**3. Frequency stability of power system – An estimation of active power unbalance**

Stability of power system refers to its ability to maintain synchronous operation of all connected synchronous generators in stationary state and for the defined initial state after disturbances occur, so that the change of the variables of state in transitional process is limited, and system structure preserved. The system should be restored to initial stationary state unless topology changes take place, that is, if there are topological changes to the system, a new stationary state should be invoked. Although the stability of power system is its unique trait, different forms of instability are easier to comprehend and analysed if stability problems are classified, that is, if "partial" stability classes are defined. Partial stability classes are usually defined for fundamental state parameters: transmission angle, voltage and frequency. Figure 4. shows classification of stability according to (IEEE/CIGRE, 2004). Detailed description of physicality of dynamics and system stability, mathematical models and techniques to resolve equations of state and stability aspect analysis can be found in many books and papers.

Fig. 4. Classification of "partial" stability of electric power system

Frequency stability is defined as the ability of power system to maintain frequency within standardized limits. Frequency instability occurs in cases when electric power system cannot permanently maintain the balance of active powers in the system, which leads to frequency collapse. In cases of high intensity disturbances or successive interrelated and mutually caused (connected) disturbances, there can be cascading deterioration of frequency stability, which, in the worst case scenario, leads to disjunction of power system to subsystems and eventual total collapse of function of isolated parts of electric power system formed in this way.

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 167

generators is done according to the criteria of electric distance from the point of unbalance

coefficients of their synchronizing powers1 *PSik*(*t*). Therefore, generators closer to the load bus *k* (those with lower initial transmission angles and bigger transmission susceptanse)

certain generators start to decelerate (Anderson & Fouad, 2002). The change of generators' angle frequency *i* is defined by a differential equation governing the motion of machine by

> <sup>2</sup> <sup>0</sup> *i i i*

1 (0 ) 2 *i Sik k*

Equation (13) provides first criterion for distribution of active power unbalance : *Initial slowing down of generators depends on a.) relative relation of coefficient of synchronising power* 

It is clear that some generators will have different initial slowdowns. Therefore, in transient process, frequencies of different generators vary. Synchronizing powers maintain generators in synchronous operation and if transient stability is maintained, oscillations of frequency and active power for a coherent group of generators have a muted character. When the system retains synchronised operation, it is possible to define system's retarding in general, that is, to define a medium value of frequency of a group of generators. To produce an equation to describe the change of medium frequency, we introduce the term "centre of

*d PP dt H <sup>P</sup>*

*PSik(t) and total synchronising system power and b.) inertia constant of generator's rotor Hi.*

(t) is expressed in the function of total unbalance , then according to

1

*Sjk*

*<sup>n</sup> <sup>i</sup>*

*j*

and angular frequency \_\_

*n*

*i n i*

, \_\_ <sup>1</sup>

 

The equation describing the moving of inertia centre according to (Anderson & Fouad, 2002)

1 Synchronising power of a multi-machine system is defined by:

and it shows the dependance of the change of electric power of *i* machine with the change of of the

1

*H*

*i*

*i i*

*H*

0

*ij*

*ij*

*ij P*

*H d <sup>P</sup> dt* 

0

(Anderson & Fouad, 2002) the aforementioned equation becomes:

0

*n*

\_\_ <sup>1</sup>

 

difference in angles *i* and *j*, provided that the angles of other machines are fixed.

*i n i*

1

*H*

*i*

*i i*

*H*

(t). Due to a sudden change in power balance,

(12)

(13)

is defined as follows:

(14)

0 0 cos sin *ij*

,

 

*s i j ij ij ij ij*

*P EE B G*

(0+) depending on

(load at node *k*). Certain generators take over a part of unbalance *Pk*

take over a bigger part of unbalance *Pi*

inertia ". The angle of inertia centre \_\_

is as follows:

the swing equation:

If unbalance *Pi*

In a normal regime, all connected synchronous generators in power system generate voltage of the same (nominal) frequency and the balance of active power is maintained. Then all voltage nods in network have a frequency of nominal value. When the system experiences permanent unbalance of active power (usually due to the breakdown of generator or load bus), power balance is impaired. Generators with less mechanic then electric power due to unbalance redistribution start slowing down. Because inertia of certain generators vary, as well as redistribution of unbalance ratio, generators start operating at different speeds and generate voltage of different frequencies. After transient process, we can assume that the system has a unique frequency again – frequency of the centre of inertia.

During long-term dynamic processes, there is a redistribution of power between generators, and subsequently redistribution of power in transmission lines, which can lead to overload of these elements. In case of the overload of elements over a longer period of time, there are overload protective device which tripping overloaded elements. This leads to cascading deterioration of system stability, and in critical cases (if interconnecting line is tripping), disjunction of system to unconnected elements – islands. In general, this scenario of disturbance propagation causes major problems in systems which have large active power unbalance and small system inertia. Usually, when these critical situations take place, under-frequency protection tripping the generators, additionally worsening the system. In border-line cases, this cascading event can lead to frequency instability, and complete collapse of system function.

