ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system Hamed Hasanvand a,n, Babak Mozafari a, Mohamad R. Aavan b, Turaj Amraee c a

Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Electrical Engineering, Malek-e-Ashtar University of Technology, Tehran, Iran c Department of Electrical and Computer Engineering, K.N. Toosi University of Technology, Tehran, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 28 January 2015 Received in revised form 15 June 2015 Accepted 5 September 2015

This paper addresses the application of a static Var compensator (SVC) to improve the damping of interarea oscillations. Optimal location and size of SVC are defined using bifurcation and modal analysis to satisfy its primary application. Furthermore, the best-input signal for damping controller is selected using Hankel singular values and right half plane-zeros. The proposed approach is aimed to design a robust PI controller based on interval plants and Kharitonov's theorem. The objective here is to determine the stability region to attain robust stability, the desired phase margin, gain margin, and bandwidth. The intersection of the resulting stability regions yields the set of kp–ki parameters. In addition, optimal multiobjective design of PI controller using particle swarm optimization (PSO) algorithm is presented. The effectiveness of the suggested controllers in damping of local and interarea oscillation modes of a multimachine power system, over a wide range of loading conditions and system configurations, is confirmed through eigenvalue analysis and nonlinear time domain simulation. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Interarea oscillations Damping controller Hankel singular values RHP-zeros Polynomial control

1. Introduction Low frequency oscillations have been observed in the interconnected power systems and impede the purposes of maximum power transmission and optimal power system security. These oscillations may result in serious consequences, such as tripping of the generators and initiating major system blackouts. One of the most important oscillations in the frequency range of 0.1–2 Hz involves many generators in the interconnected system, commonly known as inter-area oscillations [1]. The stability of a power system can be improved by applying an additional feedback control loop in the automatic voltage regulator (AVR) system of selected generating units. This additional controller is called a power system stabilizer (PSS). During the recent years, the fast progresses in the field of power electronics have opened new opportunities for the application of the Flexible AC Transmission System (FACTS) devices as one of the most effective ways to improve both power system controllability and power transfer limits. Shunt FACTS controllers are used in transmission network so as to provide dynamic voltage control and to improve power flow control [2]. As a promising FACTS device, static Var n

Corresponding author. E-mail addresses: [email protected], [email protected] (H. Hasanvand).

compensator (SVC) is applied to control voltage at the point of connection to the grid, and to maintain bus voltage approximately near to a constant level. In addition to voltage control, if a supplementary damping controller (SDC) is implemented, the SVC could be utilized for damping of electromechanical oscillations. As known, the SDC input signal must be opted accurately to be effective in damping of oscillations. A Residue method is widely exploited based on controllability/observability indices to select the input signal and location of FACTS devices [3]. A technique based on Hankel singular value (HSV) and right half plane zeros (RHP-zeros) for input signal selection has been presented in [4,5]. The RHP-zeros technique investigates various input-output combinations of transfer function zeros in both pre-fault and post-fault conditions whereas the HSV method uses the concept of joint controllability and observability indices. In [6], the relative residue error covariance matrix is calculated for each signal, and then the best stabilization signal associated with the smallest relative residue error covariance matrix is opted. In the present paper, the proper stabilization input signal for the SDC is selected using the HSV and RHP-zeros. The main problem encountered in the SDC design is that power systems undergo frequent changes in operating conditions. These changes results from variations in the power consumption, generation and transmission device structure, network configuration, and the number of operating generation units. The continuous variation in operating points is the source of structured

http://dx.doi.org/10.1016/j.isatra.2015.09.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

uncertainties in power systems [7]. Robust control theory is a powerful tool that is applied to cope with the uncertainties into account at the controller design process, because such uncertainties are not systemically considered by conventional control methods [8,9]. A multi-objective mixed H 2 =H 1 output feedback control with the regional pole placement to design a wide area robust damping controller has been represented in [3,10]. Application of interval polynomials and Kharitonov's theorem in design of a robust PSS to mitigate low frequency oscillations in the single machine infinite bus and 4-machine-2-area test systems has been presented in [7,11,12]. Soliman (2014) [13] has presented a simple analytical method for computing the set of robust PID and lead– lag controllers based on three-term stabilizing PSSs for a single machine infinite bus test system. Although the results are reliable and the designed PSS can stabilize the power system under a wide range of parametric uncertainties, the test systems are generally easy. In [14–16], application of Kharitonov's theorem has been reported for designing the FACTS devices based on SDC. In [15], Kharitonov's theorem and linear matrix inequality (LMI) approach are applied to characterize a fixed order robust controller for satisfying desired damping. The results in the aforementioned paper demonstrate that the Kharitonov's polynomial method is useful in synthesizing robust and simple power oscillation damping control. In [16], authors use Kharitonov's theorem for designing a robust SVC-SDC that would be able to mitigate electromechanical oscillations in 4-machine-2-area test system. The system under test is very simple and only the load variation is considered as a source of uncertainty. In this paper a robust PI controller considering robust stability and robust performance (desired Gm, Pm, and bandwidth) by illustrating the boundary locus of kp–ki plane has been designed. The proposed method is simple and straightforward for the PI controller design. The effectiveness of the proposed controller in damping of interarea oscillations has been tested by the eigenvalue analysis and nonlinear time-domain simulation for a large-scale power system.

