Modal analysis of power systems to mitigate harmonic resonance considering load models

ARTICLE IN PRESS Energy 33 (2008) 1361– 1368

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Modal analysis of power systems to mitigate harmonic resonance considering load models Masoud Esmaili , Heidar Ali Shayanfar, Alireza Jalilian Centre of Excellence for Power System Automation and Operation, Iran University of Science and Technology, Narmak, Tehran, Iran

a r t i c l e in fo

abstract

Article history: Received 25 February 2008

To detect and alleviate harmonic resonance in a power system have been a delicate issue. In this paper, the effect of load modeling on resonance behavior of power systems employing eigenvalue sensitivity analysis is investigated. The most influencing parameters to mitigate resonance modes are detected using the sensitivity of critical eigenvalues with respect to the network various components in a frequency range. Also, results inferred from the criteria of driving point impedance and bus participation factors are compared with those of the sensitivity analysis with different load models. Where to locate capacitors and filters to mitigate the resonance modes is obtained using these criteria. The methods are tested on the well-known New Zealand as well as IEEE-30 bus test systems. Simulation results are discussed in detail to investigate the methods’ efficiency and capability. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Modal impedance Eigenvalue Harmonics Frequency scan analysis Resonance modes Load model

1. Introduction Harmonic analysis of power systems has been one of the most vital problems due to increasing non-linear loads in recent power systems over past decades. Nowadays, to conserve energy and provide better control of both traditional and new processes, power networks have experienced a tremendous growth in the applications of solid-state power electronic devices at all voltage levels. At transmission levels, (Flexible Alternating Current Transmission System) FACTS devices are being installed to control power flow, or (High Voltage Direct Current) HVDC links are used to transfer energy between AC systems and large industrial consumers. At the distribution level, the increasing use of heat pumps, microwave ovens, personal computers, printers, televisions, compact fluorescent lamps and commercial premises represent a dramatic increase in small harmonic sources. Then, harmonic generating loads are widely present anywhere throughout power systems. Power quality problems like the harmonic pollution and resonance appear as a result of using such loads. Resonance is a reactive power exchange between capacitive and inductive components of the network at a given frequency. A power system should be controlled by the operator to retain its stability as well as energy quality. In fact, power systems can tolerate imposed  Corresponding author. Postal address: No. 32, Amani Street, North Sepahbod Gharani Avenue, Tehran 1598866515, Iran. Tel.: +98 912 2167727; fax: +98 21 88731293. E-mail addresses: [email protected], [email protected] (M. Esmaili), [email protected] (H. Ali Shayanfar), [email protected] (A. Jalilian).

0360-5442/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2008.05.005

harmonics as far as there is no resonance. In case of a resonance, a small non-linear load may lead to produce a great harmonic voltage, a harmful phenomenon causing more problems. The (Institute of Electrical and Electronics Engineers) IEEE 519 standard [1] has established guidelines for non-linear loads with limits on harmonic generation as well as some recommendations to measure, analyze, and control polluting loads. Harmonic treating techniques are divided into three distinctive procedures including filtering harmonic currents, improving the performance of non-linear loads, and making changes to the network structure or parameters. Making changes to power systems is more suitable than reducing harmonic distortions in large-scale systems with several non-linear loads [2]. Among the techniques for harmonic analysis, the frequency scan is the most commonly one used to characterize the response of a power system as a function of frequency [3]. At the bus of interest, the driving point impedance is obtained for the frequency range up to usually the 50th harmonic. The frequency scan analysis can identify the existence of possible resonances with their frequency. A Newton–Raphson method is proposed in Ref. [2] to shift poles and zeros to more suitable locations to solve harmonic problems. The pole and zero shifts are carried out by appropriate changes in the system parameters (e.g., capacitor or reactor banks). Eigenvalue sensitivity coefficients are used to determine the most cost-effective element changes as well as to compute Jacobian elements for the Newton method. Using eigenvalue analysis, the smallest eigenvalue of the admittance matrix is observed [4]. By means of the analysis, it is possible to determine changes of the smallest eigenvalue with

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frequency as well as changes of bus participation factors with resonance modes [5]. Therefore, the sensitivity of the critical eigenvalue can be determined with respect to the network parameters [6]. A modal sensitivity coefficient with one type of observability index are introduced in Ref. [7] to solve harmonic problems. It employs the modal analysis to compute the system poles, transfer function zeros as well as their sensitivities with respect to changes in the network parameters. To remedy harmonic effects, mitigation techniques including filters and relocating shunt capacitors are used. It is a critical task to locate shunt capacitors and filters so that a new resonance mode is not created. In Ref. [8], an approach is proposed to determine strategic buses for harmonic filter placement by correlating a new sensitivity index. The index is based on the inherent structure theory network and harmonic current injection at buses of the distribution network. Installing capacitors in distribution/transmission networks is a common practice. Where to locate the capacitors depends on the load modeling. The more practical load model, the more realistic solution for harmonic problems. Indeed, various scenarios are devised for resonance severity and location while different load models are used. The load models most commonly used in studies are conventional and (International Council on Large Electric Systems) CIGRE load models [9]. In this paper, the eigenvalue sensitivity analysis is implemented using different load models.

