (states), the SMA-TS approach based on TS identifies the loading directions which are most sensitive to oscillatory instabilities.

Multiparameter analysis of TS allows computing trajectory sensitivities with respect to NP parameters in a power system [17] at the same time, as explained in Section 3.3, and can be used to find out the influence of the different loading directions on the SSS around a HB point. For this purpose, Figure 6 shows the sensitivity norm SN through the time with respect to the three embedded loads in the system, where the highest peaks indicate the maximum influence of the corresponding load in the oscillatory behavior. The load demand corresponds to the base case as provided in [7].

values allow to assess and to determine the EP stability as well as its most associated states because of the loading increase. Figure 2 shows the TS with respect to P<sup>5</sup> ¼ 4:3 pu where the sensitivity oscillations are damped. This agrees with the corresponding critical eigenvalue λcrit ¼ �0:5395 � 6:8512i which indicates that the system is not too close to the HB point. However, Figure 3 shows sensitivity oscillations with a very small damping when P<sup>5</sup> ¼ 4:4 pu, where the critical eigenvalue λcrit ¼ �0:0305 � 6:1462i is very close to the imaginary axis in the complex plane and hence close to a HB point. This means that a very small variation in the load parameter could steer the system to operate in an unstable EP. Figure 4 shows the TS behavior when P<sup>5</sup> ¼ 4:41 pu, which corresponds to an unstable EP after a HB point. From Figures 2–4, it is clear that the highest peaks of the TS oscillations in all cases correspond to generator 2. Therefore, such a generator is the most influential in the EP stability according to the most associated states reported in Table 1. However, this correspondence between associated states and TS is not always kept

Parameter sensitivities with respect to PL<sup>5</sup> ¼ 4:41 pu, λcrit ¼ 0:0462 � 6:1105i, [30].

It should be noted from the figures that the load demand at bus 5 increases as the

In order to observe the effect of the time-step integration Δt in the computational burden, Figure 5 shows the rotor angle sensitivity of generator 2 with respect to the active power embedded at bus 5. Such a trajectory was computed using three different time-step values. The figure shows sensitivity oscillations considering a

It is important to observe that the big difference between Δt ¼ 0:001 sec and Δt ¼ 0:01 sec results in negligible differences in the resulting trajectory sensitivities.

peaks of TS are higher. This qualitative information indicates that such a load variation has a major impact in the load angle and speed of generator 2 (red lines) than the state variables associated with generator 3 (blue lines). Note that this line of reasoning also applies for unstable EPs, i.e., after passing the HB point, as can be observed in Figure 4. Thus, a short simulation in time is enough for this approach to

determine the EP stability and their most influencing generators.

as will be shown in the upcoming sections.

5.2 Simulation efficiency

Figure 4.

Power System Stability

period of 14 seconds.

Figure 6. Loads' effect on the EP's stability (WSCC system), [30].


#### Table 2.

Sensitivity norm and Hopf bifurcation (WSCC system).

The oscillation of SN<sup>ρ</sup> shows that the load embedded at bus 8 ð Þ PL<sup>8</sup> is the most sensitive for the EP. The load at bus 6 ð Þ PL<sup>6</sup> is the next most sensitive and finally the load PL5. In order to validate the information provided by the SN<sup>ρ</sup> in Figure 6, one parametric study at a time was carried out for the active powers PL<sup>8</sup> and PL<sup>6</sup> as performed in Table 1 for the load PL5. The results are reported in Table 2 as follows: the first column indicates the load nodes, and columns 2 and 3 present the measured powers in the base case and the active power increment from the base case to the HB point to each loading direction, respectively. The fourth column provides the active power magnitude at which a HB point occurs, and lastly, column 5 presents the obtained values of SNρ. It is important to observe in Table 2 that the smallest load increment matches the highest SN value and vice versa, i.e., the highest SN indicates a major change in the EP stability with respect to the corresponding load variation, which determines a shorter way to the HB point. Thus, the smallest increment ΔPL<sup>8</sup> ¼ 299 MW proves that the highest SN value indicates this load takes the system to the HB faster than ΔPL<sup>6</sup> ¼ 314 MW and this in turn faster than ΔPL<sup>5</sup> ¼ 316 MW. This agrees with the SN<sup>ρ</sup> reported in Figure 6.

## 6. Trajectory sensitivity analysis: the Mexican system

In this section, the study consisted of computing the TS norm for 91 loads embedded in a reduced equivalent of the Mexican energy system, which consists of 190 nodes and 46 generators. The transmission components are divided into 180 transmission lines and 83 power transformers. Lastly, the system contains 26

capacitive compensators in shunt connection. The unifilar diagram of the power

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

DOI: http://dx.doi.org/10.5772/intechopen.81490

In order to assess the effect of the system loads on the system's dynamic performance, the sensitivity norms with respect to 91 loads were computed. Figure 8 shows the effect of these sensitivities on a critically stable EP. Note that the active power demanded by loads connected at buses from 150 to 152 is the most sensitive in the EP stability. Therefore, according to the reasoning used into the previous

system is shown in Figure 7.

Loads' effect on the equilibrium point stability (Mexican system), [30].

Figure 7.

Figure 8.

27

The Mexican power system.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

Figure 7. The Mexican power system.

The oscillation of SN<sup>ρ</sup> shows that the load embedded at bus 8 ð Þ PL<sup>8</sup> is the most sensitive for the EP. The load at bus 6 ð Þ PL<sup>6</sup> is the next most sensitive and finally the load PL5. In order to validate the information provided by the SN<sup>ρ</sup> in Figure 6, one parametric study at a time was carried out for the active powers PL<sup>8</sup> and PL<sup>6</sup> as performed in Table 1 for the load PL5. The results are reported in Table 2 as follows: the first column indicates the load nodes, and columns 2 and 3 present the measured powers in the base case and the active power increment from the base case to the HB point to each loading direction, respectively. The fourth column provides the active power magnitude at which a HB point occurs, and lastly, column 5 presents the obtained values of SNρ. It is important to observe in Table 2 that the smallest load increment matches the highest SN value and vice versa, i.e., the highest SN indicates a major change in the EP stability with respect to the corresponding load variation, which determines a shorter way to the HB point. Thus, the smallest increment ΔPL<sup>8</sup> ¼ 299 MW proves that the highest SN value indicates this load takes the system to the HB faster than ΔPL<sup>6</sup> ¼ 314 MW and this in turn faster than

Node Pbase (MW) ΔPHB (MW) PHB (MW) SN 100.0 299 399 1.723 90.0 314 404 0.997 125.0 316 441 0.739

ΔPL<sup>5</sup> ¼ 316 MW. This agrees with the SN<sup>ρ</sup> reported in Figure 6.

Figure 6.

Power System Stability

Table 2.

26

Loads' effect on the EP's stability (WSCC system), [30].

Sensitivity norm and Hopf bifurcation (WSCC system).

6. Trajectory sensitivity analysis: the Mexican system

In this section, the study consisted of computing the TS norm for 91 loads embedded in a reduced equivalent of the Mexican energy system, which consists of 190 nodes and 46 generators. The transmission components are divided into 180 transmission lines and 83 power transformers. Lastly, the system contains 26

Figure 8.

Loads' effect on the equilibrium point stability (Mexican system), [30].

capacitive compensators in shunt connection. The unifilar diagram of the power system is shown in Figure 7.

In order to assess the effect of the system loads on the system's dynamic performance, the sensitivity norms with respect to 91 loads were computed. Figure 8 shows the effect of these sensitivities on a critically stable EP. Note that the active power demanded by loads connected at buses from 150 to 152 is the most sensitive in the EP stability. Therefore, according to the reasoning used into the previous

section, the loading increase in such directions will steer in a faster way the system to a HB than the rest of the system loads. It must be pointed out that the computation of the sensitivity norms for the 91 system loads were carried out by using one sole time-domain simulation, which corresponds to solving 91 sensitivity DAE systems, with each one consisting of 702 equations and variables. Thus, the assessment of the 91 loads is equivalent to solving 63882 equations in the same simulation at the same time. However, considering the linearity of the sensitivity systems, the same time-invariant Jacobian matrix is used during the whole time-domain simulation, which considerably reduces the computational burden.

In order to validate the load ranking influence via the sensitivity norm, Table 3 shows how the increments in the most sensitive loading directions influence the SSS, as well as the proximity to the HB point. Column 1 (Node) indicates the most sensitive loads resulted from the TS analysis shown in Figure 8. Column 2 ð Þ λcrit provides the critical eigenvalue for the new EP resulting from such an increment. Columns 3 and 4 show the measured value of active power in the analyzed base case ð Þ Pbase corresponding to Figure 9 and the increment in the specified loading direction ð Þ ΔP ð Þ 60 MW , respectively. Lastly, in columns 5 and 6, ð Þ ΔPHB and ð Þ PHB indicate the increased amount and the value of the active power where the system

It is important to outline that the load effect in the EP stability is not only dependent on the magnitude but also on the topologic location of loads. For example, the power demand embedded at bus 120 is 17 times larger than the load at bus 151; however, the load at 151 resulted in being more sensitive than the load embedded at bus 120, as shown in Table 3, column 3. It must be observed that the most sensitive loads (loads 152–147) provided a major change in the critical eigenvalue and thus in the SSS. The same increment in the most sensitive loading directions (buses 152–147) led the system to oscillatory instability due to a HB point, whereas with the increment in the least sensitive loading directions, the system remained stable. Then, the stability margins in the most sensitive loading directions become more reduced; therefore, according to the sensitivity ranking in Table 3, as the most sensitive loads were increased, the appearance of the HB was found faster as can be observed in column 5. Once more the SMA-TS approach has been successfully proved by determining that the most sensitive loads indicate the shorter ways toward the small-signal instability of the electric

In this chapter an alternative approach for monitoring the Hopf bifurcations along variations in multidimensional loading directions by using a time-domain method is presented, which is based on trajectory sensitivities. This approach, SMA-TS, is general and flexible, i.e., the size of the power systems, as well as the complexity of their mathematical modeling, does not represent any restriction. SMA-TS allows to identify the critical loading directions that steer the system to Hopf bifurcation points. Such an approach was tested in the 9-buses, 3-generators system as well as in 190-buses, 46-generators system. Regardless of the number of sensitivity parameters and system dimensions, SMA-TS requires only one simulation. Such a method keeps constant the Jacobian matrix of the system, requiring only one evaluation and factorization during the whole simulation. The computational effort then consists of performing just one forward/backward substitution at each time step. Furthermore, the approach can handle a very large integration step to drastically reduce the computational effort. Lastly, its application is suitable for real-time monitoring and security assessment in energy man-

The authors thank the University of Guanajuato for providing the financial

support with PFCE 2018 resource for this publication.

crosses a HB point by following the corresponding loading directions.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

DOI: http://dx.doi.org/10.5772/intechopen.81490

power systems.

7. Conclusions

agement systems.

29

Acknowledgements

Once the critical loads have been identified from Figure 8, it is possible to know the most affected generators by the most sensitive loads. Figure 9 shows the TS with respect to the active power at bus 152, which resulted as the most sensitive in the sensitivity norm assessment. The damped oscillations in the TS indicate that the EP is stable and the operation point is not at a HB, which agrees with the corresponding critical eigenvalue of the EP λ ¼ �0:0501 � 7:8518i. It must be observed from Figure 9 that the highest rotor angle sensitivities ∂δ32=∂PL<sup>152</sup> and ∂δ33=∂PL<sup>152</sup> have identified generators 32 and 33 as the most influenced by the active power embedded at bus 152.

Figure 9. Evolution of parameter sensitivities with respect to PL152, λcrit ¼ �0:05 � 7:85i, [30].


Table 3.

Loads' sensitivity norm to Hopf bifurcation in the Mexican system.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

In order to validate the load ranking influence via the sensitivity norm, Table 3 shows how the increments in the most sensitive loading directions influence the SSS, as well as the proximity to the HB point. Column 1 (Node) indicates the most sensitive loads resulted from the TS analysis shown in Figure 8. Column 2 ð Þ λcrit provides the critical eigenvalue for the new EP resulting from such an increment. Columns 3 and 4 show the measured value of active power in the analyzed base case ð Þ Pbase corresponding to Figure 9 and the increment in the specified loading direction ð Þ ΔP ð Þ 60 MW , respectively. Lastly, in columns 5 and 6, ð Þ ΔPHB and ð Þ PHB indicate the increased amount and the value of the active power where the system crosses a HB point by following the corresponding loading directions.

It is important to outline that the load effect in the EP stability is not only dependent on the magnitude but also on the topologic location of loads. For example, the power demand embedded at bus 120 is 17 times larger than the load at bus 151; however, the load at 151 resulted in being more sensitive than the load embedded at bus 120, as shown in Table 3, column 3. It must be observed that the most sensitive loads (loads 152–147) provided a major change in the critical eigenvalue and thus in the SSS. The same increment in the most sensitive loading directions (buses 152–147) led the system to oscillatory instability due to a HB point, whereas with the increment in the least sensitive loading directions, the system remained stable. Then, the stability margins in the most sensitive loading directions become more reduced; therefore, according to the sensitivity ranking in Table 3, as the most sensitive loads were increased, the appearance of the HB was found faster as can be observed in column 5. Once more the SMA-TS approach has been successfully proved by determining that the most sensitive loads indicate the shorter ways toward the small-signal instability of the electric power systems.

### 7. Conclusions

section, the loading increase in such directions will steer in a faster way the system to a HB than the rest of the system loads. It must be pointed out that the computation of the sensitivity norms for the 91 system loads were carried out by using one sole time-domain simulation, which corresponds to solving 91 sensitivity DAE systems, with each one consisting of 702 equations and variables. Thus, the assessment of the 91 loads is equivalent to solving 63882 equations in the same simulation at the same time. However, considering the linearity of the sensitivity systems, the same time-invariant Jacobian matrix is used during the whole time-domain simula-

Once the critical loads have been identified from Figure 8, it is possible to know the most affected generators by the most sensitive loads. Figure 9 shows the TS with respect to the active power at bus 152, which resulted as the most sensitive in the sensitivity norm assessment. The damped oscillations in the TS indicate that the

tion, which considerably reduces the computational burden.

power embedded at bus 152.

Power System Stability

Figure 9.

Table 3.

28

EP is stable and the operation point is not at a HB, which agrees with the corresponding critical eigenvalue of the EP λ ¼ �0:0501 � 7:8518i. It must be observed from Figure 9 that the highest rotor angle sensitivities ∂δ32=∂PL<sup>152</sup> and ∂δ33=∂PL<sup>152</sup> have identified generators 32 and 33 as the most influenced by the active

Evolution of parameter sensitivities with respect to PL152, λcrit ¼ �0:05 � 7:85i, [30].

Loads' sensitivity norm to Hopf bifurcation in the Mexican system.

