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Research article
Model based PI power system stabilizer design for damping low
frequency oscillations in power systems
Apr...
algebra based intelligent PID control in Ref. [10], time-delay esti-
mation based intelligent PI control in Refs. [11,12],...
Considering, nðsÞ ¼ G3ðsÞ½1 þ G1ðsÞG2ðsÞK6ðsÞ Š;
dðsÞ ¼ 1 þ G3ðsÞ G4ðsÞK1 À G1ðsÞ G2ðsÞ G3ðsÞ G4ðsÞK2K5 À
G2ðsÞ G3ðsÞ G4ðs...
assumed to be divided into separate sub-systems. When the sub-
systems are interconnected, the interactions act as perturb...
performance comparison 0.05 p. u. step change in DTm has been
considered here. Performance comparison of the closed-loop c...
the other methods in terms of settling time ðtsÞ, time for peak
overshoot ðtpÞ, peak overshoot ðypÞ, total variance of con...
For the chosen reference model, the settling time ðtsÞ is 1.095 s,
the peak overshoot ðypÞ is 3:670 Â 10À4s and time for p...
Fig. 7. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other or both machines (Exampl...
3.4.1. Design of the local controllers
The system is decomposed into three machines as discussed
below:
Machine 1: With
h
...
pair of conjugate complex poles are considered with z ¼ 0:215 and
un ¼ 4:484 rad/s and a zero at origin as given by:
Mref ...
Fig. 8. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other machines (Example 4).
A....
4. Conclusion
An enhancement of the small-signal stability of the power sys-
tem in terms of improving the damping factor ...
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Model based PI power system stabilizer design for damping low frequency oscillations in power systems

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This paper explores a two-level control strategy by blending a local controller with a centralized controller for the low frequency oscillations in a power system. The proposed control scheme provides stabilization of local modes using a local controller and minimizes the effect of inter-connection of sub-systems performance through a centralized control. For designing the local controllers in the form of proportional-integral power system stabilizer (PI-PSS), a simple and straight forward frequency domain direct synthesis method is considered that works on use of a suitable reference model which is based on the desired requirements. Several examples both on one machine infinite bus and multi-machine systems taken from the literature are illustrated to show the efficacy of the proposed PI-PSS. The effective damping of the systems is found to be increased remarkably which is reflected in the time-responses; even unstable operation has been stabilized with improved damping after applying the proposed controller. The proposed controllers give remarkable improvement in damping the oscillations in all the illustrations considered here and as for example, the value of damping factor has been increased from 0.0217 to 0.666 in Example 1. The simulation results obtained by the proposed control strategy are favorably compared with some controllers prevalent in the literature.

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Model based PI power system stabilizer design for damping low frequency oscillations in power systems

  1. 1. Research article Model based PI power system stabilizer design for damping low frequency oscillations in power systems Aprajita Salgotra* , Somnath Pan Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India a r t i c l e i n f o Article history: Received 7 December 2017 Received in revised form 22 February 2018 Accepted 18 March 2018 Available online 26 March 2018 Keywords: Frequency response matching Reference model Direct synthesis Low frequency oscillation One machine infinite bus (OMIB) system Proportional-integral power system stabilizer (PI-PSS) a b s t r a c t This paper explores a two-level control strategy by blending local controller with centralized controller for the low frequency oscillations in a power system. The proposed control scheme provides stabilization of local modes using a local controller and minimizes the effect of inter-connection of sub-systems performance through a centralized control. For designing the local controllers in the form of proportional-integral power system stabilizer (PI-PSS), a simple and straight forward frequency domain direct synthesis method is considered that works on use of a suitable reference model which is based on the desired requirements. Several examples both on one machine infinite bus and multi-machine sys- tems taken from the literature are illustrated to show the efficacy of the proposed PI-PSS. The effective damping of the systems is found to be increased remarkably which is reflected in the time-responses; even unstable operation has been stabilized with improved damping after applying the proposed controller. The proposed controllers give remarkable improvement in damping the oscillations in all the illustrations considered here and as for example, the value of damping factor has been increased from 0.0217 to 0.666 in Example 