2. The problem formulation 2.1. Eigenvalue decomposition of harmonic admittance matrix The response of a power system to harmonic distortions can be evaluated by analyzing network admittance matrix at a given harmonic frequency. To study the behavior of the admittance matrix, it is lucrative to establish a framework by its eigenvalue and eigenvectors. Based on the idea elaborated in Ref. [6], the admittance matrix at the frequency f, not necessarily an integer value, can be factorized using eigenvalue decomposition as Y f ¼ Rf Lf Lf ,

(1)

f

f

f

where R is a matrix whose columns are right eigenvectors of Y , L is a diagonal matrix whose diagonal elements are eigenvalues of Yf, and Lf is a matrix whose rows are transposed left eigenvectors of Yf. If matrix Yf is symmetrical, a situation occurring when there is no phase-shifting transformer installed in the network, following equations can also be satisfied: Lf ¼ ðRf Þ1 ;

Lfk Rfk ¼ 1.

(2)

f

Three factors of Y as stated above also satisfy the following equations: Y f Rf ¼ Rf Lf ;

Lf Y f ¼ Lf Lf .

Z f ¼ Rf diag

1

! ðRf Þ1 .

lfi

(6)

Then, harmonic voltage developed at each bus as a result of harmonic nodal current injections is given by [8]: V f ¼ Rf diag

1

! Lf If .

lfi

(7)

The equation can be rewritten as Vf ¼

n X 1 i¼1

lfi

Rfi ðLfi ÞT If ,

(8)

where If is a vector containing harmonic current injections at buses with the frequency f. According to Eq. (8), the harmonic voltage produced at the bus associated with the smallest eigenvalue can be considerable even if its harmonic current injection is not so great. Then, the smallest eigenvalue is the most critical mode of the power system and modifying it could lead to improving the whole resonance behavior of the system. The sensitivity of eigenvalues, particularly the smallest one, with respect to (admittance matrix) Ybus entries is given by the following n  n sensitivity matrix [10]: SlðkÞ ¼ Rfk ðLfk ÞT ¼



qlk qY ij

T

.

In other words, the sensitivity of the kth eigenvalue with respect to Ybus entries is given by the product of the kth right eigenvector and transposed left eigenvector of Ybus. Bus participation factors are defined as the participation of buses to an individual resonance mode. The participation factor of bus i to the eigenvalue of lk is given by the entry of qlk =qY ii of the sensitivity matrix [5]. In next sections, the sensitivity of the most critical eigenvalues with respect to network parameters is derived.

2.2. Load modeling The variation of load impedances against harmonics is modeled using two most frequently used models. In Fig. 1, the conventional and CIGRE load models are shown. As shown in Fig. 1, impedances of two models vary differently with increasing harmonic frequency. In the conventional model, widely used in studies, the load is simply represented by a parallel R and X. In CIGRE model, suggested by CIGRE Working Group 36-05, a more detailed model is used. Using different load models leads to different admittance matrices. Therefore, the network will produce different responses for the two models.

(3)

To find the individual left eigenvector, right eigenvector, and eigenvalue, the following equations are used: Y

f

Rfk

¼

lfk Rfk ;

Lfk Y f

¼

lfk Lfk ,

(4)

where Rfk ; Lfk ; lfk represent the kth right eigenvector, kth left eigenvector, and kth eigenvalue of the matrix, respectively. Consequently, the symmetrical harmonic admittance and impedance matrix can be rewritten as f

Y ¼R

f

diagðlfi ÞðRf Þ1 ,

(9)

(5)

Xs = 0.073hR

R = U2/P

X = U2/(hQ)

Xp = R = U2/P

Conventional Load Model

CIGRE Load Model

Fig. 1. CIGRE and conventional load models.

hR 6.7(Q /P)–0.74

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previously derived to be

2.3. Eigenvalue sensitivity with respect to shunt components

qjlk j ¼ m; qjG

qjlk j ¼ n. qjB

(18)

By differentiating Eq. (3) with respect to a parameter like a and supposing the symmetrical Ybus, the sensitivity of the ith eigenvalue with respect to the parameter a can be written as

By applying the chain derivative rule:

qli qY f R. ¼ Lfi qa i qa

qjlk j mðX 2  R2 Þ þ 2nRX ¼ , qR jZj2

(19)

qjlk j 2mRX þ nðX 2  R2 Þ ¼ . qX jZj2

(20)