Node λcrit Pbase (MW) ΔP<sup>60</sup> (60 MW) ΔPHB (MW) PHB (MW) 152 0.021 � 5.02i 172.64 232.64 53.0 225.64 150 0.018 � 5.03i 188.24 248.64 54.0 242.24 151 0.015 � 5.03i 18.72 78.72 54.0 72.72 147 0.012 � 5.05i 104.00 164.00 56.0 160.00 153 �0.011 � 8.83i 78.00 138.00 63.0 141.00 145 �0.015 � 5.13i 83.20 143.20 72.0 155.20 120 �0.050 � 7.85i 308.88 368.88 189.0 497.88

In this chapter an alternative approach for monitoring the Hopf bifurcations along variations in multidimensional loading directions by using a time-domain method is presented, which is based on trajectory sensitivities. This approach, SMA-TS, is general and flexible, i.e., the size of the power systems, as well as the complexity of their mathematical modeling, does not represent any restriction. SMA-TS allows to identify the critical loading directions that steer the system to Hopf bifurcation points. Such an approach was tested in the 9-buses, 3-generators system as well as in 190-buses, 46-generators system. Regardless of the number of sensitivity parameters and system dimensions, SMA-TS requires only one simulation. Such a method keeps constant the Jacobian matrix of the system, requiring only one evaluation and factorization during the whole simulation. The computational effort then consists of performing just one forward/backward substitution at each time step. Furthermore, the approach can handle a very large integration step to drastically reduce the computational effort. Lastly, its application is suitable for real-time monitoring and security assessment in energy management systems.

### Acknowledgements

The authors thank the University of Guanajuato for providing the financial support with PFCE 2018 resource for this publication.

## Conflict of interest

All the authors of this chapter declare to have no any conflict of interest related with any person, company, institution, etc.

References

[1] IEEE/CIGRE Joint Task Force on Stability Terms and Definitions. Definition and classification of power system stability. IEEE Transactions on Power Systems. 2004;19:1387-1401. DOI: 10.1109/TPWRS.2004.825981

DOI: http://dx.doi.org/10.5772/intechopen.81490

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis

[10] Ajjarapu V, Lee B. Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system. IEEE Transactions on Power Systems. 1992;7:424-431. DOI:

[11] Gu W, Milano F, Jiang P, Tang G. Hopf bifurcations induced by SVC controllers: A didactic example. Electric Power Systems Research. 2007;77: 234-240. DOI: 10.1016/j.epsr.

[12] Kwatny H, Fischl R, Nwankpa C. Local bifurcation in power systems: Theory, computation and applications. Proceedings of the IEEE. 1995;83:

[13] Xiaoyu W. A novel approach for identification and tracing of oscillatory stability and damping ratio margin boundaries [thesis]. Ames: Iowa State

[14] Joorabian M, Ramandi N, Ebadi M.

International Conference on Power and Energy (PECon 08); 1–3 December 2008; Johor Baharu, Malaysia. 2008.

[16] Sharma C, Singh P. Contribution of loads to low frequency in power system operation. In: Proceedings of the iREP Symposium-Bulk Power System Dynamics and Control-VII; 19–24 August 2007; Charleston, SC, USA.

Optimal location of static VAR compensator (SVC) based on small signal stability of power system. In: Proceedings of the 2nd IEEE

[15] Gupta A, Sharma P. Optimal location of SVC for dynamic stability enhancement based on eigenvalue analysis. Electrical and Electronics Engineering: An International Journal

(ELELIJ). 2014;3:25-37

10.1109/59.141738

2006.03.001

1456-1483

University; 2005

pp. 1333-1338

2007

[2] Chen C. Linear System Theory and Design. 3rd ed. New York, NY: Oxford University Press Inc.; 1998. 352 p

[3] Lyapunov A. Stability of motion. In:

Mathematics in Science and Engineering. Vol. 30. New York/ London: Academic Press Inc.; 1968

[4] Pérez-Arriaga I, Verghese G, Schweppe F. Selective modal analysis with applications to electric power systems. Part I: Heuristic introduction. IEEE Transactions on Power Apparatus and Systems. 1982;101:3117-3125. DOI:

10.1109/TPAS.1982.317524

1982.317525

Inc.; 1994. 1178 p

[5] Verghese G, Pérez-Arriaga I, Schweppe F. Selective modal analysis with applications to electric power systems, Part II: The dynamic stability problem. IEEE Transactions on Power Apparatus and Systems. 1982;101: 3126-3134. DOI: 10.1109/TPAS.

[6] Kundur P. Power System Stability and Control. California: McGraw Hill

[7] Sauer P, Pai M. Power System Dynamics and Stability. Upper, Saddle,

[8] Nayfeh A, Balachandran B. Applied Nonlinear Dynamics. New York, NY: John Wiley & Sons; 1995. 700 p

assessment and Control. New York, NY:

River, NJ: Prentice Hall; 1998

[9] Ajjarapu V. Computational Techniques for Voltage Stability

Springer; 2006. 250 p

31

## Author details

Enrique Arnoldo Zamora Cárdenas<sup>1</sup> \*, Alejandro Pizano Martínez<sup>1</sup> and Claudio Rubén Fuerte Esquivel<sup>2</sup>

1 Universidad de Guanajuato, Salamanca, Mexico

2 Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

\*Address all correspondence to: ezamora@ugto.mx

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Application of the Trajectory Sensitivity Theory to Small Signal Stability Analysis DOI: http://dx.doi.org/10.5772/intechopen.81490

## References

Conflict of interest

Power System Stability

Author details

30

Enrique Arnoldo Zamora Cárdenas<sup>1</sup>

1 Universidad de Guanajuato, Salamanca, Mexico

\*Address all correspondence to: ezamora@ugto.mx

provided the original work is properly cited.

2 Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Claudio Rubén Fuerte Esquivel<sup>2</sup>

with any person, company, institution, etc.

All the authors of this chapter declare to have no any conflict of interest related

\*, Alejandro Pizano Martínez<sup>1</sup> and

[1] IEEE/CIGRE Joint Task Force on Stability Terms and Definitions. Definition and classification of power system stability. IEEE Transactions on Power Systems. 2004;19:1387-1401. DOI: 10.1109/TPWRS.2004.825981

[2] Chen C. Linear System Theory and Design. 3rd ed. New York, NY: Oxford University Press Inc.; 1998. 352 p

[3] Lyapunov A. Stability of motion. In: Mathematics in Science and Engineering. Vol. 30. New York/ London: Academic Press Inc.; 1968

[4] Pérez-Arriaga I, Verghese G, Schweppe F. Selective modal analysis with applications to electric power systems. Part I: Heuristic introduction. IEEE Transactions on Power Apparatus and Systems. 1982;101:3117-3125. DOI: 10.1109/TPAS.1982.317524

[5] Verghese G, Pérez-Arriaga I, Schweppe F. Selective modal analysis with applications to electric power systems, Part II: The dynamic stability problem. IEEE Transactions on Power Apparatus and Systems. 1982;101: 3126-3134. DOI: 10.1109/TPAS. 1982.317525

[6] Kundur P. Power System Stability and Control. California: McGraw Hill Inc.; 1994. 1178 p

[7] Sauer P, Pai M. Power System Dynamics and Stability. Upper, Saddle, River, NJ: Prentice Hall; 1998

[8] Nayfeh A, Balachandran B. Applied Nonlinear Dynamics. New York, NY: John Wiley & Sons; 1995. 700 p

[9] Ajjarapu V. Computational Techniques for Voltage Stability assessment and Control. New York, NY: Springer; 2006. 250 p

[10] Ajjarapu V, Lee B. Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system. IEEE Transactions on Power Systems. 1992;7:424-431. DOI: 10.1109/59.141738

[11] Gu W, Milano F, Jiang P, Tang G. Hopf bifurcations induced by SVC controllers: A didactic example. Electric Power Systems Research. 2007;77: 234-240. DOI: 10.1016/j.epsr. 2006.03.001

[12] Kwatny H, Fischl R, Nwankpa C. Local bifurcation in power systems: Theory, computation and applications. Proceedings of the IEEE. 1995;83: 1456-1483

[13] Xiaoyu W. A novel approach for identification and tracing of oscillatory stability and damping ratio margin boundaries [thesis]. Ames: Iowa State University; 2005

[14] Joorabian M, Ramandi N, Ebadi M. Optimal location of static VAR compensator (SVC) based on small signal stability of power system. In: Proceedings of the 2nd IEEE International Conference on Power and Energy (PECon 08); 1–3 December 2008; Johor Baharu, Malaysia. 2008. pp. 1333-1338

[15] Gupta A, Sharma P. Optimal location of SVC for dynamic stability enhancement based on eigenvalue analysis. Electrical and Electronics Engineering: An International Journal (ELELIJ). 2014;3:25-37

[16] Sharma C, Singh P. Contribution of loads to low frequency in power system operation. In: Proceedings of the iREP Symposium-Bulk Power System Dynamics and Control-VII; 19–24 August 2007; Charleston, SC, USA. 2007

[17] Frank P. Introduction to System Sensitivity Theory. 1st ed. New York: Academic Press; 1978. p. 286

[18] Tomovic R, Vucobratovic M. General Sensitivity Theory. New York: North-Holland; 1972. p. 266

[19] Laufenberg M, Pai M. A new approach to dynamic security assessment using trajectory sensitivities. IEEE Transactions on Power Systems. 1998;13:953-958. DOI: 10.1109/ 59.709082

[20] Nguyen T, Pai M. Trajectory sensitivity analysis for dynamic security assessment and other applications in power systems. In: Savulescu S, editor. Real-Time Stability in Power Systems. 2nd ed. Switzerland: Springer; 2014. DOI: 10.1007/978-3-319-06680-6\_11

[21] Hiskens I, Pai M. Trajectory sensitivity analysis of hybrid systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2000;47:204-220. DOI: 10.1109/81.828574

[22] Hong Z, Shrirang A, Emil C, Mihai A. Discrete adjoint sensitivity analysis of hybrid dynamical systems with switching [Discrete adjoint sensitivity analysis of hybrid dynamical systems]. IEEE Transactions on Circuits and Systems I: Regular Papers. 2017. DOI: 10.1109/TCSI.2017.2651683

[23] Nguyen T, Pai M, Hiskens I. Sensitivity approaches for direct computation of critical parameters in a power system. International Journal of Electrical Power and Energy Systems. 2002;24:337-343. DOI: 10.1016/ S0142-0615(01)00050-3

[24] Chatterjee D, Ghosh A. Transient stability assessment of power system containing series and shunt compensators. IEEE Transactions on Power Systems. 2007;22:1210-1220. DOI: 10.1109/TPWRS.2007.901455

[25] Suguna R, Jalaja S, Pradeep M, Senthil R, SrikrishnaKumar S, Sugavanam K. Transient stability improvement using shunt and series compensators. Indian Journal of Science and Technology. 2016;9:1-11. DOI: 10.17485/ijst/2016/v9i11/89402

Chapter 3

Cai Hui

Abstract

Power System Small-Signal

Grid-Connected SmartPark

Large-scale smart charging stations can effectively satisfy and control the charging demands of tremendous plug-in electric vehicles (PEVs). But, simultaneously, their penetrations inevitably induce new challenges to the operation of power systems. In this chapter, damping torque analysis (DTA) was employed to examine the effects of the integration of smart charging station on the dynamic stability of the transmission system. A single-machine infinite-bus power system with a smart charging station that denoted the equivalent of several ones was used for analysis. The results obtained from DTA reveal that in view of the damping ratio, the optimal charging capacity is better to be considered in the design of the smart charging station. Under the proposed charging capacity, the power system can achieve the best maintained dynamic stability, and the damping ratio can reach the crest value. Phase compensation method was utilized to design the stabilizer via the active and reactive power regulators of the smart charging station respectively. With the help of the stabilizers, damping of the system oscillation under certain operating conditions can be significantly improved, and the power oscillation in the

Keywords: smart charging station, plug-in electric vehicles (PEVs), power system oscillations, small-signal stability, damping torque analysis (DTA), stabilizer design

The growing concern of carbon dioxide emission, greenhouse effect, and rapid depletion of fossil energy drives the demand for the revolutionary changes in the automobile industry. Much effort has been put into developing a new high-

efficient, environment-friendly, and safe transportation vehicle that can replace the conventional ones. The utilization of plug-in electric vehicles (PEVs) as the most suitable solution has been promoted in many countries. China is expected to have 5 million electric vehicles (EVs) by 2020 according to its Development Plan for Energy-saving and Renewable Energy Vehicles. However, a prediction by State Grid Corporation of China (SGCC) illustrates that the number will be 5–10 million due to the fast development of EVs at present in China. As EV-related technologies have been making progress and many national and local incentives have been created for EV purchases, the total number of EVs is likely to be 30 million by

Stability as Affected by

tie-line can be suppressed more quickly.

1. Introduction

2030 [1–3].

33

[26] Zamora-Cárdenas A, Fuerte-Esquivel C. Multi-parameter trajectory sensitivity approach for location of series-connected controllers to enhance power system transient stability. Electric Power Systems Research. 2010; 80:1096-1103. DOI: 10.1016/j. epsr.2010.02.002

[27] Chatterjee D, Ghosh A. TCSC control design for transient stability improvement of a multi-machine power system using trajectory sensitivity. Electric Power Systems Research. 2007; 77:470-483. DOI: 10.1016/j. epsr.2010.02.002

[28] Alexander J. Oscillatory solutions of a model system of nonlinear swing equations. International Journal of Electrical Power and Energy Systems. 1986;8:130-136. DOI: 10.1016/ 0142-0615(86)90027-X

[29] Guo T, Schlueter R. Identification of generic bifurcation and stability problems in power system differentialalgebraic model. IEEE Transactions on Power Systems. 1994;9:1032-1044. DOI: 10.1109/59.317640

[30] Zamora-Cárdenas E, Fuerte-Esquivel C. Computation of multiparameter sensitivities of equilibrium points in electric power systems. Electric Power Systems Research. 2013; 96:246-254. DOI: 10.1016/j. epsr.2012.11.013

## Chapter 3

[17] Frank P. Introduction to System Sensitivity Theory. 1st ed. New York: [25] Suguna R, Jalaja S, Pradeep M, Senthil R, SrikrishnaKumar S, Sugavanam K. Transient stability improvement using shunt and series compensators. Indian Journal of Science and Technology. 2016;9:1-11. DOI: 10.17485/ijst/2016/v9i11/89402

[26] Zamora-Cárdenas A, Fuerte-Esquivel C. Multi-parameter trajectory sensitivity approach for location of series-connected controllers to enhance power system transient stability. Electric Power Systems Research. 2010;

80:1096-1103. DOI: 10.1016/j.

77:470-483. DOI: 10.1016/j.

1986;8:130-136. DOI: 10.1016/ 0142-0615(86)90027-X

generic bifurcation and stability problems in power system differentialalgebraic model. IEEE Transactions on Power Systems. 1994;9:1032-1044. DOI:

[30] Zamora-Cárdenas E, Fuerte-Esquivel C. Computation of multiparameter sensitivities of equilibrium points in electric power systems. Electric Power Systems Research. 2013;

96:246-254. DOI: 10.1016/j.

[27] Chatterjee D, Ghosh A. TCSC control design for transient stability improvement of a multi-machine power system using trajectory sensitivity. Electric Power Systems Research. 2007;

[28] Alexander J. Oscillatory solutions of a model system of nonlinear swing equations. International Journal of Electrical Power and Energy Systems.