2. Kharitonov's stability theory During the recent decades, remarkable results regarding the stability analysis of uncertain systems have been presented [17,18]. The Kharitonov's theorem provides a necessary and sufficient analysis test for determining the robust stability of polynomials with the perturbed coefficients [19]. The characteristic equation of uncertain system is as follows: PðsÞ ¼

n X

ax s x

; ax r ax rax

ð1Þ

x¼0

where the “_’’ and ‘‘ ’’ show the minimum and maximum bounds of the polynomial coefficients, respectively. The overall system is stable, if and only if the four following Kharitonov's polynomials are Hurwitz. P 1 ðsÞ ¼ a0 þ a1 s þ a2 s2 þ a3 s3 þ a4 s4 þ a5 s5 þ⋯ P 2 ðsÞ ¼ a0 þ a1 s þ a2 s2 þ a3 s3 þ a4 s4 þ a5 s5 þ ⋯ P 3 ðsÞ ¼ a0 þ a1 s þ a2 s2 þ a3 s3 þ a4 s4 þ a5 s5 þ ⋯ P 4 ðsÞ ¼ a0 þ a1 s þ a2 s2 þ a3 s3 þ a4 s4 þ a5 s5 þ ⋯

Fig. 1. Uncertain plant with controller.

ð2Þ

Fig. 1 shows a single-input single-output (SISO) control system. The plant is represented using transfer function GðsÞ ¼ NðsÞ=DðsÞ, which the numerator and denominator are presented by: NðsÞ ¼

m X

nx sx ; nx r nx r nx ;

DðsÞ ¼

x¼0

n X

dy sy ; dy r dy r dy

ð3Þ

y¼0

A set of sixteen Kharitonov transfer functions can be expressed based on the four Kharitonov polynomials as follows: Gxy ðsÞ ¼

N x ðsÞ ; Dy ðsÞ

x; y ¼ 1; 2; 3; 4

ð4Þ

The fixed parameter PI controller can be assumed to be as : CðsÞ ¼ kp þ ðki =sÞ ¼ ðkp s þki Þ=s. The closed loop system presented in Fig. 1 with the PI controller is robustly stable if only if the sixteen characteristic polynomials of the following set are stable [16]: F xy ¼ Dy ðsÞs þ ðkp s þ ki ÞN x ðsÞ;

x; y ¼ 1; 2; 3; 4

ð5Þ

3. Problem statement Fig. 2 shows the 68-bus interconnected New York power system and New England test system (NYPS-NETS) [8]. The first eight machines have slow excitation system whereas the ninth one has a fast excitation system equipped with a conventional PSS. All system generators are considered using six-order model. Loads are constant power (CP) loads. The system equations and data are presented in [20] and [8]. 3.1. Optimal location and size of SVC Voltage stability is the capability of a power system to maintain the voltages at all system buses within an acceptable range under normal operating condition and following a disturbance [5]. In this paper, a technique based on modal analysis associated with bifurcation theory is used for analysis of voltage stability. In this technique, using continuation power flow (CPF) method [1,5], the system loads are gradually increased in order to reach the voltage collapse point because of the lack of steady-state solutions arises from system controls reaching limits (e.g., generator reactive power limits). This case is known as limit-induced bifurcations. Then, near this point, the modal analysis is carried out and the weakest system bus according to the eigenvalues of the reduced Jacobian matrix is defined [5]. The main conclusion from this method is that power system cannot support any combination of reactive power demand [5]. Based on QV analysis, the system is voltage stable if all the eigenvalues of Jacobian matrix are positive, and voltage unstable if at least one of the eigenvalues is negative. The smaller the magnitude of the eigenvalue, the closer the corresponding modal voltage is to being voltage unstable [1]. If the eigenvalue is zero the system is on the verge of voltage instability. Therefore, the most associated bus to the worst eigenvalue (smallest eigenvalue) is the best candidate for compensation. Table 1 shows the worst eigenvalues and associated buses of the test system. As can be seen in Table 1, since the eigenvalue associated with bus 64 (λ ¼1.7209) is the smallest eigenvalue, therefore, bus 64 is the best location for installing SVC to the grid. After determining the best location of SVC, its optimal size will be calP culated through minimizing the defined objective function nbus n¼1 3 abs ðV n  1Þ using PSO algorithm. The main objective is to bring the voltage of all buses approximately near 1 pu. The Vn is voltage of the nth bus and nbus is total number of the system buses. In the proposed objective function, the small voltage deviation becomes negligible while large deviations become relatively larger [5]. Therefore, this objective function will check the improvement of the voltage, only at critical buses resulting from the placement of

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

3

Fig. 2. NYPS-NETS power system model.

Table 1 The worst eigenvalues and associated buses. Eigenvalue no.

Most associated bus

Real part

Imaginary part

41 42 43 44 45 46 47 48 49 50

64 39 40 28 49 28 39 05 50 08

1.7209 4.7002 7.2893 8.6396 9.6528 10.9657 17.2037 18.1191 21.52 25.42

0 0 0 0 0 0 0 0 0 0

combination of the input and output contains more information on the system internal states, one possible approach is to evaluate observabillity and controllability indices, such as HSV that reflects the joint controllability–observability of the states [22]. The HSV σ i can be calculated as follows: 

σ i ¼ λi ðPQ Þ

1=2

where the controllability and observability Gramian matrices (P and Q respectively) are computed by solving the following Lyapunov equations [22]: AP þPAT þ BBT ¼ 0 AT Q þ Q A þ C T C ¼ 0

Voltage Magnitude Profile 1.5 After placing a 48.74 Mvar SVC at bus 64 Before placing SVC

V [p.u.]