1. The simulation results obtained by the proposed control strategy are favourably compared with some controllers prevalent in the literature. © 2018 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The low frequency oscillations in power systems had been noticeably observed in 1920s in the form of spontaneous oscilla- tions and became more prominent when synchronous generators are interconnected to build-up more capacity. These oscillations are undesirable even at low frequencies (i.e. 0.1e2 Hz) as they reduce the power transfer capability of the transmission line [1]. The ef- forts to transfer bulk power across weak transmission line may lead to the low frequency oscillations [2]. It may be due to any distur- bance such as sudden change in transmission line parameters or fluctuations in the turbine output power. In order to improve the transient stability, the gain is kept high for fast excitation that further aggravates the problem. To address this problem, a viable solution has been widely accepted by the power industries in the form of power system stabilizer (PSS). It adds a stabilizing signal to the excitation system to compensate the phase lag resulting from the voltage regulator, exciter and synchronous generator. In effect of this, the overall damping of the system is improved. The two-level scheme has been observed in Ref. [3] where the overall system is decomposed into different sub-systems and an optimal controller is designed for each sub-system. This method involves tedious computation for obtaining the optimal gain of order n as nðn þ 1Þ=2 Riccatti equations are to be solved. Proportional-integral (PI) controllers (analog and digital) for one machine infinite bus (OMIB) and multi-machine systems have been designed in Ref. [4], however, interactions among the machines are not considered. Design of two-level PSS has been discussed in Ref. [5] where the effects of one machine over the other machines following a change in the mechanical torque in one or both the machines are simulated. A dynamic pole assignment technique has been proposed in Ref. [6] that uses pre-specified eigen-structure. In Ref. [7] a power system stabilizer is designed using the output feedback control based on the reduced-order model derived by the balanced truncation method. A two-level control strategy com- bined with order-reduction has been addressed in Ref. [8] to ensure fast convergence of the designed results. However, an approximate model matching technique for obtaining the parameters of PID controller may be observed in Ref. [9]. This technique may be applied without model-reduction, even for high-order systems. Moreover, model free control has been employed for differential * Corresponding author. E-mail addresses: aprajita.salgotra3@gmail.com (A. Salgotra), somnathpan123@ gmail.com (S. Pan). Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans https://doi.org/10.1016/j.isatra.2018.03.013 0019-0578/© 2018 ISA. Published by Elsevier Ltd. All rights reserved. ISA Transactions 76 (2018) 110e121
  2. 2. algebra based intelligent PID control in Ref. [10], time-delay esti- mation based intelligent PI control in Refs. [11,12], sliding mode control based PI control in Ref. [13]. In Ref. [14], a model matching approach is used to solve a H∞ problem and the parameters are determined by closed-loop shaping. Recently, Yaghooti et al. [14] has designed a coordinated PSS using model reference adaptive technique. For multi-machine system, the mathematical modeling and block diagram showing the interactions among various sub- systems has been explained elaborately in Ref. [15]. Among the efforts to address the low frequency oscillations problem, energy storage devices offer a viable solution to maintain the power sys- tem stability and its role has been elaborated in Ref. [16]. Apart from these methods, some authors have used the soft computing techniques that do not require a mathematical model of the system. These methods include the artificial intelligence tech- niques such as artificial neural networks [17], self-tuned fuzzy logic [18], bacteria foraging optimization [19] and the heuristic searching algorithm such as genetic algorithm [20], differential evolution [21], particle swarm optimization [22], harmony search algorithm [23], etc. These methods rely on iterative procedure and most of these techniques have high computation burden. Furthermore, many methods prevalent in the literature require reduction of the system model to design the controllers. With all these in view, always there is a need for finding a simple method which will result in a simple but effective controller for operation and performance. In this paper, a PI-PSS is proposed owing to its simplicity in structure, robustness, ease of implementation and maintenance. A suitable phase-lead may be obtained by employing a PI controller to compensate the phase-lag introduced between the exciter input and the electrical torque [4]. Here, the aim is to propose a design procedure that is simple in mathematics, less involved in compu- tation