[29] Guo T, Schlueter R. Identification of

epsr.2010.02.002

epsr.2010.02.002

10.1109/59.317640

epsr.2012.11.013

Academic Press; 1978. p. 286

Power System Stability

North-Holland; 1972. p. 266

59.709082

[18] Tomovic R, Vucobratovic M. General Sensitivity Theory. New York:

[19] Laufenberg M, Pai M. A new approach to dynamic security

[20] Nguyen T, Pai M. Trajectory sensitivity analysis for dynamic security assessment and other applications in power systems. In: Savulescu S, editor. Real-Time Stability in Power Systems. 2nd ed. Switzerland: Springer; 2014. DOI: 10.1007/978-3-319-06680-6\_11

[21] Hiskens I, Pai M. Trajectory sensitivity analysis of hybrid systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2000;47:204-220. DOI:

[22] Hong Z, Shrirang A, Emil C, Mihai A. Discrete adjoint sensitivity analysis of

hybrid dynamical systems with switching [Discrete adjoint sensitivity analysis of hybrid dynamical systems]. IEEE Transactions on Circuits and Systems I: Regular Papers. 2017. DOI:

10.1109/TCSI.2017.2651683

[23] Nguyen T, Pai M, Hiskens I. Sensitivity approaches for direct computation of critical parameters in a power system. International Journal of Electrical Power and Energy Systems. 2002;24:337-343. DOI: 10.1016/ S0142-0615(01)00050-3

[24] Chatterjee D, Ghosh A. Transient stability assessment of power system

compensators. IEEE Transactions on Power Systems. 2007;22:1210-1220. DOI: 10.1109/TPWRS.2007.901455

containing series and shunt

32

10.1109/81.828574

assessment using trajectory sensitivities. IEEE Transactions on Power Systems. 1998;13:953-958. DOI: 10.1109/

## Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

Cai Hui

## Abstract

Large-scale smart charging stations can effectively satisfy and control the charging demands of tremendous plug-in electric vehicles (PEVs). But, simultaneously, their penetrations inevitably induce new challenges to the operation of power systems. In this chapter, damping torque analysis (DTA) was employed to examine the effects of the integration of smart charging station on the dynamic stability of the transmission system. A single-machine infinite-bus power system with a smart charging station that denoted the equivalent of several ones was used for analysis. The results obtained from DTA reveal that in view of the damping ratio, the optimal charging capacity is better to be considered in the design of the smart charging station. Under the proposed charging capacity, the power system can achieve the best maintained dynamic stability, and the damping ratio can reach the crest value. Phase compensation method was utilized to design the stabilizer via the active and reactive power regulators of the smart charging station respectively. With the help of the stabilizers, damping of the system oscillation under certain operating conditions can be significantly improved, and the power oscillation in the tie-line can be suppressed more quickly.

Keywords: smart charging station, plug-in electric vehicles (PEVs), power system oscillations, small-signal stability, damping torque analysis (DTA), stabilizer design

## 1. Introduction

The growing concern of carbon dioxide emission, greenhouse effect, and rapid depletion of fossil energy drives the demand for the revolutionary changes in the automobile industry. Much effort has been put into developing a new highefficient, environment-friendly, and safe transportation vehicle that can replace the conventional ones. The utilization of plug-in electric vehicles (PEVs) as the most suitable solution has been promoted in many countries. China is expected to have 5 million electric vehicles (EVs) by 2020 according to its Development Plan for Energy-saving and Renewable Energy Vehicles. However, a prediction by State Grid Corporation of China (SGCC) illustrates that the number will be 5–10 million due to the fast development of EVs at present in China. As EV-related technologies have been making progress and many national and local incentives have been created for EV purchases, the total number of EVs is likely to be 30 million by 2030 [1–3].

With the significant increase of PEVs, the corresponding parking lots termed as smart charging stations in [4–8] will be established to charge the tremendous PEVs. A typical city will contain several SmartParks, as the aggregator of numerous EV charging stations, distributed throughout the city one to few miles apart in the distribution system. The newly established smart charging stations are preferred to be connected to an additional bus of the transmission system [3–5]. The vehicle-to-grid (V2G) technology supplies the bidirectional communication between the parked vehicles and connected grid. The vehicles parked in the smart charging station can not only simply absorb active power (AP) from the grid for charging, but also participate in power regulation during discharging mode. Ref. [7] shows that most personal vehicles in the U.S. were parked more than 95% of the day and generally followed a daily schedule. The huge number of parked PEVs has the assignable potential to impact the frequency stability, voltage stability, and rotor-angle stability of the system [9–16]. Nowadays, the capacity of smart charging stations is much lower than that of the conventional power plants. Only frequency stability and voltage stability of the distribution systems attract much attention in recent years. However, if only half of the 230 million gasoline-powered cars, sport utility vehicles, and light trucks in U.S. are converted to or replaced with the electric vehicles, they would have 20 times the power capacity of all electricity in the country [17]. The impacts of SmartParks on the stability of the transmission systems cannot be ignored any more. Power system oscillations, as one section of the dynamic stability, occur inherently due to the rotor inertia of synchronous generators such that it takes time for them to respond to the sudden lack or excess of active power in a power system. The increased amplitude or weakly damped power oscillations via the tie-lines will lead to wrong activation of the automatic protection devices, splitting of the system, or even collapse. While the timevarying nature of the load-flow condition in the power system with highpenetrated smart charging stations is the common reason for the appearance of power oscillations, damping torque of the system is extremely interesting.

the optimal charging capacity is better to be considered during the design of the smart charging station. In Section 4, the phase compensation method is utilized to design the stabilizer via the active power and reactive power (RP) regulators of the smart charging station, respectively. A single-machine infinite-bus power system integrated with a smart charging station is presented as an example in Section 5. Results of numerical computation, non-linear simulations, and eigenvalue calculations at different system operating conditions are given. These simulations and results demonstrate and confirm the presented theoretical analysis, and verify the effectiveness of the designed stabilizer. Another four-machine power system is employed to show that the conclusions obtained in the singlemachine power system are also available in the multi-machine power system.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

2. A linearized model of single-machine infinite-bus system with a

The local electricity generation systems, besides supporting the regional loads, can be used to charge a smart charging station under usual conditions. Comparing with 1000-million-kilowatt capacity of outer power systems, the abundant active power generation with 10,000-kilowatt capacity in a regional system can be simplified as an equivalent synchronous machine. The inertia of the equivalent synchronous machine denotes the dynamic stability of the regional system. Because every individual smart charging station has the same dynamic behavior during the transient procedure, the smart charging stations can be regarded as an equivalent one with higher active power and reactive power capacities. A smart charging station usually is connected to the transmission system through a step-up transformer, which is seen as a reactance in this

In this chapter, the research focuses on the power oscillation which lasts for mostly 10–20 s. Uncertainties during EV charging such as the alternations of charging strategies or vehicle numbers have little effect on this analysis. EV charging demand or discharging supply during this dynamic procedure is considered as determinate power from the start time and following seconds. For a simple analysis, a constant power charging/discharging strategy is utilized to estimate the optimal

Figure 1 shows the configuration of a single-machine infinite-bus power system, where a smart charging station is connected at a busbar denoted by subscript s. The linearized models of network equations and synchronous machine are presented in

EV charging numbers for smart charging station design.

Conclusions are summarized in Section 6.

DOI: http://dx.doi.org/10.5772/intechopen.80721

smart charging station

chapter.

[18, 22, 23].

Figure 1.

35

Simplified model of the power system in a city.

Under different operating conditions, smart charging stations can vary from the adjustable load in the charging mode to the regulable generator in the discharging mode and vice versa with the voltage control strategy [4–6]. It is significantly valuable to examine how and why it may interact with the conventional power generation, hence affecting power system small-signal stability. In order to gain a good understanding on and clear insight into this interaction through theoretical analysis, a single-machine infinite-bus power system is adopted in this chapter. A smart charging station is connected to the system and theoretical damping torque analysis is carried out to check how and why the smart charging station interacts with the single generator so as to affect the power oscillation. It is expected that the analytical conclusions obtained in the chapter can be used to guide further work on a more complicated case of the oscillations in multi-machine power systems.

The organization of this chapter is as follows: in Section 2, a comprehensive model of a single-machine infinite-bus power system integrated with a smart charging station is established. After that, damping torque analysis (DTA) [18–21] is employed to examine the effect of joint operation of a smart charging station and a conventional synchronous machine on the system's small-signal stability in Section 3. The result of the damping torque analysis indicates that the smart charging station does not contribute an extra mode of electromechanical oscillation. It affects power system small-signal stability by supplying either positive or negative damping torque to the conventional power plant varying with the system operating conditions. Analysis also reveals that from the view of damping ratio,

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

the optimal charging capacity is better to be considered during the design of the smart charging station. In Section 4, the phase compensation method is utilized to design the stabilizer via the active power and reactive power (RP) regulators of the smart charging station, respectively. A single-machine infinite-bus power system integrated with a smart charging station is presented as an example in Section 5. Results of numerical computation, non-linear simulations, and eigenvalue calculations at different system operating conditions are given. These simulations and results demonstrate and confirm the presented theoretical analysis, and verify the effectiveness of the designed stabilizer. Another four-machine power system is employed to show that the conclusions obtained in the singlemachine power system are also available in the multi-machine power system. Conclusions are summarized in Section 6.

## 2. A linearized model of single-machine infinite-bus system with a smart charging station

The local electricity generation systems, besides supporting the regional loads, can be used to charge a smart charging station under usual conditions. Comparing with 1000-million-kilowatt capacity of outer power systems, the abundant active power generation with 10,000-kilowatt capacity in a regional system can be simplified as an equivalent synchronous machine. The inertia of the equivalent synchronous machine denotes the dynamic stability of the regional system. Because every individual smart charging station has the same dynamic behavior during the transient procedure, the smart charging stations can be regarded as an equivalent one with higher active power and reactive power capacities. A smart charging station usually is connected to the transmission system through a step-up transformer, which is seen as a reactance in this chapter.

In this chapter, the research focuses on the power oscillation which lasts for mostly 10–20 s. Uncertainties during EV charging such as the alternations of charging strategies or vehicle numbers have little effect on this analysis. EV charging demand or discharging supply during this dynamic procedure is considered as determinate power from the start time and following seconds. For a simple analysis, a constant power charging/discharging strategy is utilized to estimate the optimal EV charging numbers for smart charging station design.

Figure 1 shows the configuration of a single-machine infinite-bus power system, where a smart charging station is connected at a busbar denoted by subscript s. The linearized models of network equations and synchronous machine are presented in [18, 22, 23].

Figure 1. Simplified model of the power system in a city.

With the significant increase of PEVs, the corresponding parking lots termed as smart charging stations in [4–8] will be established to charge the tremendous PEVs. A typical city will contain several SmartParks, as the aggregator of numerous EV charging stations, distributed throughout the city one to few miles apart in the distribution system. The newly established smart charging stations are preferred to be connected to an additional bus of the transmission system [3–5]. The vehicle-to-grid (V2G) technology supplies the bidirectional communication between the parked vehicles and connected grid. The vehicles parked in the smart charging station can not only simply absorb active power (AP) from the grid for charging, but also participate in power regulation during discharging mode. Ref. [7] shows that most personal vehicles in the U.S. were parked more than 95% of the day and generally followed a daily schedule. The huge number of parked PEVs has the assignable potential to impact the frequency stability, voltage stability, and rotor-angle stability of the system [9–16]. Nowadays, the capacity of smart charging stations is much lower than that of the conventional power plants. Only frequency stability and voltage stability of the distribution systems attract much attention in recent years. However, if only half of the 230 million gasoline-powered cars, sport utility vehicles, and light trucks in U.S. are converted to or replaced with the electric vehicles, they would have 20 times the power capacity of all electricity in the country [17]. The impacts of SmartParks on the stability of the transmission systems cannot be ignored any more. Power system oscillations, as one section of the dynamic stability, occur inherently due to the rotor inertia of synchronous generators such that it takes time for them to respond to the sudden lack or excess of active power in a power system. The increased amplitude or weakly damped power oscillations via the tie-lines will lead to wrong activation of the automatic protection devices, splitting of the system, or even collapse. While the timevarying nature of the load-flow condition in the power system with highpenetrated smart charging stations is the common reason for the appearance of power oscillations, damping torque of the system is extremely interesting. Under different operating conditions, smart charging stations can vary from

the adjustable load in the charging mode to the regulable generator in the discharging mode and vice versa with the voltage control strategy [4–6]. It is significantly valuable to examine how and why it may interact with the conventional power generation, hence affecting power system small-signal stability. In order to gain a good understanding on and clear insight into this interaction through theoretical analysis, a single-machine infinite-bus power system is adopted in this chapter. A smart charging station is connected to the system and theoretical damping torque analysis is carried out to check how and why the smart charging station interacts with the single generator so as to affect the power oscillation. It is expected that the analytical conclusions obtained in the chapter can be used to guide further work on a more complicated case of the oscillations in

The organization of this chapter is as follows: in Section 2, a comprehensive model of a single-machine infinite-bus power system integrated with a smart charging station is established. After that, damping torque analysis (DTA) [18–21] is employed to examine the effect of joint operation of a smart charging station and a conventional synchronous machine on the system's small-signal stability in Section 3. The result of the damping torque analysis indicates that the smart charging station does not contribute an extra mode of electromechanical oscillation. It affects power system small-signal stability by supplying either positive or negative damping torque to the conventional power plant varying with the system operating conditions. Analysis also reveals that from the view of damping ratio,

multi-machine power systems.

Power System Stability

<sup>Δ</sup>Ud<sup>c</sup> <sup>¼</sup> <sup>K</sup>sp‐UdcE′

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

8 < :

DOI: http://dx.doi.org/10.5772/intechopen.80721

expressions for coefficients are not specifically listed.

control strategy can be obtained as Figure 4. From Figures 3 and 4, we have:

<sup>Δ</sup>Tet‐ex <sup>¼</sup> <sup>Δ</sup>Tst‐ex <sup>þ</sup> <sup>j</sup>ΔTdt‐ex <sup>¼</sup> <sup>K</sup>PE′

� <sup>K</sup><sup>E</sup>qUqcKsp‐Uqc<sup>δ</sup> <sup>þ</sup> <sup>K</sup><sup>E</sup>q<sup>δ</sup> � <sup>K</sup><sup>a</sup>

model

T′

Figure 3.

37

<sup>d</sup>0s þ K<sup>E</sup>qE′

q � � <sup>þ</sup> <sup>K</sup><sup>E</sup>qUqcKsp‐UqcE′

<sup>Δ</sup>Uq<sup>c</sup> <sup>¼</sup> <sup>K</sup>sp‐UqcE′

q ΔE′

q ΔE′

3. Analysis of damping torque contribution from the Phillips-Heffron

The Phillips-Heffron model of a smart charging station, which is based on the linearization of the system and describes the relationships between all variables, assessed to the single-machine infinite-busbar (SMIB) power system can be obtained as Figure 3, where the Phillips-Heffron model of SMIB only is referred to Refs. [22–24]. From Eqs. (2) and (3), the model of the smart charging station and its

<sup>Δ</sup>Tet‐sp <sup>¼</sup> <sup>Δ</sup>Tst‐sp <sup>þ</sup> <sup>j</sup>ΔTdt‐sp <sup>¼</sup> <sup>K</sup>PUdcΔUd<sup>c</sup> <sup>þ</sup> <sup>K</sup>PUqcΔUq<sup>c</sup>

h i � �

<sup>1</sup>þsT<sup>a</sup> <sup>K</sup>UtU<sup>d</sup>cKsp‐UdcE′

<sup>¼</sup> <sup>K</sup>PUdcKsp‐Udc<sup>δ</sup> <sup>þ</sup> <sup>K</sup>PUqcKsp‐Uqc<sup>δ</sup>

q

<sup>q</sup> � <sup>K</sup><sup>a</sup>

Phillips-Heffron model of the single-machine infinite-busbar system with a smart charging station.