1

0.5

0

0

10

20

30

40

50

ð6Þ

60

70

Bus # Fig. 3. Bus magnitude voltage profile of stressed system before and after placing a 48.74 Mvar SVC at bus 64.

the SVC. Using this technique, the optimal size of SVC is calculated to be 48.74 Mvar. The maximum loading parameter is increased from 1.351 to 1.3577. The power system voltage profile before and after placing the SVC is shown in Fig. 3. By optimally sizing and locating the SVC at bus 64, the power system voltage profile is improved as shown in Fig. 3. This figure shows the significant improvement of the voltages at buses 30–70 compared to before placing SVC. 3.2. Feedback signal selection Controllability and observability play an important role in input–output signals selection process. In order to specify which

ð7Þ

It must be noted that the value of each singular value σ i is associated with the state xi, and the size of the σ i is a relative measure of the contribution that the corresponding state makes to the input–output behaviors of the system. Hence, if σ i c σ i þ 1 , then the state xi affects the input–output behavior more than that xi þ 1 does [5]. The active/reactive power, current magnitude signals of some transmission lines, bus voltages and generators speed (as five groups) are listed to be analyzed using the HSV technique in preand post-fault conditions. In each category, the signals with the higher HSVs are selected and held to analyze further using RHPzeros in the next step. The RHP-zeros of pre-fault and post-fault system are calculated for each group of the selected signals in HSVs analysis step. If there is one single signal that does not produce RHP-zeros, this signal is the SDC stablizing signal. Otherwise, a more accurate investigation will be carried out on the zeros of the related transfer functions, to evaluate which signal has been produced more stable zeros with greater real magnitude part. The initial signals are summarized as follows: Active power: P62-63, P62-65, P58-63, P57-56, P57-58, P66-56, P65-66, P66-67, reactive power: Q62-63, Q62-65, Q58-63, Q57-56, Q57-58, Q66-56, Q65-66, Q66-67, line current: I62-63, I62-65, I58-63, I57-56, I57-58, I66-56, I65-66, I66-67, generator speed: wG3, wG13, wG14, wG15, wG16, bus voltage: V19, V28, V32,V60, V65. The HSV analysis is implemented in each category for both prefault and post-fault conditions when a three-phase fault occurred at bus 46. As mentioned before, the HSVs of the similar signals should be compared. It is clear from Fig. 4 that the following signals associated with largest HSVs (in pre-fault and post fault

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

Post-fault

Pre-fault 0.2

0.1 0.05

0

5

10

P62-63 P62-65 P58-63 P57-56 P57-58 P66-56 P65-66 P66-67

0.15 HSV

HSV

0.15

0

0.2

P62-63 P62-65 P58-63 P57-56 P57-58 P66-56 P65-66 P66-67

0.1 0.05 0

15

0

5

Pre-fault

Post-fault Q62-65 Q57-56 Q58-63 Q62-63 Q66-67 Q65-66 Q66-56 Q57-58

0.2 0.1

0

5

10

Q66-67 Q66-56 Q58-63 Q57-58 Q65-66 Q57-56 Q62-65 Q62-63

0.2 HSV

HSV

0.3

0.15 0.1 0.05 0

15

0

5

State number

0.015 0.01 0.005

0.01 0.005 0

10

I62-63 I58-63 I57-56 I66-56 I62-65 I65-66 I66-67 I57-58

0.015 HSV

HSV

0.02

I62-63 I62-65 I58-63 I57-56 I57-58 I66-56 I65-66 I66-67

0.02

5

15

0

5

0.25

HSV

0.15 0.1

V65 V60 V19 V32 V28

0.2 HSV

V65 V60 V19 V32 V28

0.2

0.05

0.15 0.1 0.05

0

5

10

0

15

0

5

State number

10

15

State number

Pre-fault

Post-fault w3 w13 w16 w15 w14

0.8 0.6 0.4

1

w3 w13 w16 w15 w14

0.8 HSV

1

HSV

15

Post-fault

Pre-fault 0.25

0.6 0.4 0.2

0.2 0

10 State number

State number

0

15

Post-fault

Pre-fault

0

10 State number

0.025

0

15

0.25

0.4

0

10 State number

State number

0

5

10

15

0

0

5

10

15

State number

State number Fig. 4. HSV diagrams.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

conditions) are saved to be analyzed in the next step by RHPzeros:

 P58-63, P62-63, P62-65, Q58-63, Q62-63, Q62-65, I58-63, I62-63, I62-65, wG3, wG14,V60, V65. In a feedback loop system, the existence of right half plane zeros limits the potential performance of the system. When contingencies occur and the dynamic system gain increases, the closed-loop poles will move toward the open loop zeros. Therefore, if these zeros are located in the right half plane of the root locus diagram, the closed loop poles could move to an unstable place and make system performance unstable. Consider an open loop system with a transfer function as G(s)¼ z/p. The closed-loop transfer function is expressed as: GC ðsÞ ¼

KGðsÞ kz z ¼ ¼ k cl 1 þ KGðsÞ p þ kz pcl

ð8Þ

4. Parameters of the uncertain-test system The coefficients of the system transfer function (between selected signal and SVC output) for each operating condition are categorized in a way that they are changed form their maximum Table 2 RHP- zeros investigation of obtained signal from HSV analysis. Selected signals for RHP-Zeros