(10)

All values in Eq. (10) are in per unit. To normalize sensitivities, derivative values are divided by their absolute values as  qli  Dl =l Dl =Da qli =qa . (11) ¼ i i¼ i ¼ li =a li =a qa Norm: Da=a a!0

a!0

Normalized sensitivities describe relative changes of eigenvalue in terms of relative changes of network parameters with respect to their absolute values. Shunt components are added to diagonal entries of the admittance matrix, so it is possible to write the diagonal entry in terms of Ysh, the admittance of the capacitor to which eigenvalue sensitivity is interested: Y ii ¼ Y sh þ Y non-sh ,

(12)

where Ysh and Ynon-sh represent shunt and non-shunt components of Yii. Using Eq. (10), it can be written as qlk qY qY qY ii ¼ Lk R ¼ Lk R ¼ Lki Rki ¼ SlðkÞii . qY sh k qY ii qY sh k qY sh

(13)

On the other hand, it is more interested to obtain the sensitivity for magnitude rather than complex values of network parameters. Suppose: SlðkÞii ¼ Sr þ jSi ; lk ¼ lr þ jli ;

Eqs. (19) and (20) give the sensitivity of an eigenvalue, particularly the critical one, with respect to R and X of the series impedance. The normalized values can be used rather than absolute values.

3. Numerical results The described methods have been examined on the New Zealand and IEEE-30 bus test systems. Sub-transient reactance of generators is assumed to be 0.25 per unit (p.u.). It is worthwhile to note that the error and accuracy of the software implemented to perform simulations are verified through testing it on the sample test systems used in Refs. [6,7]. Results of tests regarding conventional load models are in accordance with the ones presented in these references. After approval, the CIGRE load model is also added to compare the effect of load models on harmonic resonance. 3.1. New Zealand test system

Y sh ¼ G þ jB, F ¼ jlk j2 .

(14)

Then: qF ¼ 2ðSr li  Si lr Þ; qB

qF ¼ 2ðSr lr þ Si li Þ. qG

(15)

By applying the chain derivative rule: qjlk j Sr li  Si lr ¼ ; qB jlk j

qjlk j Sr lr þ Si li ¼ . qG jlk j

(16)

Then, the sensitivity of eigenvalue magnitude with respect to shunt parameters is obtained. 2.4. Eigenvalue sensitivity with respect to series components To obtain the sensitivity of an eigenvalue with respect to the admittance of a series component, like a line or a transformer, it can be written as

The test system, a well-known small practical power network, includes 17 buses, 20 transmission lines, 6 transformers, and an HVDC converter modeled as a PQ load. The single line diagram is shown in Fig. 2. This test system, which is really the New Zealand South Island’s network, has voltage levels of 11, 14, 16, and 220 kV and its data can be found in Ref. [11]. To investigate the harmonic behavior of the system, shunt capacitors are added to buses of ROXB-011, MANAP014, AVIEM011, and OHAUSYST, so that they compensate generators reactive power to meet the unity power factor. Driving point impedance for the New Zealand test system using conventional load model is shown in Table 1. As seen in the table, the network presents resonance behavior at some buses when 1 p.u. current with varying frequency is injected into buses. All results are in the per unit frequency defined as h ¼ f/f0, where f0 is the system rated frequency in Hz. Indeed, per unit frequency represents the harmonic order.

qlk qY ¼ Lk R qY se k qY se   qY qY ii qY qY jj qY qY ij qY qY ji ¼ Lk þ þ þ R qY ii qY se qY jj qY se qY ij qY se qY ji qY se k   qY qY qY qY ¼ Lk þ   R . qY ii qY jj qY ij qY ji k qlk ¼ SlðkÞii þ SlðkÞjj  SlðkÞij  SlðkÞji . qY se

(17)

According to Eq. (17), the sensitivity of an eigenvalue with respect to the admittance of series admittance between buses i– j is determined as a combination of four entries of the sensitivity matrix associated with buses i and j. To obtain the sensitivity of eigenvalue magnitude with respect to series impedance of Zse ¼ R+jX ¼ 1/Yse ¼ 1/(G+jB), assume the values obtained for the series admittance parts as the shunt one

Fig. 2. Single line diagram of the New Zealand test system.

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Table 1 Driving point impedances for the New Zealand test system Conventional load model

CIGRE load model

Bus

Frequency (p.u.)

Impedance (p.u.)

Bus

Frequency (p.u.)

Impedance (p.u.)

AVIEM220 TEKAP220 TEKAP011 MANAP220 OHAUSYST

45.65 27.4 27.4 14.1 4.4

109.1 21.1 14.0 7.7 3.7

TEKAP220 TEKAP011 AVIEM220 OHAUSYST BENMO016

27.4 27.35 40.4 4.3 39.25

14.41 9.68 4.58 2.27 2.01

60

Modal Impadance mag (p.u.)