While this chapter focuses on the analysis of the impact from the grid-connected smart charging station to the system small-signal stability, linearized processes and

<sup>q</sup> <sup>þ</sup> <sup>K</sup>sp‐Udc<sup>δ</sup>Δ<sup>δ</sup>

(4)

<sup>q</sup> <sup>þ</sup> <sup>K</sup>sp‐Uqc<sup>δ</sup>Δ<sup>δ</sup>

� �Δ<sup>δ</sup> (5)

<sup>1</sup>þsT<sup>a</sup> <sup>K</sup>UtU<sup>d</sup>cKsp‐Udc<sup>δ</sup> <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐Uqc<sup>δ</sup> <sup>þ</sup> <sup>K</sup>Ut<sup>δ</sup>

<sup>q</sup> <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐UqcE′

� �Δ<sup>δ</sup>

<sup>q</sup> þ KUtE′ q

(6)

Figure 2.

Control strategy of a smart charging station.

Here, U<sup>s</sup> is the voltage at the high-voltage-level busbar where the smart charging station locates; U<sup>b</sup> and U<sup>c</sup> are the voltages at infinite busbar and the low-voltagelevel busbar connected with the smart charging station; Its, Is, and Isb are the line currents as indicated in Figure 1; and Xts, Xsb, and X<sup>s</sup> are line reactances as indicated in Figure 1.

The control strategy of smart charging stations is shown in Figure 2.

The objective control is to command the currents corresponding to the fast change in demanded active and reactive power. The equations of smart charging stations according to Figure 2 are obtained as:

$$\begin{cases} \begin{cases} P\_1 = (K\_{\rm PP} + K\_{\rm IP}/s)(P\_{\rm PE}\text{ref} - P\_{\rm PE}) \\ I\_{q\rm ref} = \frac{P\_1 + P\_{10}}{U\_s} \\\\ U\_{q\rm c1} = \left(K\_{\rm PL\_q} + K\_{\rm II\_q}/s\right)(I\_{q\rm ref} - I\_{q\rm s}) \\\ U\_{q\rm c} = U\_{q\rm c0} + U\_{q\rm c1} \\\end{cases} \\\begin{cases} Q\_1 = (K\_{\rm PQ} + K\_{\rm IQ}/s)(Q\_{\rm PE\rm ref} - Q\_{\rm PE}) \\\ I\_{d\rm ref} = \frac{Q\_1 + Q\_{10}}{U\_s} \\\ U\_{d\rm c1} = \left(K\_{\rm PL\_d} + K\_{\rm II\_d}/s\right)(I\_{d\rm ref} - I\_{d\rm s}) \\\ U\_{d\rm c} = U\_{d\rm c0} + U\_{d\rm c1} \end{cases} \end{cases} \tag{1}$$

where

PPEV and QPEV are the demanded active and reactive power by the smart charging station and

<sup>K</sup>PP <sup>þ</sup> <sup>K</sup>IP <sup>s</sup> , KPQ <sup>þ</sup> <sup>K</sup>IQ <sup>s</sup> , KPI<sup>d</sup> þ KII<sup>d</sup> <sup>s</sup> , KPI<sup>q</sup> þ KII<sup>q</sup> <sup>s</sup> are the proportional-integral controllers in the smart charging station.

Linearized from Eq. (1),

$$\begin{cases} \Delta U\_{d\mathbf{c}} = K\_{\mathrm{U}\_{dc}\mathrm{U}\_{dc}}\Delta U\_{d\mathbf{c}} + K\_{\mathrm{U}\_{dc}\mathrm{U}\_{qc}}\Delta U\_{q\mathbf{c}} + K\_{\mathrm{U}\_{dc}\mathrm{E}\_{q}^{'}}\Delta \mathbf{E}\_{q}^{'} + K\_{\mathrm{U}\_{dc}\delta}\Delta \delta\\ \Delta U\_{q\mathbf{c}} = K\_{\mathrm{U}\_{qc}\mathrm{U}\_{dc}}\Delta U\_{d\mathbf{c}} + K\_{\mathrm{U}\_{qc}\mathrm{U}\_{qc}}\Delta U\_{q\mathbf{c}} + K\_{\mathrm{U}\_{qc}\mathrm{E}\_{q}^{'}}\Delta \mathbf{E}\_{q}^{'} + K\_{\mathrm{U}\_{qc}\delta}\Delta \delta \end{cases} \tag{2}$$

where

$$\begin{cases} \Delta P\_{\rm PE} = K\_{\rm PPU\_{dc}} \Delta U\_{dc} + K\_{\rm PPU\_{qc}} \Delta U\_{qc} + K\_{\rm PPE\_q'} \Delta E\_q' + K\_{\rm PPS} \Delta \delta \\\\ \Delta Q\_{\rm PE} = K\_{\rm QPU\_{dc}} \Delta U\_{dc} + K\_{\rm QPU\_{qc}} \Delta U\_{qc} + K\_{\rm QPE\_q'} \Delta E\_q' + K\_{\rm QP\delta} \Delta \delta \\\\ \Delta U\_s = K\_{\rm U\_\*U\_{dc}} \Delta U\_{dc} + K\_{\rm U\_\*U\_{qc}} \Delta U\_{qc} + K\_{\rm U\_\*E\_q'} \Delta E\_q' + K\_{\rm U\_\*\delta} \Delta \delta \end{cases} \tag{3}$$

Arranging from Eq. (2),

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

$$\begin{cases} \Delta U\_{d\mathbf{c}} = K\_{\mathrm{sp\cdot U}\_{d\mathbf{c}}\mathbf{E}\_{q}^{'}} \Delta \mathbf{E}\_{q}^{'} + K\_{\mathrm{sp\cdot U}\_{d\mathbf{c}}\delta \Delta \delta} \\ \Delta U\_{q\mathbf{c}} = K\_{\mathrm{sp\cdot U}\_{q\mathbf{c}}\mathbf{E}\_{q}^{'}} \Delta \mathbf{E}\_{q}^{'} + K\_{\mathrm{sp\cdot U}\_{q\mathbf{c}}\delta \Delta \delta} \end{cases} \tag{4}$$

While this chapter focuses on the analysis of the impact from the grid-connected smart charging station to the system small-signal stability, linearized processes and expressions for coefficients are not specifically listed.

## 3. Analysis of damping torque contribution from the Phillips-Heffron model

The Phillips-Heffron model of a smart charging station, which is based on the linearization of the system and describes the relationships between all variables, assessed to the single-machine infinite-busbar (SMIB) power system can be obtained as Figure 3, where the Phillips-Heffron model of SMIB only is referred to Refs. [22–24]. From Eqs. (2) and (3), the model of the smart charging station and its control strategy can be obtained as Figure 4.

From Figures 3 and 4, we have:

Here, U<sup>s</sup> is the voltage at the high-voltage-level busbar where the smart charging station locates; U<sup>b</sup> and U<sup>c</sup> are the voltages at infinite busbar and the low-voltagelevel busbar connected with the smart charging station; Its, Is, and Isb are the line currents as indicated in Figure 1; and Xts, Xsb, and X<sup>s</sup> are line reactances as indicated

P<sup>1</sup> ¼ ð Þ KPP þ KIP=s ð Þ PPEVref � PPEV

<sup>Q</sup><sup>1</sup> <sup>¼</sup> <sup>K</sup>PQ <sup>þ</sup> <sup>K</sup>IQ <sup>=</sup><sup>s</sup> � �ð Þ <sup>Q</sup>PEVref � <sup>Q</sup>PEV

Udc1 <sup>¼</sup> <sup>K</sup>PI<sup>d</sup> <sup>þ</sup> <sup>K</sup>II<sup>d</sup> <sup>=</sup><sup>s</sup> � �ð Þ Idsref � Id<sup>s</sup>

PPEV and QPEV are the demanded active and reactive power by the smart charg-

KII<sup>q</sup>

Iqsref � Iq<sup>s</sup> � �

<sup>s</sup> are the proportional-integral control-

<sup>q</sup> þ K<sup>U</sup>dcδΔδ

<sup>q</sup> þ K<sup>U</sup>qcδΔδ

<sup>q</sup> þ KPPδΔδ

<sup>q</sup> þ KUsδΔδ

<sup>q</sup> þ KQPδΔδ

q ΔE′

q ΔE′

q ΔE′

q ΔE′

q ΔE′ (1)

(2)

(3)

� �

The control strategy of smart charging stations is shown in Figure 2. The objective control is to command the currents corresponding to the fast change in demanded active and reactive power. The equations of smart charging

> Iqsref <sup>¼</sup> <sup>P</sup><sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>10</sup> Us

Uqc1 ¼ KPI<sup>q</sup> þ KII<sup>q</sup> =s

Uq<sup>c</sup> ¼ Uqc0 þ Uqc1

Idsref <sup>¼</sup> <sup>Q</sup><sup>1</sup> <sup>þ</sup> <sup>Q</sup><sup>10</sup> Us

Ud<sup>c</sup> ¼ Udc0 þ Udc1

KII<sup>d</sup>

<sup>s</sup> , KPI<sup>q</sup> þ

ΔUd<sup>c</sup> ¼ K<sup>U</sup>dcU<sup>d</sup>cΔUd<sup>c</sup> þ K<sup>U</sup>dcU<sup>q</sup>cΔUq<sup>c</sup> þ K<sup>U</sup>dcE′

ΔUq<sup>c</sup> ¼ K<sup>U</sup>qcU<sup>d</sup>cΔUd<sup>c</sup> þ K<sup>U</sup>qcU<sup>q</sup>cΔUq<sup>c</sup> þ K<sup>U</sup>qcE′

ΔPPEV ¼ KPPUdcΔUd<sup>c</sup> þ KPPUqcΔUqc þ KPPE′

ΔQPEV ¼ KQPUdcΔUd<sup>c</sup> þ KQPUqcΔUq<sup>c</sup> þ KQPE′

ΔU<sup>s</sup> ¼ KUsU<sup>d</sup>cΔUd<sup>c</sup> þ KUsU<sup>q</sup>cΔUq<sup>c</sup> þ KUsE′

stations according to Figure 2 are obtained as:

8 >>>>>>><

8

Control strategy of a smart charging station.

>>>>>>>>>>>>>>>>>>>>><

>>>>>>>:

8 >>>>>>><

>>>>>>>:

<sup>s</sup> , KPI<sup>d</sup> þ

>>>>>>>>>>>>>>>>>>>>>:

<sup>s</sup> , KPQ <sup>þ</sup> <sup>K</sup>IQ

lers in the smart charging station. Linearized from Eq. (1),

> 8 < :

8 >>><

>>>:

Arranging from Eq. (2),

in Figure 1.

Figure 2.

Power System Stability

where

ing station and <sup>K</sup>PP <sup>þ</sup> <sup>K</sup>IP

where

$$\begin{split} \Delta T\_{\text{et-sp}} &= \Delta T\_{\text{st-sp}} + j\Delta T\_{\text{dt-sp}} = K\_{\text{PU}\_{\text{dc}}} \Delta U\_{\text{dc}} + K\_{\text{PU}\_{\text{qc}}} \Delta U\_{\text{qc}} \\ &= \left( K\_{\text{PU}\_{\text{dc}}} K\_{\text{sp-U}\_{\text{dc}}\delta} + K\_{\text{PU}\_{\text{qc}}} K\_{\text{sp-U}\_{\text{qc}}\delta} \right) \Delta \delta \end{split} \tag{5}$$

$$
\Delta T\_{\rm et-ex} = \Delta T\_{\rm st-ex} + j\Delta T\_{\rm dt-ex} = K\_{\rm PE\_q}
$$

$$
\frac{-\left[K\_{\rm E\_q \rm U\_{qc}}K\_{\rm sp \cdot \rm U\_q \rm s} + K\_{\rm E\_q \rm s} - \frac{K\_{\rm s}}{1+\tau T\_{\rm s}}\left(K\_{\rm U \rm U\_{dc}}K\_{\rm sp \cdot \rm U\_{dc} \rm s} + K\_{\rm U \rm U\_{qr}}K\_{\rm sp \cdot \rm U\_{qr} \rm s} + K\_{\rm U \rm s}\right)\right]}{\left(T\_{d0}^{'}\rm s + K\_{\rm E\_q \rm E\_q'}\right) + K\_{\rm E\_q \rm U\_{qr}}K\_{\rm sp \cdot \rm U\_{qr} \rm E\_q'} - \frac{K\_{\rm s}}{1+\tau T\_{\rm s}}\left(K\_{\rm U \rm U\_{dc}}K\_{\rm sp \cdot \rm U\_{dc} \rm E\_q'} + K\_{\rm U\_l \rm U\_{qr}}K\_{\rm sp \cdot \rm U\_{qr} \rm E\_q'} + K\_{\rm U\_l \rm E\_q}\right)}\Delta\delta\tag{6}
$$

Figure 3. Phillips-Heffron model of the single-machine infinite-busbar system with a smart charging station.

$$
\Delta T\_{\rm et} = \Delta T\_{\rm st} + j\Delta T\_{\rm dt} = \Delta T\_{\rm et\-sp} + \Delta T\_{\rm et\-ex} = \left(\Delta T\_{\rm st\-sp} + \Delta T\_{\rm st\-ex}\right) + j\left(\Delta T\_{\rm dt\-sp} + \Delta T\_{\rm dt\-ex}\right) \tag{7}
$$

ΔTdt-ex, and ΔTdt are dependent on the output power of the synchronous machine

1. The proportional controls K<sup>p</sup> in the smart charging station mainly induce the synchronous torque into the oscillation loop, while for ΔTet-sp in Eq. (5), only its real part is related to Δδ; and the majority of damping torque is introduced by integral controls Ki/s, because 1/s induces the imaginary part in ΔTet-sp

2. Because the signals ΔUdc and ΔUqc through path a and path b in Figure 3 are significantly attenuated by lag loops before they form one part of the damping torque through the excitation system [24], the damping torque contribution from them can be neglected for simplified analysis. ΔTdt-sp represents the main damping torque supplied by the smart charging station in damping torque analysis, and ΔTdt-ex mainly expresses the torque supplied by the excitation

3. In this chapter, the '�'sign indicates the vehicles are selling power to the grid, that is, they are in discharging mode and the '+'sign indicates that they are buying power from the gird, denoting that the vehicles are in charging mode.

The optimal operation point when the system has the biggest damping torque

When the output power of the synchronous machine is fixed, the positive or negative damping torque supplied by the excitation system of the synchronous machine is only slightly changed. At the optimal operation point, the total damping torque ΔTdt of the system and ΔTdt-sp contributed from the smart charging station

When the active power in the tie-line is fixed, the output power of the synchronous machine is changed corresponding to the absorbed or injected power of the smart charging station. Both damping torques contributed by the smart charging station and the excitation system need to be considered. The total damping torque ΔTdt of the system and ΔTdt-sp contributed from the smart charging station reach

4. Design for the stabilizer attached to the smart charging station

Under the operation conditions that the total damping torques supplied by the smart charging station and the excitation system are not enough to suppress the oscillations, additional damping torques need to be added. Compared with the installation and coordinated parameter setting for power system stabilizers (PSSs) in synchronous machines, smart charging stations can be simply utilized to suppress the grid's active power oscillation with little infrastructure cost. Only a centralized stabilizer will be required at the smart charging station to maintain system stability. The stabilizer added via the active power (AP) or reactive power (RP) control

<sup>¼</sup> 0& <sup>∂</sup><sup>2</sup>

ΔTdt ∂P<sup>2</sup> PEV

≤0 whenPPEV>0 (8)

and the absorbed or injected power of the smart charging station.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

relating to Δω.

can be calculated as:

∂ΔTdt ∂PPEV

both reach their maximum values.

loop is shown in Figure 5.

39

system of the synchronous machine.