and independent of order (without requiring system order reduction) and structure of the system. These objectives are ach- ieved by blending two controllers, local and centralized e that re- sults in effective achievement for the damping of the system. The local controller is designed using the concept of a method as in Ref. [9] which is involved in finding only two parameters of the PI controller against finding six parameters of the conventional lead- lag PSS. Unlike in Ref. [9], proposed method investigates in finding a suitable reference model considering oscillatory/unstable dy- namics of the nominal system along with the consideration of desired response. This local controller ensures a minimum system performance even when the centralized controller becomes inef- fective in the event of any contingency. The centralized controller is proposed to design using the method as in Refs. [5,8], for reduction of the interactions among the sub-systems. Examples are taken from the literature on one machine infinite bus (with IEEE type- DC1 and IEEE type-ST1 exciters) and multi-machine power systems. The rest of the paper is organized as follows. The design methodology is shown in Section 2. In Section 3 simulation of four examples taken from the literature is illustrated. Finally, conclusion is drawn in Section 4. 2. Design methodology The small perturbation based block diagram of the one machine infinite bus (OMIB) system with IEEE type DC excitation as shown in Fig. 1 is considered here [24] for design of the PI-PSS as the local controller. 2.1. Design of the local PI controller The design method presented in Ref. [9] is followed here for design of the PI controller for mitigating the low frequency oscil- lations. The overall transfer function of the control system with the unknown controller has been derived analytically considering the mechanical torque deviation ðDTmÞ as input and the speed devia- tion ðDuÞ as output as given by: where, HðsÞ is the controller to be designed. GAðsÞ; GUðsÞ and GSðsÞ represents the transfer function of the AVR, the saturation compensation and the AVR stabilization circuit, respectively, while G1ðsÞ consisting of all these three transfer functions represents the IEEE Type-DC1 exciter system [25]. Nomenclature xd; xq Direct and quadrature axis synchronous reactance, in order x0 d Direct axis transient reactance xe; re Transmission line reactance and resistance, in order vd; id; fd Direct axis voltage, current and flux linkage, in order vq; iq; fq Quadrature axis voltage, current and flux linkage, in order ifd Field current E0 q Q-axis generator internal voltage EFD Field voltage V∞, Vt Infinite bus and terminal voltage, in order u Angular velocity KA, TA Exciter gain and time constant, in order T0 d0 Open-circuit generator field time constant M Inertia constant D Prime mover damping Tm, Te Prime mover and Electrical torque, in order q Angular position of direct axis w.r.t. stator D Small excursions about an initial operating point d Angle between quadrature axis and infinite bus voltage KE, TE Exciter gain and time constant, in order KF ;TF Regulating stabilizing circuit gain and time constant, in order VF Stabilizing feedback signal of IEEE Type-DC1 excitation system TCðsÞ ¼ G3ðsÞ½1 þ G1ðsÞG2ðsÞK6ðsÞ Š 1 þ G3ðsÞG4ðsÞK1 À G1ðsÞG2ðsÞG3ðsÞG4ðsÞK2K5 À G2ðsÞG3ðsÞG4ðsÞK2K4 þG1ðsÞG2ðsÞK6 þ G1ðsÞG2ðsÞG3ðsÞK2HðsÞ þ G1ðsÞG2ðsÞG3ðsÞG4ðsÞK1K6 (1) A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 111
  3. 3. Considering, nðsÞ ¼ G3ðsÞ½1 þ G1ðsÞG2ðsÞK6ðsÞ Š; dðsÞ ¼ 1 þ G3ðsÞ G4ðsÞK1 À G1ðsÞ G2ðsÞ G3ðsÞ G4ðsÞK2K5 À G2ðsÞ G3ðsÞ G4ðsÞK2K4 þ G1ðsÞG2ðsÞK6 þ G1ðsÞ G2ðsÞ G3ðsÞ G4ðsÞK1K6 and gpðsÞ ¼ G1ðsÞG2ðsÞG3ðsÞK2 the closed-loop transfer function along with the controller HðsÞ may be written as: TcðsÞ ¼ nðsÞ dðsÞ þ gpðsÞHðsÞ (2) The procedure for designing the local controller may be stated in the following steps. Step (a) From the step response of the open-loop system the peak overshoot ðypÞ, the settling time ðtsÞ, damping factor ðzÞ, un- damped natural frequency ðunÞ, etc. are determined. Step (b) An achievable reference model, Mref is chosen based on the desired requirements and the plant behavior. Step (c) The transfer function of the closed-loop system along with the unknown controller is derived as given by eqn. (2). Step (d) Here, the closed-loop control system TcðsÞ is considered to be following the performance ofMref ðsÞ. Hence, it may be written that TcðsÞyMref ðsÞ wherefrom, the expression of the controller is analytically derived as: HðsÞ ¼ nðsÞ À Mref ðsÞ Â dðsÞ Mref ðsÞ Â gpðsÞ (3) Step (e) The controller HðsÞ is now approximated in the form of a PI controller, CðsÞ in terms of the frequency response using the divided difference calculus as in Ref. [26] to obtained the following relations: CðsÞyHðsÞ where C ðsÞ ¼ KP þ KI s (4) CRðuÞju¼uk yHRðuÞju¼uk ; k2½0; N À 1Š CIðuÞju¼uk yHIðuÞju¼uk ; k2½0; N À 1Š (5) where, CRðuÞ; CIðuÞ; HRðuÞ and HIðuÞ are the real and the imaginary parts of CðsÞ and HðsÞ, respectively, and are all real functions of u. Here, uk