#### **3.1 Power system response to active power unbalance**

In order to understand the essence of dynamic response of power system, one must be familiar with the physicality of the process, that is, one must do the quality analysis of dynamic response. An example of quality analysis of dynamic response of a coherent group of the effect of a sudden application at *t=0* of a small load change *Pk* at node *k* is analyzed in (Anderson & Fouad, 2002). The analysis was carried out on a linear model of system response to a forced (small) disturbance of active power balance. Although it is an approximatization, the analysis helps understand physicality of the process of dynamic response of power system to active power unbalance . This chapter provides main conclusions of the aforementioned analysis.

Distribution of the forced power unbalance *Pk* (0+) between generators during system response is done in accordance with different criteria. When the synchronous operation of generators is maintained (stability of synchronous group is maintained), a new stationary state is established in the system after transient process, namely, new power balance. If criteria for disturbance distribution differ for generators (which is mostly the case), transient process has an oscillatory-muted character. Oscillations of the parameters of state, mostly active power, angles and frequency of generators, reflect transition between certain criteria for unbalance distribution. Generally, three quality criteria for unbalance distribution can be distinguished:

Immediately before unbalance (in t=0+) power balance in the system is maintained on the basis of accumulated electromagnetic energy of generators. Distribution of balance between

In a normal regime, all connected synchronous generators in power system generate voltage of the same (nominal) frequency and the balance of active power is maintained. Then all voltage nods in network have a frequency of nominal value. When the system experiences permanent unbalance of active power (usually due to the breakdown of generator or load bus), power balance is impaired. Generators with less mechanic then electric power due to unbalance redistribution start slowing down. Because inertia of certain generators vary, as well as redistribution of unbalance ratio, generators start operating at different speeds and generate voltage of different frequencies. After transient process, we can assume that the system has a unique frequency again – frequency of the

During long-term dynamic processes, there is a redistribution of power between generators, and subsequently redistribution of power in transmission lines, which can lead to overload of these elements. In case of the overload of elements over a longer period of time, there are overload protective device which tripping overloaded elements. This leads to cascading deterioration of system stability, and in critical cases (if interconnecting line is tripping), disjunction of system to unconnected elements – islands. In general, this scenario of disturbance propagation causes major problems in systems which have large active power unbalance and small system inertia. Usually, when these critical situations take place, under-frequency protection tripping the generators, additionally worsening the system. In border-line cases, this cascading event can lead to frequency instability, and complete

In order to understand the essence of dynamic response of power system, one must be familiar with the physicality of the process, that is, one must do the quality analysis of dynamic response. An example of quality analysis of dynamic response of a coherent group

in (Anderson & Fouad, 2002). The analysis was carried out on a linear model of system response to a forced (small) disturbance of active power balance. Although it is an approximatization, the analysis helps understand physicality of the process of dynamic response of power system to active power unbalance . This chapter provides main

response is done in accordance with different criteria. When the synchronous operation of generators is maintained (stability of synchronous group is maintained), a new stationary state is established in the system after transient process, namely, new power balance. If criteria for disturbance distribution differ for generators (which is mostly the case), transient process has an oscillatory-muted character. Oscillations of the parameters of state, mostly active power, angles and frequency of generators, reflect transition between certain criteria for unbalance distribution. Generally, three quality criteria for unbalance distribution can

Immediately before unbalance (in t=0+) power balance in the system is maintained on the basis of accumulated electromagnetic energy of generators. Distribution of balance between

(0+) between generators during system

at node *k* is analyzed

centre of inertia.

collapse of system function.

be distinguished:

**3.1 Power system response to active power unbalance** 

conclusions of the aforementioned analysis.

Distribution of the forced power unbalance *Pk*

of the effect of a sudden application at *t=0* of a small load change *Pk*

generators is done according to the criteria of electric distance from the point of unbalance (load at node *k*). Certain generators take over a part of unbalance *Pk* (0+) depending on coefficients of their synchronizing powers1 *PSik*(*t*). Therefore, generators closer to the load bus *k* (those with lower initial transmission angles and bigger transmission susceptanse) take over a bigger part of unbalance *Pi*(t). Due to a sudden change in power balance, certain generators start to decelerate (Anderson & Fouad, 2002). The change of generators' angle frequency *i* is defined by a differential equation governing the motion of machine by the swing equation:

$$\frac{2H\_i}{a\rho\_0}\frac{d\alpha\_{i\Lambda}}{dt} + P\_{i\Lambda} = 0\tag{12}$$

If unbalance *Pi* (t) is expressed in the function of total unbalance , then according to (Anderson & Fouad, 2002) the aforementioned equation becomes:

$$\frac{1}{\alpha\_0} \frac{d\alpha\_{i\Lambda}}{dt} = -\frac{P\_{Sik}}{2H\_i} \frac{P\_{k\Lambda}(0^+)}{\sum\_{j=1}^n P\_{Sjk}} \tag{13}$$

Equation (13) provides first criterion for distribution of active power unbalance : *Initial slowing down of generators depends on a.) relative relation of coefficient of synchronising power PSik(t) and total synchronising system power and b.) inertia constant of generator's rotor Hi.*