DOI: http://dx.doi.org/10.5772/intechopen.80721

<sup>¼</sup> <sup>∂</sup>ΔTdt‐sp <sup>þ</sup> <sup>∂</sup>ΔTdt‐ex ∂PPEV

their maximum values at different operation points.

From Eqs. (5)–(7), the conclusions can be summarized as follows:

Figures 3 and 4 clearly show the dynamic interaction between the smart charging station and the conventional synchronous generator. Figure 3 is very similar to the conventional Phillips-Heffron model based on which the DTA was proposed and developed. It shows that the smart charging station interacts closely with the generator by contributing the electric torque to the electromechanical oscillation loop of the generator. The contribution of electric torque is comprised of two parts, viz. ΔTet-sp which relates to Δδ and directly affects the oscillation loop, and ΔTet-ex which relates to <sup>Δ</sup>E′<sup>q</sup> and functions through the excitation system, as indicated in Figure 3. According to DTA, the electric torque can be decomposed into two components, viz. the synchronizing torque and the damping torque as shown in Eq. (7). The damping torque contributions ΔTdt-sp, ΔTdt-ex, and ΔTdt determine the influences on the damping of power system oscillation.

With certain output power of the synchronous machine and absorbed or injected power of the smart charging station, the bus voltages and currents of the corresponding operation condition can be determined. The damping torques ΔTdt-sp,

Figure 4. Linearized model of a smart charging station and its control.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

ΔTdt-ex, and ΔTdt are dependent on the output power of the synchronous machine and the absorbed or injected power of the smart charging station. From Eqs. (5)–(7), the conclusions can be summarized as follows:


The optimal operation point when the system has the biggest damping torque can be calculated as:

$$\frac{\partial \Delta T\_{\rm dt}}{\partial P\_{\rm PEV}} = \frac{\partial \Delta T\_{\rm dt \cdot \rm sp} + \partial \Delta T\_{\rm dt \cdot \rm ex}}{\partial P\_{\rm PEV}} = 0 \& \frac{\partial^2 \Delta T\_{\rm dt}}{\partial P\_{\rm PEV}^2} \le 0 \text{ when } P\_{\rm PEV} > 0 \tag{8}$$

When the output power of the synchronous machine is fixed, the positive or negative damping torque supplied by the excitation system of the synchronous machine is only slightly changed. At the optimal operation point, the total damping torque ΔTdt of the system and ΔTdt-sp contributed from the smart charging station both reach their maximum values.

When the active power in the tie-line is fixed, the output power of the synchronous machine is changed corresponding to the absorbed or injected power of the smart charging station. Both damping torques contributed by the smart charging station and the excitation system need to be considered. The total damping torque ΔTdt of the system and ΔTdt-sp contributed from the smart charging station reach their maximum values at different operation points.

### 4. Design for the stabilizer attached to the smart charging station

Under the operation conditions that the total damping torques supplied by the smart charging station and the excitation system are not enough to suppress the oscillations, additional damping torques need to be added. Compared with the installation and coordinated parameter setting for power system stabilizers (PSSs) in synchronous machines, smart charging stations can be simply utilized to suppress the grid's active power oscillation with little infrastructure cost. Only a centralized stabilizer will be required at the smart charging station to maintain system stability.

The stabilizer added via the active power (AP) or reactive power (RP) control loop is shown in Figure 5.

<sup>Δ</sup>Tet <sup>¼</sup> <sup>Δ</sup>Tst <sup>þ</sup> <sup>j</sup>ΔTdt <sup>¼</sup> <sup>Δ</sup>Tet‐sp <sup>þ</sup> <sup>Δ</sup>Tet‐ex <sup>¼</sup> <sup>Δ</sup>Tst‐sp <sup>þ</sup> <sup>Δ</sup>Tst‐ex

Power System Stability

influences on the damping of power system oscillation.

Figure 4.

38

Linearized model of a smart charging station and its control.

Figures 3 and 4 clearly show the dynamic interaction between the smart charging station and the conventional synchronous generator. Figure 3 is very similar to the conventional Phillips-Heffron model based on which the DTA was proposed and developed. It shows that the smart charging station interacts closely with the generator by contributing the electric torque to the electromechanical oscillation loop of the generator. The contribution of electric torque is comprised of two parts, viz. ΔTet-sp which relates to Δδ and directly affects the oscillation loop, and ΔTet-ex which relates to <sup>Δ</sup>E′<sup>q</sup> and functions through the excitation system, as indicated in Figure 3. According to DTA, the electric torque can be decomposed into two components, viz. the synchronizing torque and the damping torque as shown in Eq. (7). The damping torque contributions ΔTdt-sp, ΔTdt-ex, and ΔTdt determine the

With certain output power of the synchronous machine and absorbed or injected

corresponding operation condition can be determined. The damping torques ΔTdt-sp,

power of the smart charging station, the bus voltages and currents of the

<sup>þ</sup> <sup>j</sup> <sup>Δ</sup>Tdt‐sp <sup>þ</sup> <sup>Δ</sup>Tdt‐ex

(7)

The transfer function of the stabilizer is

DOI: http://dx.doi.org/10.5772/intechopen.80721

5. Case study

charging station.

Table 1.

41

fixing load-flow in the tie-line

station. From Table 1, it can be concluded that:

5.1 Case description

GPEV ¼ K<sup>w</sup>

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

1 þ sT<sup>2</sup> 1 þ sT<sup>1</sup>

The stabilizers are designed to compensate the lagging or leading angle of the forward path, in order to supply maximum positive damping into the system. The phase compensation method is used to design the parameters of the stabilizers.

Two example cases are employed in this section. From Case A to Case D, a single-machine infinite-busbar power system is used. The parameters of the system are given in Appendix A.1. Under different capacities of the smart charging station, computational results of the damping torque contribution from the smart charging station and the excitation system to the electromechanical oscillation loop of the single synchronous generator are obtained and confirmed by the eigenvalue of the system's oscillation mode. The critical point in which the system has the biggest damping torque is highlighted. In Case E, a four-machine power system is

presented. The parameters are given in Appendix A.2. The eigenvalue related to the inter-area oscillation mode is concerned under different capacities of the smart

5.2 Case A: utilizing only proportional control in the smart charging station and

With the load-flow in the tie-line fixed at 10 MW, the comparison is done when only proportional control is utilized in the smart charging station under its different charging or discharging power capacities. The computational results of the example system are shown in Table 1, when only P control is utilized in the smart charging

Pt/(10 MW) PPEV/(10 MW) ΔTdt/pu ΔTdt-sp/pu ΔTdt-ex/pu Frequency/Hz Damping

4.0 3.0 0.6340 0.0006 0.6334 1.83 3.19 3.5 2.5 0.7842 0.0004 0.7838 1.82 3.47 3.0 2.0 0.4368 0.0002 0.4366 1.74 2.99 2.5 1.5 �0.1877 �0.0003 �0.1874 1.68 1.92 2.0 1.0 �0.5730 �0.0005 �0.5725 1.61 1.24 1.5 0.5 �0.1870 �0.0002 �0.1868 1.68 1.92 1.0 0.0 0.4345 0.0003 0.4342 1.74 2.99 0.5 �0.5 0.7851 0.0005 0.7846 1.82 3.47 0.0 �1.0 0.6348 0.0007 0.6341 1.83 3.19

Computational results of the example system when only P control is utilized in the smart charging station.

1 þ sT<sup>4</sup> 1 þ sT<sup>3</sup>

(12)

ratio/%

Figure 5. Control strategy of the stabilizer added via AP and RP control loop, respectively.

The forward path function which describes the way from output signal of the stabilizer to the additional damping torque into the electromechanical oscillation loop can be obtained:

$$\begin{cases} F\_{\text{pas}}(\boldsymbol{\nu}) = \frac{\partial \Delta T\_{\text{pas}}}{\partial \Delta \mathbf{u}\_{\text{pas}}} = K\_{\text{FU}\_{\text{d}}} K\_{\text{sp},\text{U}\_{\text{d}} \text{u}\_{\text{pas}}} + K\_{\text{FU}\_{\text{p}}} K\_{\text{p},\text{U}\_{\text{p}} \text{u}\_{\text{pas}}} + \\ \quad \begin{aligned} & \quad - \left[ K\_{\text{E}\_{\text{l}} \text{u}\_{\text{r}}} K\_{\text{p} \text{u}\_{\text{r}} \text{u}\_{\text{pas}}} - \frac{K\_{\text{a}}}{\mathbf{1} + s \mathcal{T}\_{\text{2}}} \left( K\_{\text{U}\_{\text{l}} \text{U}\_{\text{d}} \text{u}\_{\text{p}} \text{U}\_{\text{d}} \text{u}\_{\text{pas}}} + K\_{\text{U}\_{\text{l}} \text{U}\_{\text{r}} \text{K}\_{\text{p}} \text{U}\_{\text{p}} \text{u}\_{\text{p}}} \right) \right] \\ & \quad \left( \overline{T}\_{\text{d}} \text{u} + K\_{\text{E}\_{\text{l}} \text{t}} \right) + K\_{\text{E}\_{\text{l}} \text{u}\_{\text{r}}} K\_{\text{sp} - \text{U}\_{\text{r}} \text{E}} - \frac{K\_{\text{a}}}{\mathbf{1} + s \mathcal{T}\_{\text{2}}} \left( K\_{\text{U}\_{\text{l}} \text{U}\_{\text{d}} \text{K}\_{\text{sp},\text{U}\_{\text{d}} \text{E}\_{\text{r}}} + K\_{\text{U}\_{\text{l}} \text{U}\_{\text{s}} \text{K}\_{\text{p}} \$$

where,

Fpssp or Fpssq is corresponding to the utilized output signal of the stabilizer upssp or upssq.

<sup>K</sup>sp‐Udcupssp , Ksp‐Uqcupssp , Ksp‐Udcupssq and <sup>K</sup>sp‐Uqcupssq are obtained from the linearization of the control strategy with the output signals of the stabilizer considered.

The active power P<sup>b</sup> in the tie-line is chosen for the feedback signals of the stabilizers via the active power regulator and the reactive power regulator.

From the linear system control theory, the active power P<sup>b</sup> can be written as a function of the rotor speed of the generator.

$$
\Delta P\_\mathrm{b} = r\_\mathrm{P}(\mathfrak{s}) \Delta w \tag{10}
$$

where rPðÞ¼ s KPb<sup>δ</sup> þ KPbE′ q K<sup>E</sup>′ qδ � � <sup>ω</sup><sup>0</sup> <sup>s</sup> is the reconstruction function for Pb. While ΔP<sup>b</sup> ¼ ΔP<sup>t</sup> � ΔPPEV, KPbδ, KPbE′ <sup>q</sup> and K<sup>E</sup>′ <sup>q</sup><sup>δ</sup> are related to the reconstruction of this feedback signal.

Considering Eqs. (9) and (10), the electric torques contributed by the stabilizer via active and reactive power regulators, respectively, are expressed as:

$$\begin{cases} \Delta \mathbf{T}(\Delta \mu\_{\text{pssp}}) = F\_{\text{pssp}}(s) r\_{\text{p}}(s) G\_{\text{PEVP}}(s) \Delta \alpha\\ \Delta \mathbf{T}(\Delta \mu\_{\text{pssq}}) = F\_{\text{pssq}}(s) r\_{\text{p}}(s) G\_{\text{PEVQ}}(s) \Delta \alpha \end{cases} \tag{11}$$

where GPEVP(s) and GPEVQ(s) are the transfer function of the stabilizer via active and reactive power regulators respectively.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

The transfer function of the stabilizer is

$$G\_{\rm PEV} = K\_{\rm w} \frac{\mathbf{1} + sT\_2}{\mathbf{1} + sT\_1} \frac{\mathbf{1} + sT\_4}{\mathbf{1} + sT\_3} \tag{12}$$

The stabilizers are designed to compensate the lagging or leading angle of the forward path, in order to supply maximum positive damping into the system. The phase compensation method is used to design the parameters of the stabilizers.

## 5. Case study

The forward path function which describes the way from output signal of the stabilizer to the additional damping torque into the electromechanical oscillation

� � � �

<sup>¼</sup> <sup>K</sup>PUdcKsp‐Udcupssp <sup>þ</sup> <sup>K</sup>PUqcKsp‐Uqcupssp<sup>þ</sup>

1 þ sT<sup>a</sup>

<sup>¼</sup> <sup>K</sup>PUdcKsp‐Udcupssq <sup>þ</sup> <sup>K</sup>PUqcKsp‐Uqcupssq <sup>þ</sup> <sup>K</sup>PE′

1 þ sT<sup>a</sup>

<sup>q</sup> � <sup>K</sup><sup>a</sup> 1 þ sT<sup>a</sup>

<sup>q</sup> � <sup>K</sup><sup>a</sup> 1 þ sT<sup>a</sup>

� � � �

Fpssp or Fpssq is corresponding to the utilized output signal of the stabilizer upssp

<sup>K</sup>sp‐Udcupssp , Ksp‐Uqcupssp , Ksp‐Udcupssq and <sup>K</sup>sp‐Uqcupssq are obtained from the linearization of the control strategy with the output signals of the stabilizer considered. The active power P<sup>b</sup> in the tie-line is chosen for the feedback signals of the stabilizers via the active power regulator and the reactive power regulator.

From the linear system control theory, the active power P<sup>b</sup> can be written as a

<sup>q</sup> and K<sup>E</sup>′

Considering Eqs. (9) and (10), the electric torques contributed by the stabilizer

� � <sup>¼</sup> <sup>F</sup>psspð Þ<sup>s</sup> <sup>r</sup>pð Þ<sup>s</sup> <sup>G</sup>PEVPð Þ<sup>s</sup> <sup>Δ</sup><sup>ω</sup>

� � <sup>¼</sup> <sup>F</sup>pssqð Þ<sup>s</sup> <sup>r</sup>pð Þ<sup>s</sup> <sup>G</sup>PEVQ ð Þ<sup>s</sup> <sup>Δ</sup><sup>ω</sup>

where GPEVP(s) and GPEVQ(s) are the transfer function of the stabilizer via active

<sup>K</sup>UtU<sup>d</sup>cKsp‐Udcupssp <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐Uqcupssp

<sup>q</sup> <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐UqcE′

� �

<sup>q</sup> <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐UqcE′

� �

ΔP<sup>b</sup> ¼ rPð Þs Δω (10)

<sup>s</sup> is the reconstruction function for Pb.

<sup>q</sup><sup>δ</sup> are related to the reconstruction

<sup>q</sup> þ KUtE′ q

(9)

(11)

<sup>q</sup> þ KUtE′ q

<sup>K</sup>UtU<sup>d</sup>cKsp‐UdcE′

q

<sup>K</sup>UtU<sup>d</sup>cKsp‐Udcupssq <sup>þ</sup> <sup>K</sup>UtU<sup>q</sup>cKsp‐Uqcupssq

<sup>K</sup>UtU<sup>d</sup>cKsp‐UdcE′

� <sup>K</sup><sup>E</sup>qUqcKsp‐Uqcupssp � <sup>K</sup><sup>a</sup>

� <sup>K</sup><sup>E</sup>qUqcKsp‐Uqcupssq � <sup>K</sup><sup>a</sup>

function of the rotor speed of the generator.

While ΔP<sup>b</sup> ¼ ΔP<sup>t</sup> � ΔPPEV, KPbδ, KPbE′

(

and reactive power regulators respectively.

q K<sup>E</sup>′ qδ

via active and reactive power regulators, respectively, are expressed as:

� � <sup>ω</sup><sup>0</sup>

ΔT Δupssp

ΔT Δupssq

where rPðÞ¼ s KPb<sup>δ</sup> þ KPbE′

of this feedback signal.