50

40

30

20

10

0

5

10

15

20 25 30 35 Frequency (p.u.)

40

45

50

Fig. 3. Modal impedances for the New Zealand test system using conventional load model.

60

50 Modal Impadance mag (p.u.)

In Table 1, bus AVIEM220 elaborates the worst resonance of all buses while a harmonic current source of 1 p.u. is injected there. That is, equivalent Norton impedance seen from bus AVIEM220 resonates at h ¼ 45.65. In other words, the harmonic voltage produced at that bus is 109.1 times as much as the harmonic current injection at that bus. This could be expected since there is a resonance path between the capacitor, generator, and transformer installed at that bus. Inasmuch as buses TEKAP220 and TEKAP011 are connected to each other, they present a unique resonance frequency of 27.4 p.u. Despite the fact that these two buses have considerable resonance impedances, they are less critical compared to the first ones. Other driving impedances are far less than the first ones. On the other hand, results are different if CIGRE load model is applied as shown in Table 1. Using this load model results in a less resonance impedance of 14.41 p.u. compared to 109.1 p.u. This means that the conventional load model gives resonance modes more severe than the CIGRE load model; indeed, the conventional load model may lead the operator to use extra shunt MVArs. Results of frequency scan analysis for the New Zealand test system is shown in Fig. 3 using the conventional load model. The figure depicts the variation of all admittance matrix eigenvalues versus frequency. Using CIGRE load model, the frequency scan results are depicted in Fig. 4. As seen graphically in the figure, resonance peaks are greatly reduced compared to those of the conventional load model. As shown in Figs. 3 and 4, there are a few dominant resonance modes for the test system. That is, some eigenvalues closes to the origin at the complex x–y plane at some frequencies. Numerical details for the critical resonance modes and participation factor of buses for both load models are presented in Table 2. All values shown in the table are in p.u. Using conventional load model, the most critical resonance frequency in Table 2 is determined as 45.65 p.u. having eigenvalue of l15 ¼ 9.141 103 p.u. In other words, the modal impedance for this mode is Zm15 ¼ 1/l15 ¼ 109.4 p.u. The most participating bus for this resonance mode is AVIEM220, where there is the largest sensitivity to the eigenvalue. In other words, from the bus participation viewpoint, bus AVIEM220 is probably the best location where remedial actions can be performed to mitigate the resonance. On the contrary, bus TEKAP220 is the best place to alleviate the resonance if the CIGRE load model is used. Moreover, the critical resonance frequency is 27.40 p.u. in this case. To do an error analysis, buses identified as the best place to mitigate resonance are obtained similarly by methods of driving point impedance and participation factor. That is, buses with large driving impedance have also a great participation factor on the harmonic. Using the CIGRE load model, the third mode resonates at the frequency of 4.30 p.u. with the modal impedance of 1/2.044  101 ¼ 4.89 p.u. The resonance frequency of 4.3 p.u. is close to the

40

30

20

10

0

5

10

15

20 25 30 35 Frequency (p.u.)

40

45

50

Fig. 4. Modal impedances for the New Zealand test system using CIGRE load model.

fifth harmonic which is probably exits throughout the network. This means that the little fifth harmonics could be amplified as much as about 4.98 times and the most critical bus is OHAUSYST located on the southeast of the power system. Hence, attention should be paid to prevent such a phenomenon. For instance, one of preventive actions could be to install a notch filter tuned at the fifth harmonic.

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Table 2 The most critical resonance modes and bus participation factors of the New Zealand test system Load model

Resonance

Critical eigenvalue

The largest bus PF

Most participating bus

Mode

Frequency

Conventional

15 14 13

45.65 27.40 10.70

9.141  103 2.722  102 8.604  102

0.99746 0.57434 0.38246

AVIEM220 (9) TEKAP220 (16) TEKAP220 (16)

CIGRE

15 14 12

27.40 27.65 4.30

3.910  102 8.473  102 2.044  101

0.56035 0.59874 0.43873

TEKAP220 (16) TEKAP220 (16) OHAUSYST (11)

Table 3 The largest sensitivity of eigenvalues with respect to Y entries for the New Zealand test system Conventional load model

CIGRE load model

Frequency (p.u.)

Largest sensitivity

Ybus entry (i, j)

Frequency (p.u.)

Largest sensitivity

Ybus entry (i, j)

45.65 27.40 10.70

0.99746 0.57434 0.38246

(9, 9) (16,16) (16,16)

27.40 27.65 4.30

0.56035 0.59874 0.43873

(16,16) (16,16) (11,11)

Table 4 The most influencing parameters to mitigate critical resonance modes of the New Zealand test system using eigenvalue sensitivity analysis with conventional load model Eigenvalue sensitivity (p.u.)

