**3.2 An estimation of active power unbalance – Computer simulation testing**

Algorhytam for identification and estimation of unbalance in electric power system presented in Refs. (Avdakovic et all, 2009, 2010) assumes availability of WAMS. Today, these systems are in force in many electric power systems worldwide, and one of their main

<sup>2</sup> Relation between arbitraty power change *ΔP* and its corresponding frequency change *Δf* , defined as K= *ΔP/ Δf* [MWs] is called regulative energy or regulative constant.

1 (0 )

This equation points out an important trait of power system: *Although some generators* 

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

1

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

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* 

*(0+) is distributed between generators depending on their constant of statism of speed* 

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

**3.2 An estimation of active power unbalance – Computer simulation testing**  Algorhytam for identification and estimation of unbalance in electric power system presented in Refs. (Avdakovic et all, 2009, 2010) assumes availability of WAMS. Today, these systems are in force in many electric power systems worldwide, and one of their main

2 Relation between arbitraty power change *ΔP* and its corresponding frequency change *Δf* , defined as

*j*

*<sup>H</sup> Pt P*

 

( ) (0 ) *<sup>i</sup> i k n*

*H*

*j*

generators is defined per criterion (Anderson & Fouad, 2002), which is as follows:

1

*d P*

1

*i*

*dt <sup>H</sup>*

2 *k n*

*i*

*/dt), which change during transient process, the system as a whole* 

(15)

(0+) at moment *t1* between

(16)

\_\_

0

*i*

> / .

system has a unique frequency. Distribution of unbalance *Pk*

*retarding at different rates (d*

occurred..

*Pk*

*regulators.* 

power systems with active power unbalance .

K= *ΔP/ Δf* [MWs] is called regulative energy or regulative constant.

*retarding at the constant rate d dt*

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 instability.

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 reactive power.

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

*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 unbalance Pk(0+).* 

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).

Fig. 5. WSCC 9-bus test system

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 171

1

yp(polyval)

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>2</sup> 59.8

The estimate of values *df/dt*, that is, values *dω/dt* for signals ωi (i=1,2,3) with the DWT application will be provided later on. Frequency range [*Fm*/2 : *Fm*] of every level of decomposition of DWT is in direct relation with signal sampling frequency, and is presented as *Fm= Fs/*2*l+1*, where *Fs* present sampling frequency and *l* present the level of

The sampling time of 0.02 sec or sampling frequency of analysed signals of 50 Hz were used in order to present this method and simulations,. Based on Nyquist theorem, the highest frequency a signal can have is *Fs/*2 or 25 Hz. Example of the fifth level of *ω1* signal decomposition from Figure 6, using Db4 wavelet function, is given in Figure 8, while frequency range of analysed signals at different levels of decomposition is given in Table 2.

> D1 [25.0 – 12.50 Hz] D2 [12.5 – 6.250 Hz] D3 [6.25 – 3.120 Hz] D4 [3.12 – 1.560 Hz] D5 [1.56 – 0.780 Hz] A5 [0.00 – 0.780 Hz]

Decomposition of signals ω2 i ω3 from Figure 6 was done in the same manner. A5 low frequency components of all three signals and centre of inertia are illustrated in Figure 9. It can be seen that the low frequency components of analysed signals are very similar to the calculated value of the centre of inertia, and therefore, suitable for defining values *df/dt*, or in this case, the analysed *dω/dt*. Estimate is given in Table 3. As can be seen, both methods provide rather good results, and estimated values are very similar to the calculated vales.

time (s)

59.85

Fig. 7. Curve fitting

decomposition.

Table 2. Frequency range of analysed signals

59.9

59.95

Frequency [Hz]

60

60.05

Connection of nominal 10 MW (0.1 pu) of active power to bus 8 as three phase short circuit circuit with active resistance 10 p.u. is simulated. The change of angle speed or frequency of some generators and centre of inertia (COI) after simulated disturbance are shown in Figure 6. and the show oscillations of machines after the disturbance and slow decrease of frequency in the system. It can be seen that some generators slow down by oscillating around medium frequency of the centre of inertia. The slow down around 0.09 Hz/s is presented as direction (ωCOI).