<sup>þ</sup> <sup>K</sup><sup>E</sup>qUqcKsp‐UqcE′

<sup>þ</sup> <sup>K</sup><sup>E</sup>qUqcKsp‐UqcE′

Control strategy of the stabilizer added via AP and RP control loop, respectively.

loop can be obtained:

Power System Stability

<sup>F</sup>psspðÞ¼ <sup>s</sup> <sup>∂</sup>ΔTpss

T′

<sup>d</sup>0s þ K<sup>E</sup>qE′ q

� �

<sup>F</sup>pssqðÞ¼ <sup>s</sup> <sup>∂</sup>ΔTpss

KPE′ q

8

Figure 5.

>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>:

T′

where,

or upssq.

40

∂Δupssp

<sup>d</sup>0s þ K<sup>E</sup>qE′ q

� �

∂Δupssq

#### 5.1 Case description

Two example cases are employed in this section. From Case A to Case D, a single-machine infinite-busbar power system is used. The parameters of the system are given in Appendix A.1. Under different capacities of the smart charging station, computational results of the damping torque contribution from the smart charging station and the excitation system to the electromechanical oscillation loop of the single synchronous generator are obtained and confirmed by the eigenvalue of the system's oscillation mode. The critical point in which the system has the biggest damping torque is highlighted. In Case E, a four-machine power system is presented. The parameters are given in Appendix A.2. The eigenvalue related to the inter-area oscillation mode is concerned under different capacities of the smart charging station.

### 5.2 Case A: utilizing only proportional control in the smart charging station and fixing load-flow in the tie-line

With the load-flow in the tie-line fixed at 10 MW, the comparison is done when only proportional control is utilized in the smart charging station under its different charging or discharging power capacities. The computational results of the example system are shown in Table 1, when only P control is utilized in the smart charging station. From Table 1, it can be concluded that:


Table 1.

Computational results of the example system when only P control is utilized in the smart charging station.

1. The total damping torque contribution ΔTdt is approximately equal to the damping torque from the excitation system ΔTdt-ex. The change of ΔTdt is mainly induced by ΔTdt-ex which is the impact from the excitation system of the synchronous machine under different output power. The smart charging station only with the proportional control functions as an adjustable load in charging mode or as a regulator generator in discharging mode.

torque and damping torque from the smart charging station coincided at the same point. It demonstrates conclusion (3) obtained in Section 3. Under this operation point, the smart charging station just consumes the electricity generated by the equivalent synchronous machine. There is no active power exchange in the tie-line. Beyond or below this point, the damping ratio of the system will decrease because of the increased load burden in the tie-line either from the synchronous machine to the infinite bus or vice versa. Considering each vehicle can draw 3.5 kW of active power [25] and always around 60% personal vehicles in the parking lots need to be charged [26], roughly 5000 personal vehicles are optimal to be accepted in this equivalent smart charging

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

DOI: http://dx.doi.org/10.5772/intechopen.80721

3. During the discharging process, the damping of the system tends to deteriorate with the increasing power injected from the smart charging station to grid.

5.4 Case C: utilizing PI control in the smart charging station and fixing load-

A comparison of the damping torques is made under different charging or discharging power capacities of the smart charging station with the fixed load-flow

1. The total damping torque contribution ΔTdt is simultaneously influenced by ΔTdt-sp which relates to Δδ and directly affects the oscillation loop, and ΔTdt-ex

excitation system of the synchronous machine respectively is complementary during the charging process. Compared with Table 1, the positive damping torque supplied by the smart charging station helps the system to improve the

ΔTdt/pu ΔTdt-sp/pu ΔTdt-ex/pu Frequency/

Hz

Damping ratio/%

which relates to <sup>Δ</sup>E′<sup>q</sup> and functions through the excitation system.

2. The damping torque supplied from the smart charging station and the

low damping capacity from 5 to 15 MW in charging mode of the smart charging station. This conclusion can be confirmed by the analysis from Eqs. (5) and (6). When the charging power of the smart charging station is between 0 and 30 MW, the product of ΔTdt-sp and ΔTdt-ex is negative.

4.0 3.0 0.3144 0.2878 0.6022 1.72 2.80 3.5 2.5 0.5527 0.2122 0.7649 1.72 3.24 3.0 2.0 0.3902 0.0082 0.3984 1.72 2.94 2.5 1.5 0.1770 0.3152 0.1382 1.73 2.53 2.0 1.0 0.0093 0.5294 0.5201 1.73 2.22 1.5 0.5 0.1693 0.3122 0.1429 1.73 2.52 1.0 0.0 0.3523 0.0582 0.4105 1.74 2.84 0.5 0.5 0.5062 0.2091 0.7153 1.73 3.14 0.0 1.0 0.2971 0.2908 0.5879 1.73 2.75

Computational results of the example system when load-flow in the tie-line is fixed at 10 MW.

station.

Pt/ (10 MW)

Table 3.

43

PPEV/ (10 MW)

flow in the tie-line

in the tie-line. The results are shown in Table 3. From Table 3, it can be concluded that:

2. Integral control in the smart charging station not only helps to reduce the steady-state error and accelerate the smart charging station to the steady operation point, but also supplies either positive or negative damping torque into the system. It demonstrates conclusion (1) obtained in Section 3.

## 5.3 Case B: utilizing PI control in the smart charging station and fixing output power of the synchronous machine

A comparison of the damping torques is made under different charging or discharging power capacities of the smart charging station with the fixed output power of the synchronous machine. The computational results of the example system are shown in Table 2, when active power supplied by the synchronous machine is fixed at 10 MW. From Table 2, it can be concluded that:



#### Table 2.

Computational results of the example system when active power supplied by the synchronous machine is fixed at 10 MW.

torque and damping torque from the smart charging station coincided at the same point. It demonstrates conclusion (3) obtained in Section 3. Under this operation point, the smart charging station just consumes the electricity generated by the equivalent synchronous machine. There is no active power exchange in the tie-line. Beyond or below this point, the damping ratio of the system will decrease because of the increased load burden in the tie-line either from the synchronous machine to the infinite bus or vice versa. Considering each vehicle can draw 3.5 kW of active power [25] and always around 60% personal vehicles in the parking lots need to be charged [26], roughly 5000 personal vehicles are optimal to be accepted in this equivalent smart charging station.

3. During the discharging process, the damping of the system tends to deteriorate with the increasing power injected from the smart charging station to grid.

## 5.4 Case C: utilizing PI control in the smart charging station and fixing loadflow in the tie-line

A comparison of the damping torques is made under different charging or discharging power capacities of the smart charging station with the fixed load-flow in the tie-line. The results are shown in Table 3.

From Table 3, it can be concluded that:



Table 3.

Computational results of the example system when load-flow in the tie-line is fixed at 10 MW.

1. The total damping torque contribution ΔTdt is approximately equal to the damping torque from the excitation system ΔTdt-ex. The change of ΔTdt is mainly induced by ΔTdt-ex which is the impact from the excitation system of the synchronous machine under different output power. The smart charging station only with the proportional control functions as an adjustable load in

2. Integral control in the smart charging station not only helps to reduce the steady-state error and accelerate the smart charging station to the steady operation point, but also supplies either positive or negative damping torque into the system. It demonstrates conclusion (1) obtained in Section 3.

5.3 Case B: utilizing PI control in the smart charging station and fixing output

A comparison of the damping torques is made under different charging or discharging power capacities of the smart charging station with the fixed output power of the synchronous machine. The computational results of the example system are shown in Table 2, when active power supplied by the synchronous

1. While the output of the synchronous machine is constant, the damping torque from the excitation system of the synchronous machine is nearly unchanged. The signals ΔUdc and ΔUqc through path a and path b only contribute slight changes to ΔTdt-ex. The variety of total damping torque contribution ΔTdt is mainly induced from ΔTdt-sp which comes from the smart charging station and directly affects the oscillation loop. It demonstrates conclusion (2) obtained in

2. The damping torque from the smart charging station changes at its different charging or discharging capacity, which is either positive or negative. The smart charging station can help to improve the damping with certain charging capacity which is between the lower and upper threshold. In charging mode, the smart charging station is preferred to operate around 10 MW which is nearly the same as 10.4 MW calculated by Eq. (8). The highest total damping

Pt/(10 MW) PPEV/(10 MW) ΔTdt/pu ΔTdt-sp/pu ΔTdt-ex/pu Frequency/Hz Damping

1.0 3.0 0.0633 0.3182 0.3815 1.69 2.38 1.0 2.5 0.1560 0.2452 0.4021 1.72 2.51 1.0 2.0 0.3742 0.0457 0.4199 1.74 2.88 1.0 1.5 0.8400 0.3798 0.4602 1.75 3.71 1.0 1.0 1.0677 0.5806 0.4871 1.76 4.10 1.0 0.5 0.8180 0.3699 0.4481 1.75 3.67 1.0 0.0 0.3523 0.0582 0.4105 1.74 2.84 1.0 0.5 0.1398 0.2593 0.3991 1.72 2.48 1.0 1.0 0.0227 0.3370 0.3597 1.70 2.29

Computational results of the example system when active power supplied by the synchronous machine is fixed

ratio/%

machine is fixed at 10 MW. From Table 2, it can be concluded that:

charging mode or as a regulator generator in discharging mode.

power of the synchronous machine

Section 3.

Power System Stability

Table 2.

42

at 10 MW.


## 5.5 Case D: stabilizer design

While the operation condition for the smart charging station varies stochastically, the stabilizer is designed and attached to the smart charging station to supply additional damping torques into the system.

The stabilizer via the active and reactive power loops is designed respectively under the condition that the equivalent synchronous machine supplies 20 MW and the smart charging station consumes 10 MW of active power. A three-phase shortcircuit fault happens in Bus s at 0.5 s and lasts for 0.1 s.

The forward path is:

$$\begin{cases} F\_{\text{pssp}}(s) = -6.875 \ 1 + \text{j6.829 } \ 3 \\ F\_{\text{pssq}}(s) = \textbf{1.7520} + \textbf{j7.284 } \ \textbf{1} \end{cases} \tag{13}$$

oscillation frequency fcritical and its corresponding attenuation factor αcritical are

Comparisons of tie-line power oscillation without and with the stabilizer via active and reactive power

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

Without the stabilizer 0.241 9 + j10.864 4 With the stabilizer via active power regulator 0.759 0 + j10.816 9 With the stabilizer via reactive power regulator 0.651 1 + j10.847 1

Eigenvalue of the oscillation mode without and with stabilizers under PPEV 10 MW, Pt 20 MW.

regulable generator in discharging mode in view of the damping ratio.

torque and integral control that introduces the damping torque into the grid.

With the load-flow in the tie-line L6-7 fixed at 70 MW, a comparison is done between the proportional controlled smart charging station and the adjustable load/ generator connected at Bus 7, respectively. The eigenvalue related to the inter-area oscillation mode is a concern. From Table 6, only the proportional controlled smart charging station functions as the adjustable load during charging period and as the

Proportional control of the smart charging station only supplies the synchronous

extracted. The eigenvalue of the critical oscillation mode is

Four-machine power system integrated with the smart charging station.

λcritical = αcritical + j2πfcritical.

Figure 6.

Table 5.

Figure 7.

45

regulators under PPEV = 10 MW, P<sup>t</sup> = 20 MW.

DOI: http://dx.doi.org/10.5772/intechopen.80721

The parameters of the designed stabilizer attached to active and reactive power regulators, respectively, are (Table 4).

With the designed stabilizer, the eigenvalue of the system with the smart charging station can be obtained as that, the effectiveness of the designed stabilizers is verified by the time-domain simulation in Figure 6 and eigenvalue calculation in Table 5. From Table 5 and Figure 6, it can be seen that the designed stabilizer attached to the smart charging station can not only help to reduce the power fluctuations for PEV charging, but also suppress the power oscillation in the tie-line.

## 5.6 Case E: analysis in the four-machine power system

The power system integrated with the smart charging station is shown in Figure 7. The Prony method is employed to analyze the time-domain simulation of the power flow in the tie-line L6-7 [27]. The critical inter-area electromechanical


Table 4.

Parameters of smart charging station-based stabilizers.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

#### Figure 6.

3. The impact of the damping torque from the synchronous machine also needs to be considered. With this impact, the highest total damping torque and damping torque from the smart charging station are obtained at different points. In this case, the optimized operation point reaches 25 MW which is nearly the same as 24.6 MW calculated by Eq. (8). It demonstrates conclusion (3) in Section 3. Under the operating conditions that the absorbed power of the smart charging station varies from 20 to 25 MW, although the smart charging station supplies the negative damping torque into the grid, the total damping torque is still positive and keeps increasing with the compensation of the damping torque from the excitation system. The smart charging station at the optimal operation point is also charged by the electricity generated by the

4.The damping of the system tends to deteriorate with the increasing power injected from the smart charging station to the grid during the discharging

While the operation condition for the smart charging station varies stochastically, the stabilizer is designed and attached to the smart charging station to supply

The stabilizer via the active and reactive power loops is designed respectively under the condition that the equivalent synchronous machine supplies 20 MW and the smart charging station consumes 10 MW of active power. A three-phase short-

> FpsspðÞ¼� s 6:875 1 þ j6:829 3 FpssqðÞ¼ s 1:7520 þ j7:284 1

The parameters of the designed stabilizer attached to active and reactive power

With the designed stabilizer, the eigenvalue of the system with the smart charging station can be obtained as that, the effectiveness of the designed stabilizers is verified by the time-domain simulation in Figure 6 and eigenvalue calculation in Table 5. From Table 5 and Figure 6, it can be seen that the designed stabilizer attached to the smart charging station can not only help to reduce the power fluctuations for PEV charging, but also suppress the power oscillation in the tie-line.

The power system integrated with the smart charging station is shown in Figure 7. The Prony method is employed to analyze the time-domain simulation of the power flow in the tie-line L6-7 [27]. The critical inter-area electromechanical

Stabilizer via active power regulator T<sup>1</sup> = T<sup>3</sup> = 0.019 5; T<sup>2</sup> = T<sup>4</sup> = 0.5; K<sup>W</sup> = 16.570 4 Stabilizer via reactive power regulator T<sup>1</sup> = T<sup>3</sup> = 0.081 0; T<sup>2</sup> = T<sup>4</sup> = 0.5; K<sup>W</sup> = 12.628 9

Stabilizer Parameters

(13)

equivalent local synchronous machine.

additional damping torques into the system.

regulators, respectively, are (Table 4).

Parameters of smart charging station-based stabilizers.

circuit fault happens in Bus s at 0.5 s and lasts for 0.1 s.

5.6 Case E: analysis in the four-machine power system

process.

Power System Stability

5.5 Case D: stabilizer design

The forward path is:

Table 4.

44

Comparisons of tie-line power oscillation without and with the stabilizer via active and reactive power regulators under PPEV = 10 MW, P<sup>t</sup> = 20 MW.


#### Table 5.

Eigenvalue of the oscillation mode without and with stabilizers under PPEV 10 MW, Pt 20 MW.

#### Figure 7.

Four-machine power system integrated with the smart charging station.

oscillation frequency fcritical and its corresponding attenuation factor αcritical are extracted. The eigenvalue of the critical oscillation mode is

λcritical = αcritical + j2πfcritical.