It is clear that some generators will have different initial slowdowns. Therefore, in transient process, frequencies of different generators vary. Synchronizing powers maintain generators in synchronous operation and if transient stability is maintained, oscillations of frequency and active power for a coherent group of generators have a muted character. When the system retains synchronised operation, it is possible to define system's retarding in general, that is, to define a medium value of frequency of a group of generators. To produce an equation to describe the change of medium frequency, we introduce the term "centre of inertia ". The angle of inertia centre \_\_ and angular frequency \_\_ is defined as follows:

$$\overline{\delta\phi} = \frac{\sum\_{i=1}^{n} H\_i \delta\_i}{\sum\_{i=1}^{n} H\_i}, \quad \overline{o} = \frac{\sum\_{i=1}^{n} H\_i o i\_i}{\sum\_{i=1}^{n} H\_i} \tag{14}$$

The equation describing the moving of inertia centre according to (Anderson & Fouad, 2002) is as follows:

<sup>1</sup> Synchronising power of a multi-machine system is defined by: 0 0 0 cos sin *ij ij ij s i j ij ij ij ij ij P P EE B G* ,

and it shows the dependance of the change of electric power of *i* machine with the change of of the difference in angles *i* and *j*, provided that the angles of other machines are fixed.

$$\frac{1}{\rho\_0} \frac{d \, \overline{o\_\Delta}}{dt} = \frac{-P\_{k\Delta}(0^+)}{\sum\_{i=1}^n 2H\_i} \tag{15}$$

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 169

functions is to identify current and potential problems in power system operation in relation to the system's safety and support to operators in control centres when making decisions to prevent disturbance propagations. Phasor Measurement Unit technology (PMU) enabled full implementation of these systems and measurement of dynamic states in wider area. Current control and running of power system is based upon local measurement of statistic values of system parameters of power system (voltage, power, frequency ...). WAMS are based on embedded devices for measuring phasor voltage and current electricity at those points in power system which are of particular importance, that is measuring amplitudes and angles in real time using PMUs. Such implemented platform enables realistic dynamic view of electric power system, more accurate measurement, rapid data exchange and implementation of algorithms which enable coordination and timely alert in case of

Depending on the nature of active power unbalance, the system disturbance can be temporary (short circuit at the transmission line with successful reclosure) or permanent (tripping generators or consumers). Disturbances with permanent power unbalance are of a particular interest. As shown earlier, dominant variables of state which define power system response to a permanent active power unbalance are the change of frequency and generator's active power. Less dominant variables, but not to be ignored, are voltage and

In short, algorithm for on-line identification of active power unbalance can be described

*Analysis of the response of change of generator's frequency ωi(t) during the period of first oscillation makes it possible to define transient stability. If transient stability is maintained, then the application of DWT (using low-frequency component of signal) makes it possible to estimate with high precision the change of the frequency of inertia centre. Furthermore, provided that the values of inertia of all generators are known as well as system inertia as a whole, it is possible to define the total forced* 

To illustrate estimate of active power unbalance in power system, WSCC 9-bus test system has been chosen (Figure 5.). Additional data on this test system can be found in (Anderson & Fouad, 2002). The following example has been analysed in details in (Anderson & Fouad, 2002).

2 3

5 6

G2 G3

7 8 9

4

1

G1

instability.

reactive power.

*unbalance Pk*

*(0+).* 

Fig. 5. WSCC 9-bus test system

as:

This equation points out an important trait of power system: *Although some generators retarding at different rates (di/dt), which change during transient process, the system as a whole retarding at the constant rate d dt* / .

Frequencies of some generators approach the frequency of inertia centre because synchronizing powers in a stable response mute oscillations. After a relatively short time (*t=t1* ), of few seconds, all generators adjust to the frequency of inertia centre, that is, the system has a unique frequency. Distribution of unbalance *Pk*(0+) at moment *t1* between generators is defined per criterion (Anderson & Fouad, 2002), which is as follows:

$$P\_{i\Lambda}(t\_1) = \frac{H\_i}{\sum\_{j=1}^n H\_j} P\_{k\Lambda}(0^+) \tag{16}$$

This equation provides second criterion for unbalance distribution: After lapse of time t1 since the unbalance occurred, the total value of unbalance Pk(0+) is distributed between generators depending on their relative inertia in relation to the total inertia of a coherent group of generators. Therefore, unbalance distribution according to this criterion does not depend on electric distance of the generator from the point at which the unbalance occurred..

Finally, if the generators' speed regulators are activated, they lead to the change in mechanical power of generator and redistribution of unbalance depending on statistic coefficients of speed regulators. After a certain period of time, an order of ten seconds (*t=t2*), the system establishes a new stationary state. Frequency in the new stationary state depends on total regulative system constante2. This leads to a third criterion for unbalance distribution: *After lapse of time t2 since the unbalance occurred, the total value of unbalance Pk(0+) is distributed between generators depending on their constant of statism of speed regulators.* 

The previous analysis, although it does not take into account the effects of load characteristics on the amount of power unbalance , credibly illustrates quality processes in power systems with active power unbalance .