Table 5 The most influencing parameters to mitigate critical resonance modes of the New Zealand test system using eigenvalue sensitivity analysis with CIGRE load model Eigenvalue sensitivity (p.u.)

Resonance frequency (p.u.) 45.65

27.40

10.70

qli =qGj

0.99712 AVIEM220 (9)

0.57233 TEKAP220 (16)

0.38246 TEKAP220 (16)

qli =qBj

0.026137 AVIEM220 (9)

0.048065 TEKAP220 (16)

0.022227 TEKAP011 (15)

qli =qRj

839.31 LIVIN220AVIEM220

8581.2 TWIZE220TEKAP220

507.62 TWIZE220OHAUSYST

qli =qX j

452.02 LIVIN220AVIEM220

2796.5 TWIZE220TEKAP220

117.33 TWIZE220OHAUSYST

The largest sensitivity of critical eigenvalue with respect to Ybus entries, qlk =qY ij , is shown in Table 3. As seen in Table 3, the largest sensitivity occurs at the same diagonal entries of admittance matrix as determined by the bus participation factors. According to results, with conventional load model, the most critical eigenvalue has the largest sensitivity to the entry (9, 9) of Ybus. In other words, it is more beneficial to change the (9, 9) entry of Ybus corresponding to bus AVIEM220 to mitigate this resonance mode. However, the entry of (16,16) corresponding to bus TEKAP220 becomes the most efficient bus to alleviate the critical resonance if the CIGRE load model is used. In Table 4, the most sensitive parameters of the network to alter critical eigenvalues are presented using the conventional load model. Results shown in the table are obtained using the sensitivity of eigenvalue with respect to G and B of shunt components and R and X of lines and transformers. It is worthwhile to note that scarcely does the system operator change transformers or lines to mitigate the resonance problems. Also, shunt devices usually do not have any conductance.

Resonance frequency (p.u.) 27.40

27.65

4.30

qli =qGj

0.55688 TEKAP220 (16)

0.27413 TEKAP220 (16)

0.43855 OHAUSYST (11)

qli =qBj

0.06224 TEKAP220 (16)

0.5323 TEKAP220 (16)

0.13355 AVIEM011 (10)

qli =qRj

8344.7 TWIZE220TEKAP220

7104.1 TWIZE220TEKAP220

117.67 TWIZE220OHAUSYST

qli =qX j

2512.1 TWIZE220TEKAP220

5890.7 TWIZE220TEKAP220

29.55 AVIEM220AVIEM011

Consequently, the most appropriate parameter to control the resonance modes is the susceptance of shunt equipment including capacitors, reactors, and filters. According to Table 4, the most efficient bus to control the first resonance mode is AVIEM220 bus with a sensitivity of 0.026137. This means that the resonance is deteriorated with increasing the capacitor installed there. The second resonance mode is improved by adding capacitors to bus TEKAP220. The first two resonance frequencies of h ¼ 45.65 and 27.40 are far from possible resonances of the system. However, the third one is close to the 11th harmonic which can present in systems. Then, the operator should take care of this mode more than the others. Its sensitivity is about that of the first mode and the resonance is improved with decreasing susceptance at bus TEKAP011. In Table 5, the most influencing parameters of the network to alleviate resonances are shown using the CIGRE load model. As a generally accepted fact, it is lucrative to increase the resistance of a circuit to greatly reduce resonance peaks. This is verified by large and positive sensitiveness to resistances in