With the load-flow in the tie-line L6-7 fixed at 70 MW, a comparison is done between the proportional controlled smart charging station and the adjustable load/ generator connected at Bus 7, respectively. The eigenvalue related to the inter-area oscillation mode is a concern. From Table 6, only the proportional controlled smart charging station functions as the adjustable load during charging period and as the regulable generator in discharging mode in view of the damping ratio.

Proportional control of the smart charging station only supplies the synchronous torque and integral control that introduces the damping torque into the grid.


\* '+' denotes the adjustable load absorbing active power from the grid; '' denotes the regulable generator injecting active power into the grid.

#### Table 6.

Comparison of eigenvalue related to inter-area oscillation mode only utilizing the proportional control of the smart charging station.

smart charging station, it is difficult to calculate the optimal charging point in

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

DOI: http://dx.doi.org/10.5772/intechopen.80721

Comparison of damping ratio related to inter-area oscillation mode utilizing the P and PI control of the smart

The chapter investigates the impacts of a grid-connected smart charging station on power system's small-signal stability based on a simple single-machine infinitebus power system integrated with a smart charging station. Damping torque analysis (DTA) is employed to examine the contribution from the smart charging station to the electromechanical oscillation loop of the generator in theoretical analysis. The analysis has concluded that, the smart charging station affects power system's smallsignal stability in light of its interaction with the synchronous machine. The proportional controls in the smart charging station mainly induce the synchronous torque into the oscillation loop and the majority of damping torque is introduced by integral controls. The damping torque supplied by the smart charging station is mainly directly induced into the oscillation loop, and the damping torque from the excitation system is almost from the synchronous machine itself. The optimal operation condition of the smart charging station is the moment when the system has the highest damping ratio. In this chapter, such an optimal operation condition is defined, indicating that the optimal charging capacity is considered for smart

Results of the damping torque computation of a single-machine power system integrated with a smart charging station, confirmed by eigenvalue calculations of system oscillation mode, are presented in the chapter. The conclusions obtained from the theoretical analysis are demonstrated and verified by these results. Under the optimal operation condition, the total damping torque supplied from the smart charging station and synchronous machine reaches its maximum value. During the discharging process, the damping of the system tends to deteriorate with the increasing power injected from the smart charging station to the grid. Another fourmachine power system is employed to manifest that the conclusions obtained in the single-machine power system are also available in the multi-machine power system. The stabilizer is designed and attached to the active or reactive power regulator of the smart charging station to supply additional positive damping into the system. The phase compensation method is used here. The effectiveness is confirmed by the

theory (Figure 8).

charging station, respectively.

Figure 8.

6. Conclusion

charging station design.

47


#### Table 7.

Comparison of eigenvalue related to the inter-area oscillation mode utilizing PI control of the smart charging station.

Conclusion (1) obtained in the single-machine infinite-busbar power system is also available in the multi-machine power system.

Both proportional and integral controls are utilized in the smart charging station. A comparison is done under different charging or discharging power capacities of the smart charging station when the load-flow in the tie-line is fixed at 70 MW. The eigenvalue related to the inter-area oscillation mode is concerned. The results are shown in Table 7.

From Table 7, the optimal charging point with the highest damping ratio in the single-machine power system can be obtained by Eq. (8) and verified by the damping torque calculation; and it also exists in the multi-machine power system. But, because of the complex interconnection of the synchronous machines and the

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

Figure 8.

Comparison of damping ratio related to inter-area oscillation mode utilizing the P and PI control of the smart charging station, respectively.

smart charging station, it is difficult to calculate the optimal charging point in theory (Figure 8).

### 6. Conclusion

The chapter investigates the impacts of a grid-connected smart charging station on power system's small-signal stability based on a simple single-machine infinitebus power system integrated with a smart charging station. Damping torque analysis (DTA) is employed to examine the contribution from the smart charging station to the electromechanical oscillation loop of the generator in theoretical analysis. The analysis has concluded that, the smart charging station affects power system's smallsignal stability in light of its interaction with the synchronous machine. The proportional controls in the smart charging station mainly induce the synchronous torque into the oscillation loop and the majority of damping torque is introduced by integral controls. The damping torque supplied by the smart charging station is mainly directly induced into the oscillation loop, and the damping torque from the excitation system is almost from the synchronous machine itself. The optimal operation condition of the smart charging station is the moment when the system has the highest damping ratio. In this chapter, such an optimal operation condition is defined, indicating that the optimal charging capacity is considered for smart charging station design.

Results of the damping torque computation of a single-machine power system integrated with a smart charging station, confirmed by eigenvalue calculations of system oscillation mode, are presented in the chapter. The conclusions obtained from the theoretical analysis are demonstrated and verified by these results. Under the optimal operation condition, the total damping torque supplied from the smart charging station and synchronous machine reaches its maximum value. During the discharging process, the damping of the system tends to deteriorate with the increasing power injected from the smart charging station to the grid. Another fourmachine power system is employed to manifest that the conclusions obtained in the single-machine power system are also available in the multi-machine power system.

The stabilizer is designed and attached to the active or reactive power regulator of the smart charging station to supply additional positive damping into the system. The phase compensation method is used here. The effectiveness is confirmed by the

Conclusion (1) obtained in the single-machine infinite-busbar power system is also

Comparison of eigenvalue related to the inter-area oscillation mode utilizing PI control of the smart charging

Both proportional and integral controls are utilized in the smart charging station. A comparison is done under different charging or discharging power capacities of the smart charging station when the load-flow in the tie-line is fixed at 70 MW. The eigenvalue related to the inter-area oscillation mode is concerned. The results are

From Table 7, the optimal charging point with the highest damping ratio in the

single-machine power system can be obtained by Eq. (8) and verified by the damping torque calculation; and it also exists in the multi-machine power system. But, because of the complex interconnection of the synchronous machines and the

available in the multi-machine power system.

shown in Table 7.

PG2/ (10 MW)

Power System Stability

\*

Table 6.

Table 7.

station.

46

active power into the grid.

smart charging station.

PPEV/ (10 MW) Frequency/ Hz

Damping ratio/%

9.0 3.0 0.4724 3.12 3.0 0.4724 3.12 8.5 2.5 0.4710 3.14 2.5 0.4710 3.14 8.0 2.0 0.4693 3.15 2.0 0.4693 3.15 7.5 1.5 0.4672 3.16 1.5 0.4672 3.16 7.0 1.0 0.4650 3.15 1.0 0.4650 3.15 6.5 0.5 0.4624 3.13 0.5 0.4624 3.13 6.0 0.0 0.4595 3.11 0.0 0.4595 3.11 5.5 0.5 0.4564 3.07 0.5 0.4564 3.07 5.0 1.0 0.4530 3.01 1.0 0.4530 3.01

'+' denotes the adjustable load absorbing active power from the grid; '' denotes the regulable generator injecting

Comparison of eigenvalue related to inter-area oscillation mode only utilizing the proportional control of the

PG2/(10 MW) PPEV/(10 MW) Frequency/Hz Damping ratio/% 9.0 3.0 0.4734 2.52 8.5 2.5 0.4778 2.64 8.0 2.0 0.4802 2.98 7.5 1.5 0.4753 2.56 7.0 1.0 0.4725 2.48 6.5 0.5 0.4683 2.34 6.0 0.0 0.4611 2.25 5.5 0.5 0.4627 2.30 5.0 1.0 0.4721 2.39

Adjustable load/ generator\*

/(10 MW)

Frequency/ Hz

Damping ratio/%

nonlinear simulations and eigenvalue calculations in the single-machine power system.

Although the configuration of the power system with the grid-connected smart charging station and the function to describe the charging/discharging behaviors of EVs adopted in this chapter are very simple, all the essential elements have been included to serve the purpose of study, which can thoroughly reveal the dynamic interaction between the equivalent smart charging station and conventional generation in the transmission system. The optimal charging capacity is better to be considered during the capacity design of the smart charging station. With the help of the designed smart charging station-based stabilizer, the small-signal stability can be effectively maintained. Studies on the interactions among several smart charging stations with the dynamic stability in distribution systems and the uncertainties and diversities of EV charging/discharging behaviors will be carried out for future researches.

## A. Appendix: the parameters of example systems

1. Single-machine infinite-busbar power system.

The parameters of the synchronous machine (The unit is in pu):

M ¼ 5:0; T′ <sup>d</sup><sup>0</sup> ¼ 5:0 s; D ¼ 1:2; Xd ¼ 0:8; Xq ¼ 0:4; X′ <sup>d</sup> ¼ 0:05; ω<sup>0</sup> ¼ 2 � 50π; U<sup>t</sup> ¼ 1:05; Utref ¼ 1:05; K<sup>a</sup> ¼ 20:0; T<sup>a</sup> ¼ 0:01 s.

The parameters of the network (The unit is in pu): Xts ¼ 0:2; Xsb ¼ 0:1; X<sup>s</sup> ¼ 0:1; U<sup>b</sup> ¼ 1:0.

The parameters for the smart charging station(The unit is in pu):

KPP ¼ 20, KIP ¼ 20, KPI<sup>q</sup> ¼ 0:3, KII<sup>q</sup> ¼ 0:3, KPQ ¼ 15, KIQ ¼ 15, KPI<sup>d</sup> ¼ 0:7, KII<sup>d</sup> ¼ 0:7.

2. Four-machine power system.

The parameters of the synchronous machine are:

TJ1 ¼ TJ2 ¼ 117; TJ3 ¼ TJ4 ¼ 111:15; T′ <sup>d</sup><sup>01</sup> ¼ T′ <sup>d</sup><sup>02</sup> ¼ T′ <sup>d</sup><sup>03</sup> ¼ T′ <sup>d</sup><sup>04</sup> ¼ 8:0 s; Dmac1 ¼ Dmac2 ¼ Dmac3 ¼ Dmac4 ¼ 5:0 pu; Xd<sup>1</sup> ¼ Xd<sup>2</sup> ¼ Xd<sup>3</sup> ¼ Xd<sup>4</sup> ¼ 0:2; Xq<sup>1</sup> ¼ Xq<sup>2</sup> ¼ Xq<sup>3</sup> ¼ Xq<sup>4</sup> ¼ 0:1889; X′ <sup>d</sup><sup>1</sup> ¼ X′ <sup>d</sup><sup>2</sup> ¼ X′ <sup>d</sup><sup>3</sup> ¼ X′ <sup>d</sup><sup>4</sup> ¼ 0:0333; ω<sup>01</sup> ¼ ω<sup>02</sup> ¼ ω<sup>03</sup> ¼ ω<sup>04</sup> ¼ 100π; Ut1 ¼ Ut3 ¼ 1:03; Ut2 ¼ Ut4 ¼ 1:01; Utref1 ¼ Utref3 ¼ 1:03; Utref2 ¼ Utref4 ¼ 1:01.

All the generators are equipped with the same AVR: K<sup>a</sup> = 50; T<sup>a</sup> = 0.55 s; The parameters of the smart charging station are (in pu):

KPP ¼ 20, KIP ¼ 20, KPI<sup>q</sup> ¼ 0:3, KII<sup>q</sup> ¼ 0:3;

KPQ ¼ 15, KIQ ¼ 15, KPI<sup>d</sup> ¼ 0:7, KII<sup>d</sup> ¼ 0:7.

The parameters of the lines are (in pu): X<sup>15</sup> ¼ X<sup>36</sup> ¼ X<sup>29</sup> ¼ X<sup>48</sup> ¼ 0:01667; X<sup>56</sup> ¼ X<sup>89</sup> ¼ 0:025; X<sup>67</sup> ¼ 0:105; X<sup>78</sup> ¼ 0:005.

Author details

State Grid Jiangsu Economic Research Institute, Nanjing, P.R. China

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

DOI: http://dx.doi.org/10.5772/intechopen.80721

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: caihui300@hotmail.com

provided the original work is properly cited.

Cai Hui

49

G1 is connected with the slack bus of the system.G3 and G4 generate 70 MW of active power, respectively. The loads at Bus 6 and Bus 8 are 100 and 200 MW accordingly.

### Acknowledgements

The authors would like to acknowledge the support of Dr. Tim Littler from Queen's University, Belfast, UK.

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

## Author details

nonlinear simulations and eigenvalue calculations in the single-machine power

charging/discharging behaviors will be carried out for future researches.

The parameters of the synchronous machine (The unit is in pu):

<sup>d</sup><sup>0</sup> ¼ 5:0 s; D ¼ 1:2; Xd ¼ 0:8; Xq ¼ 0:4; X′

The parameters for the smart charging station(The unit is in pu):

The parameters of the network (The unit is in pu): Xts ¼ 0:2; Xsb ¼ 0:1;

KPP ¼ 20, KIP ¼ 20, KPI<sup>q</sup> ¼ 0:3, KII<sup>q</sup> ¼ 0:3, KPQ ¼ 15, KIQ ¼ 15, KPI<sup>d</sup> ¼ 0:7,

Dmac1 ¼ Dmac2 ¼ Dmac3 ¼ Dmac4 ¼ 5:0 pu; Xd<sup>1</sup> ¼ Xd<sup>2</sup> ¼ Xd<sup>3</sup> ¼ Xd<sup>4</sup> ¼ 0:2;

ω<sup>01</sup> ¼ ω<sup>02</sup> ¼ ω<sup>03</sup> ¼ ω<sup>04</sup> ¼ 100π; Ut1 ¼ Ut3 ¼ 1:03; Ut2 ¼ Ut4 ¼ 1:01;

All the generators are equipped with the same AVR: K<sup>a</sup> = 50; T<sup>a</sup> = 0.55 s;

The parameters of the lines are (in pu): X<sup>15</sup> ¼ X<sup>36</sup> ¼ X<sup>29</sup> ¼ X<sup>48</sup> ¼ 0:01667;

active power, respectively. The loads at Bus 6 and Bus 8 are 100 and 200 MW

The authors would like to acknowledge the support of Dr. Tim Littler from

G1 is connected with the slack bus of the system.G3 and G4 generate 70 MW of

<sup>d</sup><sup>1</sup> ¼ X′

<sup>d</sup><sup>01</sup> ¼ T′

<sup>d</sup><sup>02</sup> ¼ T′

<sup>d</sup><sup>3</sup> ¼ X′

<sup>d</sup><sup>2</sup> ¼ X′

<sup>d</sup><sup>03</sup> ¼ T′

<sup>d</sup> ¼ 0:05; ω<sup>0</sup> ¼ 2 � 50π;

<sup>d</sup><sup>04</sup> ¼ 8:0 s;

<sup>d</sup><sup>4</sup> ¼ 0:0333;

A. Appendix: the parameters of example systems

U<sup>t</sup> ¼ 1:05; Utref ¼ 1:05; K<sup>a</sup> ¼ 20:0; T<sup>a</sup> ¼ 0:01 s.

The parameters of the synchronous machine are:

Utref1 ¼ Utref3 ¼ 1:03; Utref2 ¼ Utref4 ¼ 1:01.

KPP ¼ 20, KIP ¼ 20, KPI<sup>q</sup> ¼ 0:3, KII<sup>q</sup> ¼ 0:3; KPQ ¼ 15, KIQ ¼ 15, KPI<sup>d</sup> ¼ 0:7, KII<sup>d</sup> ¼ 0:7.

X<sup>56</sup> ¼ X<sup>89</sup> ¼ 0:025; X<sup>67</sup> ¼ 0:105; X<sup>78</sup> ¼ 0:005.

The parameters of the smart charging station are (in pu):

TJ1 ¼ TJ2 ¼ 117; TJ3 ¼ TJ4 ¼ 111:15; T′

Xq<sup>1</sup> ¼ Xq<sup>2</sup> ¼ Xq<sup>3</sup> ¼ Xq<sup>4</sup> ¼ 0:1889; X′

1. Single-machine infinite-busbar power system.

M ¼ 5:0; T′

KII<sup>d</sup> ¼ 0:7.

accordingly.