P58-63 P62-63 P62-65 Q58-63 Q62-63 Q62-65 I58-63 I62-63 I62-65 V65 V60 wG3 wG14

and minimum coefficient across all the operating conditions. As a result, a finite number of operating conditions is considered and the transfer function of original system is calculated. Then, the original system (139 states) is reduced to 5 states using balanced truncation approach and the transfer function is calculated for the reduced system. It is assumed that the reduced system has the transfer function as follows: Gr ðsÞ ¼

nx nx dy dy

where z and p are the open loop zeros and poles whereas zcl and pcl are closed-loop zeros and poles respectively. In the above closed-loop transfer function, when the systems gain increases, the location of the zeros is not changed while the closed-loop poles location is changeable. Accordingly, the system input–output signals should be selected in a way that a minimum number of the open loop zeros in the right half plane of root locus diagram appears [22]. Accordingly, selection of the system input–output is accomplished in a way that a minimum number and more stable real part (greater in absolute magnitude) of zeros in the s-plane of the root locus diagram appears. The obtained signals from the HSV analysis process are tested to investigate whether they produce RHP-zeros in pre-fault and post-fault conditions when a threephase fault occurred at bus 46. The results are summarized in Table 2. As evident in Table 2, I62-63 and I62-65 do not produce any RHP-zeros. Therefore, a more detailed examination should be carried out to choose the best signal. The zero plots of transfer functions relevant to two mentioned signals in the pre-fault and post-fault conditions are shown in Fig. 5. From the obtained results, it can be concluded that the transfer function contains more stable open loop zeros for I62-63 compared to I62-65 for both pre- and post-fault conditions. Although both of these signals are suitable, Fig. 5 depicts that I62-63 is more preferable than I62-65.

Pre-fault

Post-fault

RHP-Zeros

RHP-Zeros

Yes Yes Yes Yes Yes Yes Yes No No Yes Yes No No

Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes

5

n4 s4 þ n3 s3 þ n2 s2 þ n1 s þ n0 d5 s5 þ d4 s4 þ d3 s3 þ d2 s2 þd1 s þd0

ð9Þ

The boundary values of parameters can be computed as: n o ¼ max n1x ; n2x ; … ; njx ; …; nJx n o ¼ min n1x ; n2x ; … ; njx ; …; nJx n o 1 2 j J ¼ max dy ; dy ; … ; dy ; …; dy n o 1 2 j J ð10Þ ¼ min dy ; dy ; … ; dy ; …; dy

where x¼0,1,…,4, y¼0,1,..,5, and J is maximal number of considered operating conditions. In the present work, 40 operating points are considered including the load variation and line outage. Some of these operating conditions have been summarized in Table 3. From this table, as well as using the Eq. (10), the upper– lower bounds of the cofficients in nominator and denominator can be easily calculated.

5. Controller synthesis PID (or PI) controllers are used in most practical industrial applications and thus the appropriate PI(D) control design is still a very important issue especially for systems under some nonlinearities, perturbations or time-variant behavior. Since, the aforementioned controllers are very popular because of their easy implementation and desired performance at the same time even under uncertain conditions, recently, some researchers have focused on the design of PI or PID controllers using Kharitonov's theorem [23–27]. A new method for the calculation of all stabilizing PI controllers has been given in [23]. The proposed method is based on plotting the stability boundary locus in the (kp,ki)plane and then computing the stabilizing values of the parameters of a PI controller. The method does not require linear programming to find the PI or PID controller parameters. Authors in [24] have proposed the Kronecker summation method for computation of all stabilizing PI controllers. The results obtained by this technique are similar to [23], but the computation process is too complex. Application of PI or PID controllers designed by Kharitonov's theorem in power system area has been presented in [13,25–27]. In [13], design of a robust PID-based PSS has been reported. Simulation results confirm the robustness and effectiveness of the proposed PID controller in damping of system low frequency oscillations subjected to the small and large disturbances. A new control structure with a tuning method to design a PID load frequency controller for single area power system has been reported in [25]. Simulation results show that the proposed controller improves the load disturbance rejection performance significantly even in the presence of the uncertainties in plant parameters. Design of a robust decentralized PI controller as a solution of the load frequency control (LFC) using Kharitonov's polynomials in a multi-area power system has been presented in [26]. Authors in [27] have utilized an internal oscillator to control system frequency and a PI controller to maintain voltage stability of a micro-grid that is tuned by Kharitonov's theorem. In this paper, a robust PI controller is designed using polynomial control and Kharitonov's theorem.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

Post-fault

Pre-fault

40

I62-65 I62-63

10

I62-65 I62-63

20 Imag

Imag

5 0

0

-5

-20

-10

-40

-0.8

-0.6

-0.4

-0.2

0

-0.6

-0.5

Real

-0.4

-0.3 -0.2 Real

-0.1

0

0.1

Fig. 5. zero plot comparison of selected signal.

By solving above equations:

Table 3 16-machine power system operating conditions.

kp ¼ 

Case no P

Q

Line outage

1 2 3 4 5 6 7 8 9 10 11 12

Base: 3097.96 MVar Base  1.03 Base  0.93 Base Base Base Base Base  1.03 Base  0.97 Base  1.01 Base Base

– – – 26-27 26-27, 18-49 26-27, 18-49, 36-34 26-27, 18-49, 36-34, 67-68 60-61 57-60 38-46 25-54 26-27, 18-49, 36-34, 67-68, 23-24

Base: 17622.7 MW Base  1.03 Base  0.93 Base Base Base Base Base  1.03 Base  0.97 Base  1.01 Base Base