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Table 5. However, as noted previously, we focus only on the susceptance in Table 5. Of three resonance modes obtained in Table 5, the third one has a frequency close to the fifth harmonic which may widely present throughout the network. Therefore, it is a vital task for the operator to mitigate this mode as compared with others. The task is possible by increasing the shunt capacitor at generator bus AVIEM011. By comparing Tables 4 and 5, it can be concluded that sensitivities to the shunt susceptance is greater if CIGRE load model is used. That is, less capacitor/inductor banks are needed using the CIGRE load model compared to the conventional model. In addition, it is also possible to control the resonance by installing series compensators, as for example FACTS devices, to change series reactance in the network. As seen from Tables 4 and 5, this way is more efficient than shunt components because of higher sensitivities with respect to series reactance. However, it is worthwhile to note that installing a series 1MVAr device costs more than a shunt 1MVAr device. To check the accuracy of the methods, as seen from Tables 4 and 5, bus AVIEM220 is identified as the best place to install a reactive bank. Also, line LIVIN220-AVIEM220 is the most effective one to install FACTS compensators. As seen, both ways detect bus AVIEM220 as a solution point for harmonic h ¼ 45.65. By looking at the results obtained by criteria of driving point impedance, bus participation factor, and eigenvalue sensitivities, it is possible to compare resonance-alleviating buses recommended by each criterion. The first two criteria only give appropriate buses, while the last one gives both buses and direction of changing eigenvalue with shunt’s susceptance. The summery of all criteria is shown in Table 6. Using the conventional load model, bus AVIEM220 in Table 6 is selected by all three criteria as the best place to mitigate the most critical resonance. For the second mode, all criteria propose the same bus, too. For the third mode, buses TEKAP220 and TEKAP011, connected to each other, are selected as the best buses to alleviate the resonance. Using the CIGRE load model, for the third mode, which is more close to existing harmonics, two first criteria propose bus OHAUSYST, located at the left-bottom of the single line diagram, as the most efficient bus to control the resonance. On the other hand, the sensitivity analysis proposes bus AVIEM011 located at the center of the single line diagram. That is, for both load models, first two methods give similar results; however, those of the sensitivity analysis are a bit different. 3.2. IEEE-30 bus test system The IEEE-30 bus test system originally represents a portion of the American Midwestern Electric Power System as of December 1961 with the voltage levels of 11, 33, and 132 kV. The system, whose single line diagram is shown in Fig. 5, is composed of 30 buses, 6 generators, 5 transformers, and 36 branches [12]. Table 6 Buses to mitigate resonance modes proposed by three criteria Load model

Frequency (p.u.)

Driving impedance

Participation factor

Eigenvalue sensitivity

Conventional 45.65 27.40 10.70

AVIEM220 TEKAP220 –

AVIEM220 TEKAP220 TEKAP220

AVIEM220 TEKAP220 TEKAP011

CIGRE

TEKAP220 – OHAUSYST

TEKAP220 TEKAP220 OHAUSYST

TEKAP220 TEKAP220 AVIEM011

27.40 27.65 4.30

Fig. 5. Single line diagram of the IEEE-30 bus test system.

In Table 7, driving point impedance is shown for the IEEE-30 bus test system. As seen, bus CLOVERDLE132 presents the largest modal impedance of 50.32 p.u. at h ¼ 22.8 if the conventional load model is used. Nevertheless, the largest modal impedance of 26.24 p.u. at h ¼ 44.25 occurs at bus KUMIS132 if the CIGRE load model is applied. This verifies the previous network results as to the conventional load model gives resonance modes more conservatively than the CIGRE model. In Table 8, the most critical resonance modes with bus participation factors are shown. Bus ROANOKE132 is determined as the most critical mode which resonates at h ¼ 38.3 as well as h ¼ 38.6. As seen in Fig. 5, bus ROANOKE132 is located at a place with the most number of transformers and connections with other buses. Also, a 19 MVAr shunt capacitor is installed at its near bus of ROANOKE33. These features make ROANOKE132 a good candidate for a resonance. Nevertheless, the resonance harmonic order is so high that it is far enough from risky harmonics existing in a power system. The other resonance occurs at h ¼ 22.8 with the most participating bus of CLOVERDLE132. This bus is directly connected to bus ROANOKE132. Then, the potential resonance portion of the network consists of ROANOKE132, CLOVERDLE132, and their connection. On the other hand, in Table 8, the worst harmonic resonance is determined at bus KUMIS132 if the CIGRE load model is used. With the CIGRE load model, dominant eigenvalues are less critical than the conventional load model as it is shown in the previous test system. In Table 9, the sensitivities of network parameters with respect to the dominant resonance modes are presented when the conventional load model is used. As elaborated previously, inasmuch as altering shunt MVAr is more practical in a power system than other parameters of the table, the sensitivity of qli =qBk is focused. A negative value for the first mode of this sensitivity means that the resonance can be mitigated by adding reactor banks at bus ROANOKE11. The resonance happens because of the capacitor bank at ROANOKE132, which is directly connected to ROANOKE11. To mitigate the resonance for two other modes, capacitor banks can be installed at CLOVERDLE132. Consequently, the operator can shift some parts of the capacitor bank from ROANOKE132 to CLOVERDLE132 to enhance all of the three resonance modes. Table 10 shows sensitivities when the CIGRE load model is applied. According to results, installing reactors is a remedy for resonances. Of course, the resonance harmonic order is so high that there is no need to take any action in view of the fact that

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Table 7 Driving point impedances for IEEE-30 bus test system Conventional load model

CIGRE load model

Bus

Frequency (p.u.)

Impedance (p.u.)

Bus

Frequency (p.u.)

Impedance (p.u.)