Specialised literature provides many techniques to estimate frequency and the level of frequency change, that is, *df/dt*. One of the methods used with estimating *df/dt* is the Method of Least Squares. It represents one of the most important and most widely used methods for data analysis. Mathematical details which elaborate this method can be found in a number of books and papers.

Fig. 6. Speed deviation following application of a 10 MW resistive load at bus 8 (Avdakovic et al., 2011)

Here, the estimation of *df/dt* was done in Matlab using polyfit and polyval functions. Figure 7 shows calculated value of polynomial at given points (yp), using values of angle frequency ω1 from Figure 6 and polynomials of third degree. The estimate of *df/dt*, that is, *dω/dt* for signals ωi (i=1,2,3), are provided in Table 2.

Fig. 7. Curve fitting

Connection of nominal 10 MW (0.1 pu) of active power to bus 8 as three phase short circuit circuit with active resistance 10 p.u. is simulated. The change of angle speed or frequency of some generators and centre of inertia (COI) after simulated disturbance are shown in Figure 6. and the show oscillations of machines after the disturbance and slow decrease of frequency in the system. It can be seen that some generators slow down by oscillating around medium frequency of the centre of inertia. The slow down around

Specialised literature provides many techniques to estimate frequency and the level of frequency change, that is, *df/dt*. One of the methods used with estimating *df/dt* is the Method of Least Squares. It represents one of the most important and most widely used methods for data analysis. Mathematical details which elaborate this method can be found in a number

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>2</sup> 0.997

Fig. 6. Speed deviation following application of a 10 MW resistive load at bus 8 (Avdakovic

Here, the estimation of *df/dt* was done in Matlab using polyfit and polyval functions. Figure 7 shows calculated value of polynomial at given points (yp), using values of angle frequency ω1 from Figure 6 and polynomials of third degree. The estimate of *df/dt*, that is, *dω/dt* for

time (s)

0.09 Hz/s is presented as direction (ωCOI).

of books and papers.

0.9975

et al., 2011)

signals ωi (i=1,2,3), are provided in Table 2.

0.998

0.9985

0.999

0.9995

1.0005

1

The estimate of values *df/dt*, that is, values *dω/dt* for signals ωi (i=1,2,3) with the DWT application will be provided later on. Frequency range [*Fm*/2 : *Fm*] of every level of decomposition of DWT is in direct relation with signal sampling frequency, and is presented as *Fm= Fs/*2*l+1*, where *Fs* present sampling frequency and *l* present the level of decomposition.

The sampling time of 0.02 sec or sampling frequency of analysed signals of 50 Hz were used in order to present this method and simulations,. Based on Nyquist theorem, the highest frequency a signal can have is *Fs/*2 or 25 Hz. Example of the fifth level of *ω1* signal decomposition from Figure 6, using Db4 wavelet function, is given in Figure 8, while frequency range of analysed signals at different levels of decomposition is given in Table 2.


Table 2. Frequency range of analysed signals

Decomposition of signals ω2 i ω3 from Figure 6 was done in the same manner. A5 low frequency components of all three signals and centre of inertia are illustrated in Figure 9. It can be seen that the low frequency components of analysed signals are very similar to the calculated value of the centre of inertia, and therefore, suitable for defining values *df/dt*, or in this case, the analysed *dω/dt*. Estimate is given in Table 3. As can be seen, both methods provide rather good results, and estimated values are very similar to the calculated vales.

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 173

A5Omega1

A5Omega2

A5Omega3

COI

MLS [Hz/s] DWT [Hz/s]

100

*1/dt* -0.0888 -0.0801

*2/dt* -0.0799 -0.0756

*3/dt* -0.0787 -0.0764 Table 3. Comparison of estimates of *df/dt*, and *dω/dt* using the Method of Least Squares and

Inertia of generators for WSCC 9 bus system is *H1*=23,64 (sec), *H2*=6,4 (sec) and *H3*=3,01 (sec), so base on the on the basis of (12), it is easy to determine distribution of unbalance of active power in the system per a generator, and subsequently, the total unbalance of active