The considered PI controller in this paper consists of a washout filter block. The input is first passed through a high pass filter (washout) so that the input's steady state value is eliminated. The time constant value of washout filter (Tw) can be in the range of 1– 20 s [1]. The Tw is set to be 10 s in this paper [1]. In addition, the controller output is subjected to an anti-windup limiter in order to limit the controller output , and its dynamic is given by a small time constant Tr ¼0.001 s [20], thus the controller output is locked if one of its limits is reached. 5.1. Robust stability Consider the SISO control system of Fig. 1 where CðsÞ is a PI controller. The problem is to compute the parameters of the PI controller that stabilize the system of Fig. 1. The substitution s ¼ jω in the Eq. (4) and subsequent decomposition of the numerator and denominator into their real and imaginary parts lead to: N x ðjωÞ N R x ðωÞ þjN I x ðωÞ ¼ ; Gxy ðjωÞ ¼ Dy ðjωÞ DR y ðωÞ þjDI y ðωÞ

x; y ¼ 1; 2; 3; 4

ð11Þ

The closed loop characteristic equation can be written as:     jω kp þ ki N ðωÞ þ jN I x ðωÞ Π ðjωÞ ¼ 1 þ G ðjωÞ C ðjωÞ ¼ 1 þ R x DR y ðωÞ þ jDI y ðωÞ jω ¼ ðjωÞ

ðDR y ðωÞ þjDI y ðωÞÞ þ

þ ðjω kp þki Þ ðNR x ðωÞ þ jN I x ðωÞÞ ¼ RΠ þ jI Π ¼ 0

ki ¼

ðDI y ðωÞN I x ðωÞ þ DR y ðωÞNR x ðωÞÞ N2R x ðωÞ þ N 2I x ðωÞ

ω ðDI y ðωÞN R x ðωÞ  DR y ðωÞN I x ðωÞÞ N 2R x ðωÞ þ N 2I x ðωÞ

ð14Þ

ð15Þ

It can be proved that if a controller is capable of satisfying stability of the 16 Kharitonov transfer functions and maintain their performance in a suitable level, it will be able to guarantee the stability and desirable performance of the uncertain system [23]. The set of all the stabilizing values of the parameters of a PI controller that stabilize the interval plant of the Eq. (4) can be written as: SðCðsÞ Gxy ðsÞÞ ¼ SðCðsÞ G11 ðsÞÞ \ SðCðsÞ G12 ðsÞÞ⋯SðCðsÞ G43 ðsÞÞ \ SðCðsÞ G44 ðsÞÞ

ð16Þ

Thus, the final area of stability for the original interval plant is given by the intersection of all 16 relevant partial areas. The final region includes the stabilizing values of kp and ki that stabilize the original system. The equations are straightforward and relatively easy to compute compared to the represented method in [23]. 5.2. Robust performance conditions In control system designs, apart from satisfying the robust stability, a desirable performance must be achieved. Desirable performance can be summarized in terms of having a desired time response, a reasonable Gm, Pm, and bandwidth that are related to upper bound of sensitivity function, restriction of maximum overshoot in the time domain response and achieving suitable settling time, respectively [26]. 5.2.1. Stabilization for specified gain–phase margin It is assumed that a gain–phase margin tester Ae  jθ is connected to the feed forward path in Fig. 1. The characteristic equation can be written as:

Π ðjωÞ ¼ 1 þAe  jθ GðjωÞ CðjωÞ ¼ 1 þ Að cos ðθÞ    NR x ðωÞ þ jN I x ðωÞ jω kp þki  j sin ðθÞÞ DR y ðωÞ þ jDI y ðωÞ jω

ð17Þ

Then, the Eq. (17) can be written as: ð12Þ

Equating the real and imaginary parts of Π ðjωÞ to zero, respectively, we have: ( ki N R x ðωÞ  kp ω N I x ðωÞ  ωDI y ðωÞ ¼ 0 ð13Þ ω DR y ðωÞ þ ωkp NR x ðωÞω þ ki N I x ðωÞ ¼ 0

Π ðjωÞ ¼ jω DR y ðωÞ  ω DI y ðωÞ þ A cos ðθÞki N R x ðωÞ þ jA cos ðθÞki NI x ðωÞ þ A cos ðθÞω kp NR x ðωÞ   A cos ðθÞ ω kp N I x ðωÞ  jA sin ðθÞki N R x ðωÞ þ A sin ðθÞki N I x ðωÞ þ jA sin ðθÞ ω kp NI x ðωÞ þ þ A sin ðθÞω kp N R x ðωÞ ¼ 0

ð18Þ

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

By equating the real and imaginary parts of Π ðjωÞ to zero, respectively: (

7

These equations provide conditions relevant to suitable Gm and Pm simultaneously.

 ωDI y ðωÞ þ A cos ðθÞki N R x ðωÞ A cos ðθÞ ω kp N I x ðωÞ þ A sin ðθÞki N I x ðωÞ þ A sin ðθÞω kp N R x ðωÞ ¼ 0 ω DR y ðωÞ þ A cos ðθÞki N I x ðωÞ þA cos ðθÞ ω kp N Rx ðωÞ  A sin ðθÞki N R x ðωÞ þ A sin ðθÞ ω kp N I x ðωÞ ¼ 0

By solving the Eq. (19), the boundary of a region in kp–ki plane in which the Gm and Pm of the system are in an agreement with the designer's desired values can be found as:

kp ¼

ki ¼

ð19Þ

5.2.2. Bandwidth analysis As known, the response speed or bandwidth of the closed loop system is very significant in the analysis and design of control

sin ðθÞ ðDI y ðωÞN R x ðωÞ  NI x ðωÞDR y ðωÞÞ  cos ðθÞ ðDI y ðωÞNI x ðωÞ þ DR y ðωÞNR x ðωÞÞ

ð20Þ

AðN 2R x ðωÞ þ N 2I x ðωÞÞ

ωð sin ðθÞ ðDR y ðωÞN R x ðωÞ þ DI y ðωÞNI x ðωÞÞ þ cos ðθÞ ðDI y ðωÞN R x ðωÞ  DR y ðωÞN I x ðωÞÞÞ AðN 2R x ðωÞ þ N2I x ðωÞÞ