CLOVERDLE132 ROANOKE132 ROANOKE11 HANCOCK132 ROANOKE1

22.8 38.3 50.0 26.3 38.2

50.32 34.18 14.13 8.46 6.59

KUMIS132 GLENLYN132 CLOVERDLE132 Bus 26 BLAINE132

44.25 28.20 32.95 50 36.7

26.24 25.82 22.02 20.68 17.57

Table 8 The most critical resonance modes and bus participation factors of IEEE-30 bus test system Load model

Resonance

Critical eigenvalue

The largest bus PF

Most participating bus

Mode

Frequency

Conventional

14 15 13

38.30 22.80 38.60

1.215  102 1.224  102 2.476  102

0.41524 0.61465 0.42892

ROANOKE132 CLOVERDLE132 ROANOKE132

CIGRE

18 19 20

44.25 45.30 45.05

1.653  102 1.766  102 2.306  102

0.3863 0.28496 0.27236

KUMIS132 KUMIS132 KUMIS132

Table 9 The most influencing parameters to mitigate critical resonance modes of IEEE-30 bus test system using eigenvalue sensitivity analysis with conventional load model

Table 10 The most influencing parameters to mitigate critical resonance modes of IEEE-30 bus test system using eigenvalue sensitivity analysis with CIGRE load model

Eigenvalue sensitivity (p.u.)

Eigenvalue sensitivity (p.u.)

Resonance frequency (p.u.) 38.30

22.80

38.60

qli =qGj

0.41523 ROANOKE132

0.61274 CLOVERDLE132

0.2157 ROANOKE132

qli =qBj

0.019945 ROANOKE11

0.048455 CLOVERDLE132

qli =qRj

363.42 ROANOKE132CLOVERDLE132

qli =qX j

213.37 ROANOKE132CLOVERDLE132

Resonance frequency (p.u.) 44.25

45.30

45.05

qli =qGj

0.38535 KUMIS132

0.2839 KUMIS132

0.19792 KUMIS132

0.37228 CLOVERDLE132

qli =qBj

0.027167 KUMIS132

0.084928 REUSENS132

0.18711 KUMIS132

50.659 ROANOKE132REUSENS132

385.4 ROANOKE132CLOVERDLE132

qli =qRj

424.89 KUMIS132HANCOCK132

345.85 KUMIS132HANCOCK132

59.992 KUMIS132HANCOCK132

25.432 ROANOKE132REUSENS132

213.74 ROANOKE132CLOVERDLE132

qli =qX j

399.14 KUMIS132HANCOCK132

337.01 KUMIS132HANCOCK132

459.43 KUMIS132HANCOCK132

hardly ever can higher harmonics be found in a power system. That the high frequency resonances are alleviated by adding inductors leads us to this result that the reactor portion of the CIGRE load model is decayed at higher frequencies. This can be verified through Fig. 1. In the CIGRE load model of Fig. 1, XS and XP increase with harmonic order and lead to a less inductive load. However, in the conventional load model the inductive part increases with the harmonic order. In Table 11, summary of the most appropriate candidates recommended by evaluated criteria to mitigate resonance is listed. As seen, for the first mode, the bus ROANOKE132 is proposed by two first criteria, where as its adjacent ROANOKE11 is proposed by the sensitivity analysis. For the second vital mode, all criteria are in agreement to propose CLOVERDLE132 as the best place to alleviate resonance. For the third mode, buses ROANOKE132 and CLOVERDLE132 directly connected to each other are the best candidates. On the other hand, if the CIGRE load model is used, bus KUMIS132, as proposed by all criteria in Table 11, is the most efficient place to add reactors to remedy harmonics. As a

Table 11 Buses to mitigate resonance modes proposed by three criteria for IEEE-30 bus test system Load model

Frequency (p.u.)

Driving impedance

Participation factor

Eigenvalue sensitivity

Conventional 38.30 22.80 38.60

ROANOKE132 ROANOKE132 ROANOKE11 CLOVERDLE132 CLOVERDLE132 CLOVERDLE132 – ROANOKE132 CLOVERDLE132

CIGRE

KUMIS132 – –

44.25 45.30 45.05

KUMIS132 KUMIS132 KUMIS132

KUMIS132 REUSENS132 KUMIS132

result from both load models, two first criteria propose similar set of buses, whereas the sensitivity proposes a slightly different area. As a result from two test systems tested in this paper, the load model changes the place where harmonics should be alleviated.

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M. Esmaili et al. / Energy 33 (2008) 1361–1368

According to the results from both test systems, the conventional load model gives resonance modes more conservatively than the CIGRE model. Therefore, using the conventional load models results in more conservative harmonic modes, a phenomenon which makes the system operator use unnecessary amounts of capacitor/reactor banks. Inasmuch as conventional load model is usually used in most harmonic studies because of its simplicity, it is necessary to use more realistic load models to follow the real power system harmonics behavior despite of a rather complexity. The methods of driving point impedance and bus participation factors give only the place where the resonance can be mitigated. Despite the fact that the sensitivity analysis is a little more complex and requires more computational burden, it gives more detailed and accurate results. Indeed, the sensitivity method gives the operator not only where the resonance should be mitigated but also how much changes in the network parameters is required to mitigate the resonance to a satisfactory level.