The aforementioned analysed example demonstrates the procedure for estimating *df/dt* value using DWT. It is possible to define (simulate) the value of forced unbalance of active power in more complex power systems in the exact same way. An example of a more detailed analysis and application of this methodology is provided in Ref. (Avdakovic et al., 2010), while simulations and analyses were done on New England 39 bus system. When analysing more complex power systems, the frequency range of low frequency electromechanic occurrences/oscillations is in the range of 5 Hz, so it is a matter of practicality to choose sampling time of 0,1 sec or 10 Hz. With further multiresolution analysis in this chosen frequency range and the availability of WAMS, it becomes possible to obtain some very important information for monitoring and control of power system. This is mostly information related to the very start of some dynamic occurrence in the power system which we obtain from the first level of decomposition of analysed signals. Since electric power systems are mostly widespread across huge geographic area, it is necessary to have information on the location of initial disturbance in the power system, which is easily

Fig. 9. COI and low frequency (A5) component of signals angular speed *ω1, ω2* and *ω<sup>3</sup>*

0

59.8

*d*

*d*

*d*

power in the observed system.

DWT

59.85

59.9

Frequency

59.95

60

50

Samples

Fig. 8. MRA analysis signal of angular speed ω<sup>1</sup>

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

Samples

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

Samples

Samples

Samples

Samples

A5

D5

D4

D3

D2

D1

Samples

59.8 59.9 60 60.1

0.01






Fig. 8. MRA analysis signal of angular speed ω<sup>1</sup>

0

Magnitude

5 x 10-3

0

Magnitude

5 x 10-3

0

Magnitude

2 x 10-3

0

Magnitude

5 x 10-3

0

Magnitude

Magnitude

Fig. 9. COI and low frequency (A5) component of signals angular speed *ω1, ω2* and *ω<sup>3</sup>*


Table 3. Comparison of estimates of *df/dt*, and *dω/dt* using the Method of Least Squares and DWT

Inertia of generators for WSCC 9 bus system is *H1*=23,64 (sec), *H2*=6,4 (sec) and *H3*=3,01 (sec), so base on the on the basis of (12), it is easy to determine distribution of unbalance of active power in the system per a generator, and subsequently, the total unbalance of active power in the observed system.

The aforementioned analysed example demonstrates the procedure for estimating *df/dt* value using DWT. It is possible to define (simulate) the value of forced unbalance of active power in more complex power systems in the exact same way. An example of a more detailed analysis and application of this methodology is provided in Ref. (Avdakovic et al., 2010), while simulations and analyses were done on New England 39 bus system. When analysing more complex power systems, the frequency range of low frequency electromechanic occurrences/oscillations is in the range of 5 Hz, so it is a matter of practicality to choose sampling time of 0,1 sec or 10 Hz. With further multiresolution analysis in this chosen frequency range and the availability of WAMS, it becomes possible to obtain some very important information for monitoring and control of power system. This is mostly information related to the very start of some dynamic occurrence in the power system which we obtain from the first level of decomposition of analysed signals. Since electric power systems are mostly widespread across huge geographic area, it is necessary to have information on the location of initial disturbance in the power system, which is easily

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 175

order to reduce the effect of disturbance, and adjust the operation of effective measures to

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identified unbalance of active power.

**5. References** 

obtained from DWT signal filters with the frequency range of 1 – 2 Hz. Frequency range of 1 – 2 Hz is the space of local oscillations in power system and by a simple comparison of power values of signals in this frequency range, analysed from multiple geographically distant locations , it is easy to establish the location of disturbance. From the power point of view, power values of local oscillations of signals measured/simulated closer to the disturbance will have higher energy power values compared to those distant from the location of disturbance. Furthermore, as we proceed to the higher levels of decomposition (or lower frequency ranges of filters) of chosen signals with sampling frequency of 0.1 sec, we enter the intra-area and inter-area of oscillations which can represent a real danger for electric power system, and should it be that they are not muted, can lead in a black-out. These signals make it possible to identify intra-area and inter-area oscillations, their character and how to mute them. Furthermore, by comparing these signals it is possible to obtain more information on the system's operation as a whole after disturbance (Avdakovic & Nuhanovic, 2009). In line with what has been demonstrated in the example, low frequency component of signal angle or frequency serves to estimate values *df/dt,* that is, to define total forced unbalance of active power in power system.