120

ð21Þ

200

100

150

60

ki

ki

80

100

40

0

50

Stablized Kp-Ki values

20 0

2

4

6

8

10

12

14

Stablized region with: Gm > 0.7

0 -5

16

0

5

10

kp

100

20

25

120 100

60

80

ki

ki

80

40

60 40

Stablized region with: Pm > 30 degree

20 0

15

kp

-2

0

2

4

6

Bandwidth condition:10 30 deg; 10 < BW < 20 Hz

5

10

15

kp Fig. 6. (a) Closed loop robust stability region (hatched area), (b) robust performance region for specified Gm, (c) for specified Pm, (d) Closed loop robust bandwidth, in kp–ki plane, and (e) Closed loop robust stability region with desired margins and bandwidth.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

systems. In [23], a relative stabilization has been attained by moving all the closed-loop poles to the desired region in the complex plane. Here, the bandwidth analysis is recommended. It is obvious that the settling time of the controlled system will be reduced by increasing the system bandwidth. It is aimed to specify a region in which all the values of kp–ki cause the system speed to remain in a desirable level. To achieve this goal, the desired bandwidth ðωBw Þ is selected based on the settling time of desired time response. The formulation is given by:

Table 4 Optimal parameters of PI controller. Parameter

kp

ki

Optimal value

3.6

35.7

  pffiffiffi   Gxy ðjωBw Þ CðjωBw Þ      ¼ 2 Gxy ðj 0Þ Cðj 0Þ  1 þ Gxy ðjω Þ Cðjω Þ 2 1 þ Gxy ðj 0Þ Cðj 0Þ Bw Bw

ð22Þ

Since a PI controller is used, the Eq. (22) can be written as:     Gxy ðjωBw Þ ðki þ jkp ωBw Þ 1   jω þ Gxy ðjω Þ ðk þ jkp ω Þ ¼ pffiffiffi 2 Bw Bw Bw i

ð23Þ

The Eq. (23) can be expressed as:

2

x 10

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

1 0 -1 0

5

10

15

x 10

Speed deviation of G13-G12 (pu)

Speed difference of G13-G16 (pu)

Fig. 7. Closed loop system interarea modes with robust and optimal PI controllers.

pffiffiffi       2 Gxy ðjωBw Þ ki þ jkp ωBw  ¼ jωBw þ Gxy ðjωBw Þ ðki þ jkp ωBw Þ

20

ð24Þ

-4

SVC without SDC SVC with robust SDC SVC with optimal SDC

5

0

-5

0

5

10

Time (s)

15

20

Ti me (s)

5 x 10

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

0

-5

0

5

10

15

Speed difference of G13-G12 (pu)

Speed difference of G13-G16 (pu)

Fig. 8. Speed difference response of G13–G16 and G13–G12 for scenario 1.

20

x 10

1

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

0.5 0 -0.5 -1

0

5

Time (s)

10

15

20

Time (s)

x 10

-3

SVC without SDC SVC with robust SDC SDC with optimal SDC

2 0 -2 0

5

10

15

20

Speed difference of G13-G12 (pu)

Speed difference of G13-G16 (pu)

Fig. 9. Speed difference response of G13–G16 and G13–G12 for scenario 2.

1

x 10

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

0.5 0 -0.5 -1

0

5

Time (s)

10

15

20

Time (s)

Fig. 10. Speed difference response of G13–G16 and G13–G12 for scenario 3.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

x 10

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

5

0

-5 0

5

10

15

Speed difference of G13-G12 (pu)

Speed difference of G13-G16 (pu)

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

20

1

x 10

9

-3

SVC without SDC SVC with robust SDC SVC with optimal SDC

0.5 0 -0.5 -1

0

5

10

Time (s)

15

20

Time (s)

Fig. 11. Speed difference response of G13–G16 and G13–G12 for scenario 4.

Scenario 2

Scenario 1 SVC with robust SDC

SVC with robust SDC

SVC wit optimal SDC

SVC with optimal SDC

0.45

B SVC (pu)

B SVC (pu)

0.5

0.38

0.4

0.37

0.36

0.35

0

5

10

15

20

0.35

0

5

10

15

20

Time (s)

Time (s) Fig. 12. SVC control signal for the scenarios 1 and 2.

Using the triangle inequality:   jωBw    ki þ jkp ωBw  r pffiffiffi   ð 2  1Þ Gxy ðjωBw Þ

ð25Þ

Let us assume that ω ¼ ωBw , therefore the Eq. (4) can be written in the following form: ^ ^ ^ xy ðjωBw Þ ¼ N x ðjωBw Þ ¼ N R x ðωBw Þ þ jNI x ðωBw Þ; G ^ I y ðωBw Þ ^ R y ðωBw Þ þ jD Dy ðjωBw Þ D

x; y ¼ 1; 2; 3; 4 ð26Þ

By substituting the Eq. (26) in the Eq. (25):   jωBw    ki þ jkp ωBw  r   pffiffiffi ^  ^ ð 2  1Þ N^ R x ðωBw Þ þ jN^ I x ðωBw Þ

ð27Þ

Then: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ω2 u  2 Bw þk2p ω2Bw ki r u pffiffiffi t ^ 2 ðω Þ ^ ðω Þ þ N N ð 2  1Þ2 ^ 2Rx Bw ^I2x Bw

ð28Þ

DR y ðωBw Þ þ jDI y ðωBw Þ

DR y ðωBw Þ þ DI y ðωBw Þ

The regions of kp–ki plane in which the desired bandwidth is in an appropriate level can be obtained using the Eq. (27).