4. Conclusion Where and how to mitigate vital resonances in a power system is the aim of this paper. Buses which can potentially produce large harmonics are detected using the implemented criteria. The first criterion of driving point impedance gives resonance modes with their modal impedance. In the second criterion, participation factors give the most participated buses for a resonance mode. The third criterion of sensitivity gives the sensitivity of critical harmonic modes with respect to power system parameters. Buses with a high value of sensitivity are selected as the best place to mitigate resonance. All methods are employed to propose the best buses to mitigate critical resonance modes. The performance of all methods are evaluated under conventional and CIGRE load models. Two test systems, namely New Zealand and IEEE-30 bus, are used to test the efficiency of the proposed methods to detect and alleviate potentially critical buses. According to the results, all methods lead to a similar set of buses to alleviate resonance if the conventional load model is used. However, if the CIGRE load model is used, the results of eigenvalue sensitivity are slightly different. As a result, sensitivity analysis, giving the sensitivity of the dominant eigenvalue to the actual system parameters, detects the mitigation area more precisely than the others. Furthermore, using the conventional load model leads to more severe resonances than the CIGRE load model. Indeed, the conventional load model makes the system operator use extra amounts of capacitor/reactor banks to mitigate the resonance. References [1] Transmission and Distribution Committee of the IEEE Power Engineering Society, USA. IEEE recommended practices and requirements for harmonic control in electrical power systems. IEEE Standard 519-1992. E-ISBN: 0-73810915-0.

[2] Varricchio SL, Martins N, Lima LTG. A Newton–Raphson method based on eigenvalue sensitivities to improve harmonic voltage performance. IEEE Trans Power Deliv Jan. 2003;18(1):334–42. [3] IEEE Task Force on Harmonics Modeling and Simulation. Modeling and simulation of the propagation of harmonics in electric power networks—Part II: Sample systems and examples. IEEE Trans Power Deliv Jan. 1996;11(1): 466–74. [4] Osowski S. SVD technique for estimation of harmonic components in a power system, a statistical approach. IEE Gen Trans Distrib 1994;141(5):473–9. [5] Xu W, Huang Z, Cui Y, Wang H. Harmonic resonance mode analysis. IEEE Trans Power Deliv Apr. 2005;20(2):1182–90. [6] Huang Z, Cui Y, Xu W. Application of modal sensitivity for power system harmonic resonance analysis. IEEE Trans Power Syst Feb. 2007;22(1):222–31. [7] Varricchio SL, Gomes Jr. S, Martins N. Modal analysis of industrial system harmonics using the s-domain approach. IEEE Trans Power Deliv Jul. 2004; 19(3):1232–7. [8] Teng Au M, Milanovic JV. Planning approaches for the strategic placement of passive harmonic filters in radial distribution networks. IEEE Trans Power Deliv Jan. 2007;22(1). [9] CIGRE Working Group 36-05 (Disturbing Loads). Harmonics, characteristic parameters, methods of study, estimates of existing values in the network. Electra 1981;77:35–54. [10] Caramia P, Russo A, Varilone P. The inherent structure theory of network for power quality issues. In: Proceedings of the IEEE power engineering society. Winter Meeting, Columbus, OH, 2001. [11] Pai MA. Computer techniques in power system analysis. Norwell: Kluwer Academic Publishers; 1986. [12] /http://www.ee.washington.edu/research/pstca/S. Masoud Esmaili was born in Sarab, Iran, in 1972. He received B.S. and M.S. degrees in Electrical Engineering from Tabriz and IUST universities, respectively. He is currently a Ph.D. student at the Iran University of Science and Technology. His research area includes power system studies particularly security, restructuring, dynamics, and power quality.

Heidar Ali Shayanfar received the B.S. and M.S.E. degrees in Electrical Engineering in 1973 and 1979, respectively. He received his Ph.D. degree in Electrical Engineering from Michigan State University, USA, in 1981. Currently, he is a Full Professor in Electrical Engineering Department of Iran University of Science and Technology, Tehran, Iran. His research interests are in the Application of Artificial Intelligence to Power System control design, Dynamic Load Modeling, Power System Observability studies, power quality, and Voltage Collapse. He is a member of Iranian Association of Electrical and Electronic Engineers and IEEE. Alireza Jalilian was born in Yazd, Iran, in 1961. He obtained his B.Sc. in electrical engineering from Iran in 1989, and his M.Sc. and Ph.D. from Australia in 1992 and 1997, respectively. He joined the power group of the department of Electrical Engineering of Iran University of Science and Technology in 1998. His research interests are power quality causes, effects and mitigations.