48

Acknowledgements

Queen's University, Belfast, UK.

X<sup>s</sup> ¼ 0:1; U<sup>b</sup> ¼ 1:0.

2. Four-machine power system.

Although the configuration of the power system with the grid-connected smart charging station and the function to describe the charging/discharging behaviors of EVs adopted in this chapter are very simple, all the essential elements have been included to serve the purpose of study, which can thoroughly reveal the dynamic interaction between the equivalent smart charging station and conventional generation in the transmission system. The optimal charging capacity is better to be considered during the capacity design of the smart charging station. With the help of the designed smart charging station-based stabilizer, the small-signal stability can be effectively maintained. Studies on the interactions among several smart charging stations with the dynamic stability in distribution systems and the uncertainties and diversities of EV

system.

Power System Stability

Cai Hui State Grid Jiangsu Economic Research Institute, Nanjing, P.R. China

\*Address all correspondence to: caihui300@hotmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Zhenya L. Electric Power and Energy in China. Beijing: China Electric Power Press; 2012. pp. 124-135 (in Chinese). ISBN: 978-7-5123-2667-5

[2] Zheng J, Mehndiratta S, Guo JY, et al. Strategic policies and demonstration program of electric vehicle in China. Transport Policy. 2012;19(1):17-25. DOI: 10.1016/j.tranpol.2011.07.006

[3] Bowen Z, Littler T. Local storage meets local demand: A technical solution to future power distribution system. IET Generation, Transmission and Distribution. 2016;10(3):704-711. DOI: 10.1049/iet-gtd.2015.0442

[4] Venayagamoorthy GK. SmartParks for short term power flow control in smart grids. In: IEEE International Electric Vehicle Conference (IEVC) 2012. Greenville, USA: IEEE; 2012. pp. 1-6. DOI: 10.1109/IEVC. 2012.6183288

[5] Mitra P, Venayagamoorthy GK. SmartPark as a virtual STATCOM. IEEE Transactions on Smart Grid. 2011;2(3): 445-455. DOI: 10.1109/TSG.2011. 2158330

[6] Mitra P, Venayagamoorthy GK. Intelligent coordinated control of a wind farm and distributed SmartParks. In: IEEE Industry Application Society Annual Meeting (IAS) 2010. Houston, USA: IEEE; 2010. pp. 1-8. DOI: 10.1109/ IAS.2010.5615930

[7] Tomic J, Kempton W. Using fleets of electric drive vehicles for grid support. Journal of Power Sources. 2007;168(2): 459-468. DOI: 10.1016/j.jpowsour. 2007.03.010

[8] Venayagamoorthy GK, Sharma RK, Gautam PK. Dynamic energy management system for a smart

microgrid. IEEE Transactions on Neural Networks and Learning Systems. 2016; 27(8):1643-1656. DOI: 10.1109/ TNNLS.2016.2514358

http://explore.bl.uk/primo\_library/ libweb/action/display.do?tabs= detailsTab&gathStatTab=true&ct=

DOI: http://dx.doi.org/10.5772/intechopen.80721

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark

ETOCRN613373549&indx=1&recIds=

[21] Lingling Y, Wenjuan D, Yizhang Y, Yanfeng G. Methods of DTA to estimate the impact of integration of DFIG on single-machine infinite-bus power system. In: 2016 China International Conference on Electricity Distribution.

ETOCRN320143870

2016. pp. 1-5. DOI: 10.1109/ CICED.2016.7576247

22(5):688-703

10.1049/cp.2011.0208

[22] Wenjuan D, Haifeng W, Liye X. Power system small-signal stability as affected by grid-connected photovoltaic generation. European Transactions on Electrical Power. 2012;

[23] Wenjuan D, Haifeng W, Hui C. Modeling a Grid-connected SOFC power plant into power systems for small-signal analysis and control. European Transactions on Electrical Power. 2012;23(3):330-341. DOI:

[24] Haifeng W, Swift FJ. The capability

damping power system oscillations. IEE

Transmission and Distribution. 1996; 143(4):353-358 Availabe from: https://explore-bl-uk.vpn.seu.edu.cn/ primo\_library/libweb/action/display. do?tabs=detailsTab&gathStatTab= true&ct=display&fn=search&doc= ETOCRN613373549&indx=1&recIds=

[25] Majeau-Bettez G, Hawkins TR,

[26] Stikes K, Gross T, Lin Z, et al. Plug-in Hybrid Electric Vehicle Market Introduction Study: Final Report.

environmental assessment of lithiumion and nickel metal hydride batteries for plug-in hybrid and battery electric vehicles. Environmental Science and Technology. 2011;45(10):4548-4554

Stroemman AH. Life-cycle

of the static var compensator in

Proceedings of Generation,

ETOCRN011774271

ETOCRN613373549&indx=1&recIds=

[15] Xiaoyan Y, Chunlin G, Xuan X, Dequan H, Zhou M. Research on large scale electric vehicles participating in the economic dispatch of wind and thermal power system. In: 2017 China International Electrical and Energy Conference. 2017. pp. 223-228. DOI: 10.1109/CIEEC.2017.8388450

[16] Hua P, Zuofang L, Qianzhong X. Economic dispatch of power system including electric vehicle and wind farm. In: 2017 IEEE Conference on Energy Internet and Energy System Integration. 2017. pp. 1-5. DOI: 10.1109/

[17] Beck LJ. V2G—101, a text about vehicle-to-grid (V2G), the technology which enables a future of clean and

[18] Yaonan Y. Electric Power System Dynamics. New York: Academic Press; 1983. pp. 79-94. ISBN: 0127748202

[19] Wenjuan D, Haifeng W, Jun C. Model and theory of PSS localized phase compensation method. Proceeding of the CSEE. 2012;32(19):36-41 (in

[20] Wenjuan D, Haifeng W, Jun C. Application of localized phase compensation method to design a stabilizer in a multi-machine power system. Proceeding of the CSEE. 2012; 32(22):73-78 (in Chinese). Available from: http://explore.bl.uk/primo\_ library/libweb/action/display.do?tabs= detailsTab&gathStatTab=true&ct=

display&fn=search&doc=

efficient electric-powered transportation. USA:V2G-101 Copyright; 2009. pp. 10-25

display&fn=search&doc=

ETOCRN313900056

EI2.2017.8245238

Chinese)

51

[9] Kempton W, Tomic J. Vehicleto-grid power fundamentals: Calculating capacity and net revenue. Journal of Power Sources. 2005;144(1):268-279. DOI: 10.1016/j.jpowsour.2004.12.025

[10] Xifan W, Chengcheng S, Xiuli W, et al. Survey of electric vehicle charging load and dispatch control strategy. Proceedings of the CSEE. 2013;33(1): 1-10 (in Chinese). Available from: http://d.wanfangdata.com.cn/ Periodical/zgdjgcxb201301001

[11] Kempton W, Tomic J. Vehicle-togrid power implementation: From stabilizing the grid to support largescale renewable energy. Journal of Power Sources. 2005;144(1):280-294. DOI: 10.1016/j.jpowsour.2004.12.022

[12] Liting T, Mingxia Z, Wang H. Evaluation and solutions for electric vehicles' impact on the grid. Proceedings of the CSEE. 2012;32(31): 43-49 (in Chinese). Available from: http://d.wanfangdata.com.cn/ Periodical/zgdjgcxb201231006

[13] Huston C, Venayagamoorthy GK, Corzine K. Intelligent scheduling of hybrid and electric vehicle storage capacity in a parking lot for profit maximization in grid power transaction. In: Proceedings of IEEE Energy 2030. Atlanta, USA: IEEE; 2008. pp. 1-8. DOI: 10.1109/ENERGY.2008.4781051

[14] Zhengshuo L, Hongbin S, Guo Q, et al. Study on wind-EV complementation on the transmission grid side considering carbon emission. Proceedings of the CSEE. 2012;32(10): 41-48 (in Chinese). Availabe from:

Power System Small-Signal Stability as Affected by Grid-Connected SmartPark DOI: http://dx.doi.org/10.5772/intechopen.80721

http://explore.bl.uk/primo\_library/ libweb/action/display.do?tabs= detailsTab&gathStatTab=true&ct= display&fn=search&doc= ETOCRN613373549&indx=1&recIds= ETOCRN313900056

References

Power System Stability

2012.6183288

2158330

IAS.2010.5615930

2007.03.010

50

ISBN: 978-7-5123-2667-5

10.1016/j.tranpol.2011.07.006

[3] Bowen Z, Littler T. Local storage meets local demand: A technical solution to future power distribution system. IET Generation, Transmission and Distribution. 2016;10(3):704-711. DOI: 10.1049/iet-gtd.2015.0442

[4] Venayagamoorthy GK. SmartParks for short term power flow control in smart grids. In: IEEE International Electric Vehicle Conference (IEVC) 2012. Greenville, USA: IEEE; 2012. pp. 1-6. DOI: 10.1109/IEVC.

[5] Mitra P, Venayagamoorthy GK. SmartPark as a virtual STATCOM. IEEE Transactions on Smart Grid. 2011;2(3): 445-455. DOI: 10.1109/TSG.2011.

[6] Mitra P, Venayagamoorthy GK. Intelligent coordinated control of a wind farm and distributed SmartParks. In: IEEE Industry Application Society Annual Meeting (IAS) 2010. Houston, USA: IEEE; 2010. pp. 1-8. DOI: 10.1109/

[7] Tomic J, Kempton W. Using fleets of electric drive vehicles for grid support. Journal of Power Sources. 2007;168(2): 459-468. DOI: 10.1016/j.jpowsour.

[8] Venayagamoorthy GK, Sharma RK,

Gautam PK. Dynamic energy management system for a smart

[1] Zhenya L. Electric Power and Energy in China. Beijing: China Electric Power Press; 2012. pp. 124-135 (in Chinese).

microgrid. IEEE Transactions on Neural Networks and Learning Systems. 2016;

27(8):1643-1656. DOI: 10.1109/

[9] Kempton W, Tomic J. Vehicleto-grid power fundamentals: Calculating capacity and net revenue. Journal of Power Sources. 2005;144(1):268-279. DOI: 10.1016/j.jpowsour.2004.12.025

[10] Xifan W, Chengcheng S, Xiuli W, et al. Survey of electric vehicle charging load and dispatch control strategy. Proceedings of the CSEE. 2013;33(1): 1-10 (in Chinese). Available from: http://d.wanfangdata.com.cn/ Periodical/zgdjgcxb201301001

[11] Kempton W, Tomic J. Vehicle-togrid power implementation: From stabilizing the grid to support largescale renewable energy. Journal of Power Sources. 2005;144(1):280-294. DOI: 10.1016/j.jpowsour.2004.12.022

[12] Liting T, Mingxia Z, Wang H. Evaluation and solutions for electric

Proceedings of the CSEE. 2012;32(31): 43-49 (in Chinese). Available from: http://d.wanfangdata.com.cn/ Periodical/zgdjgcxb201231006

[13] Huston C, Venayagamoorthy GK, Corzine K. Intelligent scheduling of hybrid and electric vehicle storage capacity in a parking lot for profit maximization in grid power transaction. In: Proceedings of IEEE Energy 2030. Atlanta, USA: IEEE; 2008. pp. 1-8. DOI:

10.1109/ENERGY.2008.4781051

et al. Study on wind-EV

[14] Zhengshuo L, Hongbin S, Guo Q,

complementation on the transmission grid side considering carbon emission. Proceedings of the CSEE. 2012;32(10): 41-48 (in Chinese). Availabe from:

vehicles' impact on the grid.

TNNLS.2016.2514358

[2] Zheng J, Mehndiratta S, Guo JY, et al. Strategic policies and demonstration program of electric vehicle in China. Transport Policy. 2012;19(1):17-25. DOI: [15] Xiaoyan Y, Chunlin G, Xuan X, Dequan H, Zhou M. Research on large scale electric vehicles participating in the economic dispatch of wind and thermal power system. In: 2017 China International Electrical and Energy Conference. 2017. pp. 223-228. DOI: 10.1109/CIEEC.2017.8388450

[16] Hua P, Zuofang L, Qianzhong X. Economic dispatch of power system including electric vehicle and wind farm. In: 2017 IEEE Conference on Energy Internet and Energy System Integration. 2017. pp. 1-5. DOI: 10.1109/ EI2.2017.8245238

[17] Beck LJ. V2G—101, a text about vehicle-to-grid (V2G), the technology which enables a future of clean and efficient electric-powered transportation. USA:V2G-101 Copyright; 2009. pp. 10-25

[18] Yaonan Y. Electric Power System Dynamics. New York: Academic Press; 1983. pp. 79-94. ISBN: 0127748202

[19] Wenjuan D, Haifeng W, Jun C. Model and theory of PSS localized phase compensation method. Proceeding of the CSEE. 2012;32(19):36-41 (in Chinese)

[20] Wenjuan D, Haifeng W, Jun C. Application of localized phase compensation method to design a stabilizer in a multi-machine power system. Proceeding of the CSEE. 2012; 32(22):73-78 (in Chinese). Available from: http://explore.bl.uk/primo\_ library/libweb/action/display.do?tabs= detailsTab&gathStatTab=true&ct= display&fn=search&doc=

ETOCRN613373549&indx=1&recIds= ETOCRN320143870

[21] Lingling Y, Wenjuan D, Yizhang Y, Yanfeng G. Methods of DTA to estimate the impact of integration of DFIG on single-machine infinite-bus power system. In: 2016 China International Conference on Electricity Distribution. 2016. pp. 1-5. DOI: 10.1109/ CICED.2016.7576247

[22] Wenjuan D, Haifeng W, Liye X. Power system small-signal stability as affected by grid-connected photovoltaic generation. European Transactions on Electrical Power. 2012; 22(5):688-703

[23] Wenjuan D, Haifeng W, Hui C. Modeling a Grid-connected SOFC power plant into power systems for small-signal analysis and control. European Transactions on Electrical Power. 2012;23(3):330-341. DOI: 10.1049/cp.2011.0208

[24] Haifeng W, Swift FJ. The capability of the static var compensator in damping power system oscillations. IEE Proceedings of Generation, Transmission and Distribution. 1996; 143(4):353-358 Availabe from: https://explore-bl-uk.vpn.seu.edu.cn/ primo\_library/libweb/action/display. do?tabs=detailsTab&gathStatTab= true&ct=display&fn=search&doc= ETOCRN613373549&indx=1&recIds= ETOCRN011774271

[25] Majeau-Bettez G, Hawkins TR, Stroemman AH. Life-cycle environmental assessment of lithiumion and nickel metal hydride batteries for plug-in hybrid and battery electric vehicles. Environmental Science and Technology. 2011;45(10):4548-4554

[26] Stikes K, Gross T, Lin Z, et al. Plug-in Hybrid Electric Vehicle Market Introduction Study: Final Report.

Washington, DC: Tech. Rep. DE2010-972306; 2010

[27] Guoqiang H, Renmu H, Huachun Y, et al. Iterative prony method based power system low frequency oscillation mode analysis and PSS design. In: 2005 IEEE/PES Asia and Pacific Transmission and Distribution Conference and Exhibition. Dalian, China: IEEE; 2005. pp. 1-6. DOI: 10.1109/ TDC.2005.1546867

Section 2

Power Oscillations and

Electrical Infrastructures

Section 2