6. Controller design In this section, the proposed robust PI controller design procedure is applied to a large-scale power system and the simulation results have been presented. The overlapped region of speciation-satisfied area for each Kharitonov's polynomial is the useful parameter area for the selection of controller parameters, where the completely uncertain plant can be stabilized. The shaded region in Fig. 6(a) shows a region of kp–ki which corresponding to PI controller, can stabilize all Gxy ðjwÞ; x; y ¼ 1; 2; 3; 4: Choosing a test point within each region, the stable region that contains the values of stabilizing kp and ki

parameters can be determined. For example, choosing a test point such as kp ¼ 5 and ki ¼120, it can be calculated that the characteristic polynomial has right half plane complex roots, and therefore, the system is unstable for these values of parameters. The closed loop poles associated with G11 are as  12.4473716.309i, 1.8383717.608i,  2.9993 and  1.2026 could be mentioned as an example. In order to achieve the robust performance, desirable features should be presented and the corresponding regions need to be extracted, accordingly. The upper limit sensitivity function must be lower than 0.4. In other words, it is necessary that ‖ð1 þ Gxy ðjωÞ CðjωÞÞ  1 ‖ o 0:4. This condition guarantees the low sensitivity of closed-loop transfer function against the open-loop transfer function variations. Therefore, the appropriate Gm will be above 0.7. To satisfy a desirable maximum percent overshoot, minimum desirable Pm will be θ ¼ ðπ =6Þ rad. The desirable bandwidth varies from 10 to 20 rad/s [28]. To obtain the stability boundary locus for a given value of Gm A, one needs to set θ ¼ 0 in Eqs. (20) and (21). On the other hand, setting A¼1 in these equations, one can obtain the stability boundary locus for a given Pm θ. Fig. 6(b) and (c) shows the shaded region in kp–ki plane, which satisfies the Gm and Pm constraints, respectively. By selecting kp and ki from the shaded area, all 16 Kharitonov polynomials are stable considering the Gm and Pm. For instance, choosing a point in the region, such as kp ¼ 4 and ki ¼ 40, the closed loop poles of G22 are  6:85677 17:349i;  4:30997 10:031i;  3:3359 and  0.4109. The Gm is equal to 2.466 and the Pm is equal to 37.5 degree. Additionally, the kp–ki region relevant to desirable speed region can be defined by using Eq. (28). Fig. 6(d) shows a shaded region, which satisfies the bandwidth constraint. Now, all controllers' coefficients situated in the common overlapped region, shown in Fig. 6(e) (overlapped shaded areas depicted in Fig. 6(a–d)), robustly stabilize the closed loop power system over all the uncertainties under consideration, and ensure the performance margins and bandwidth conditions.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

the system oscillations by changing the SVC susceptance during transient operationtime.

7. Simulation results In order to evaluate the dynamic performance of the proposed controllers on interarea oscillation damping, eigenvalue analysis and non-linear time domain simulations are carried out considering four case studies. The studied operating conditions as four scenarios are summarized below: Scenario 1: A three-phase fault occurs at bus 46 at t¼1 and clears after 100 ms. Scenario 2: Outage of line 26–27 at t¼1.05 s. Scenario 3: A three-phase fault occurs at bus 28 at t¼ 1 s and it is removed by the operation of the circuit breaker line 28–29 after 50 ms (outage of line 28–29). Scenario 4: Heavy loading (1.5 times nominal load), A threephase fault occurs at bus 28 and outage of line 28–29 at t ¼1 s. In order to illustrate the effectiveness of the proposed controller in damping of oscillations, nonlinear time domain simulations have been performed. In the case of SDC, it is assumed that kp ¼5 and ki ¼40. Therefore, the designed controller is as CðsÞ ¼ 5 þ ð40=sÞ. To show the effectiveness of the proposed controller, the results are compared to the optimal PI controller. Since the decaying rate of oscillations is dominated by the maximum damping factor and the amplitude of each oscillation mode is determined by its damping ratio, thus, a proper objective function should include the damping factor and the damping ratio in the formulation of optimal PI design. The proposed objective function is introduced in the following equation: Min F ¼ F 1 þ ϖ F 2 ¼

X 

max

σ i Z σ 0 1 r q r nq

σq  σ0

2

þϖ

X  ζi Z ζ0

ζ 0  min ζ q

2

8. Conclusion A novel method for designing a robust SVC-based SDC for a large-scale power system has been presented in this paper. The uncertainty has been described using a transfer function with interval polynomials in numerator and denominator. The analyzed uncertainty is the variation in operating conditions resulting from load variation and changes in system configuration. The proposed robust controller is designed based on interval polynomial control methodology and Kharitonov's theorem. The designed controller (PI controller) is low-order fixed-parameter controller which satisfy the robust stability and robust performance (desired Gm, Pm, and bandwidth). In design process, the region of stability (parameters space) is determined where the robust stability and robust performance are analyzed. In addition, to show the effectiveness of proposed robust PI controller in damping of oscillations, an optimal PI controller has been also designed. The simulation results confirmed that the designed controllers are robust and damp out electromechanical oscillations at different operating conditions. In addition, it can be concluded that the designed PI controller has better performance in damping of oscillations compared to optimal PI controller.

References

1 r q r nq

ð29Þ where max σ q and min ζ q are the maximum real part of the eigenvalues and the minimum of the damping ratio, respectively. σ 0 and ζ 0 are expected damping factor and expected damping ratio, respectively. q¼ 1,2,…, nq is the index of eigenvalues. Also, ϖ is the weight constant. The main reason to define this objective function is this fact that beside unstable or lightly damped oscillation modes, other oscillation modes can also be shifted to the left side of complex plane. The constraints for PI controller parameters are expressed as follows: ( 0 r kp r 10 ð30Þ 0 r ki r100 The PSO algorithm is used to find the optimal parameters of PI controller by minimizing the introduced objective function in the Eq. (29) [21]. The maximum iteration number (Itermax) and the population size are set to 100 and 50, respectively. Optimal parameters of PI controller are listed in Table 4. The open loop interarea modes as well as closed loop interarea modes are presented in Fig. 7 when the aforementioned controllers (robust and optimal PI controllers) are applied to the full order system. The simulation results show that designed controller effectively shifts the interarea oscillations modes to the desired side in the s-plane. Figs. 8– 11 shows the speed deviation response of G13–G16 (representative of interarea mode) and G13–G12 (representative of local mode) for the mentioned case. As can be seen, the designed controllers can damp out interarea and local oscillations satisfactorily. In addition, the simulation results demonstrate the effectiveness of the proposed robust PI controller in reduction of overshoot and settling time of oscillations over the whole range of operating conditions compared to optimal PI controller. Fig. 12 presents the SVC control signal for the scenarios 1 and 2. The result shows effectiveness of the designed controllers in mitigating

[1] Kundur P. Power system stability, and control. New York: McGraw-Hill; 1994. [2] Padiyar KR. FACTS: controllers in power transmission and distribution. Anshan; 2009. [3] Zhang Y, Bose A. Design of wide-area damping controllers for interarea oscillations. IEEE Trans Power Syst 2008;23:1136–43. [4] Farsangi MM, Song YH, Lee KY. Choice of FACTS device control inputs for damping interarea oscillations. IEEE Trans Power Syst 2004;19:1135–43. [5] Farsangi MM, Nezamabadi-pour H, Song YH, Lee KY. Placement of SVCs and selection of stabilizing signals in power systems. IEEE Trans Power Syst 2007;22:1061–71. [6] Kunjumuhammed LP, Singh R, Pal BC. , Robust signal selection for damping of inter-area oscillations. IET Gener Transm Distrib 2012;6:404–16. [7] Rigatos G, Siano P. Design of robust electric power system stabilizers using Kharitonov's theorem. Math Comput Simul 2011;82:181–91. [8] Pal BC, Chaudhuri B. Robust control in power systems. New York: Springer; 2005. [9] Poznyak AS. Advanced mathematical tools for automatic control engineers: volume 1. Deterministic techniques. 1st ed.Elsevier; 2008. [10] Li Y, Rehtanz C, Rüberg S, Luo L, Cao Y. Wide-area robust coordination approach of HVDC and FACTS controllers for damping multiple interarea oscillations. IEEE Trans Power Syst 2012;27:1096–105. [11] Soliman HM, Elshafei AL, Shaltout AA, Morsi MF. Robust power system stabilizer. IEE Proc – Electr Power Appl 2000;147:285–91. [12] Soliman M. Robust non-fragile power system stabilizer. Electr Power Energy Syst 2015;64:626–34. [13] Soliman M. Parameterization of robust three-term power system stabilizers. Elect Power Syst Res 2014;117:172–84. [14] Soliman HM, Bayoumi HE, Awadallah MA. Reconfigurable fault-tolerant PSS and FACTS controllers. Elect Power Compon Systs 2010;38:1446–68. [15] Simfukwe DD, Pal BC. Robust and low order power oscillation damper design through polynomial control. IEEE Trans Power Syst 2013;28:1599–608. [16] Robak S. Robust SVC controller design and analysis for uncertain power systems. Control Eng Pract 2009;17:1280–90. [17] Gu DW, Petkov P, Konstantinov MM. Robust control design with MATLAB. Springer; 2005. [18] Zhou K. Essentials of robust control; 1999. [19] Doyle JC, Francis BA, Tannenbaum A. Feedback control theory. New York: Dover Publications; 2009. [20] Milano F. Power system analysis toolbox – documentation for PSAT version 2.0.0; 2007. [21] Hasanvand H, Bakhshideh Zad B, Mozafari B, Feizifar B. Damping of lowfrequency oscillations using an SVC-based supplementary controller. IEEJ Trans Electr Electron Eng 2013;8:550–6. [22] Skogestad S, Postlethwaite I. Multi variable feedback control, analysis and design. New York: John Wiley and Sons; 1996.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i

H. Hasanvand et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [23] Tan N, Kaya I, Yeroglu C, Atherton DP. Computation of stabilizing PI and PID controllers using the stability boundary locus. Energy Convers Manag 2006;47:3045–58. [24] Fang J, Zheng Da, Ren Z. , Computation of stabilizing PI and PID controllers by using Kronecker summation method. Energy Convers Manag 2009;50:1821–7. [25] Padhan DG, Majhi S. A new control scheme for PID load frequency controller of single-area and multi-area power systems. ISA Trans 2013;52:242–51. [26] Toulabi MR, Shiroei M, Ranjbar AM. Robust analysis and design of power system load frequency control using the Kharitonov's theorem. Electr Power Energy Syst 2014;55:51–8.

11

[27] Habibi F, Hesami Naghshbandy A, Bevrani H. Robust voltage controller design for an isolated micro-grid using Kharitonov's theorem and D-stability concept. Electr Power Energy Syst 2013;44:656–65. [28] Shiau JK, Taranto GN, Chow JH, Boukarim G. Power swing damping controller design using an iterative linear matrix inequality algorithm. IEEE Trans Control Syst Technol 1999;7(3):371–81.

Please cite this article as: Hasanvand H, et al. Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.005i