are sufficiently small frequency points around u ¼ 0. In this case, for designing a PI controller with two unknowns only one low frequency point is required. Step (f) A suitable frequency point of matching u0 is chosen from the frequency response of the open-loop system for the purpose of matching. It is evident from eqn. (5) that a value of u0 resulting in two linear algebraic real equations with the unknown parameters ðKp and KiÞ of the controller. The following expressions are obtained: aK ¼ b a ¼ 2 6 4 1 0 0 À 1 u0 3 7 5; K ¼ ½ KP KI ŠT b ¼ ½ HRðu0Þ HIðu0Þ Š 9 = ; (6) 2.2. Criteria for the selection of a suitable reference model As the nominal system has low damping factor, the step response of the system is highly oscillatory. In such case, designing a control system aiming to have enormously high damping factor without any oscillation would be difficult and may lead to degra- dation of other performances of the system such as the controller output, etc. This is more stringent in case of unstable system that has negative damping factor. Therefore, a pair of complex conjugate poles (allowing some oscillations in step response) is taken for the reference model which would lead to an easily achievable perfor- mance for the control system to be designed. Obviously, the damping factor of the reference model is chosen a higher value than that of the nominal system. In order to have zero steady-state speed deviation, one zero at origin in the s-plane is considered. In the sequel, good overall responses have been obtained that are illustrated through the examples taken from the literature. 2.3. Design of the centralized controller For multi-machine systems, the overall power system is Fig. 1. Control configuration of OMIB with IEEE Type DC1 excitation system and PI-PSS. A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121112
  4. 4. assumed to be divided into separate sub-systems. When the sub- systems are interconnected, the interactions act as perturbations [27]. In order to reduce the effects of interactions, a centralized controller is utilized and the strategy shown in Refs. [5,8], has been adopted here. For the purpose of minimizing the effect of in- teractions among various sub-systems in a multi-machine power system (with the state-space matrices A; B; C), the global gain matrix G is obtained as follows: G ¼ ÀB0 H where Hij ¼ Aij; isj ¼ 0; i ¼ j (7) where, B0 is the pseudo-inverse of B and Aij represents the off- diagonal elements of state matrix. The elements of matrix H are obtained by replacing the diagonal elements of the state matrix A by zero. Now, the appropriate entries in the centralized controller matrix are replaced by the local controller parameters to finally obtain the overall controller that incorporate the effect of both the controllers in the interconnected system. 3. Simulation results 3.1. Example 1: an OMIB with IEEE type-DC1 excitation system An OMIB with IEEE Type-DC1 excitation system (Fig. 1) is considered here for which the linearized state-space model is available in Refs. [4,6]. As the proposed design procedure works with the transfer function model, the transfer function of the nominal system (with DTm, the mechanical torque deviation as input and Du, the speed deviation as output) has been derived as: From (8), the open-loop poles are at À8:132±j8:985; À0:234±j10:773; À3:0854; and À1:55: The damp- ing factor ðzÞ and natural frequency ðunÞ corresponding to the dominant poles ðÀ0:234±j10:773Þ are 0.0217 and 10.8 rad/s, respectively. The step response of the open-loop system with 0.05 pu input and the bode plot are shown in Fig. 2. It is observed that the step response is highly oscillatory. According to the criteria for selection of the achievable reference model (given in Section 2.2), a pair of conjugate complex poles with z ¼ 0:53 and un ¼ 10:8 rad/s are considered along with a zero at origin of the s-plane to ensure zero steady-state speed deviation. Hence, the reference model, Mref ðsÞ is chosen as: Mref ðsÞ ¼ 0:0193s ½1 þ sð0:052 þ 0:84jފ ½1 þ sð0:052 þ 0:84jފ ¼ 0:0193s 0:0099s2 þ 0:1049s þ 1 (9) For the chosen reference model, the settling time ðtsÞ is 0.859 s, the peak overshoot ðypÞ is 5:163  10À4s and time for peak over- shoot ðtpÞ is 0.1212 s. According to the design procedure, a fre- quency point of u0 ¼ 0:05 rad=s is chosen and a PI controller has been obtained as: CðsÞ ¼ À21:068 À 0:163 s (10) Then, the closed-loop control system becomes: TcðsÞ ¼ 0:211s5 þ4:509s4 þ50:26s3 þ183:9s2 þ157:9s s6 þ21:37s5 þ353:4s4 þ3301s3 þ19160s2 þ56000sþ57520 (11) The closed-loop poles are À4:66 ± j5:22; À 3:08±j11:8; À3.80 and À 2:09. The z and un corresponding to the dominant pole are 0.666 and 7 rad/s, respectively which shows a significant improvement on the damping factor. For the purpose of Fig. 2. Open-loop bode plot and step response (Example 1). ToðsÞ ¼ 0:211s5 þ 4:509s4 þ 50:26s3 þ 183:9s2 þ 157:9s s6 þ 21:37s5 þ 353:4s4 þ 3301s3 þ 27520s2 þ 88740s þ 81900 (8) A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 113
  5. 5. performance comparison 0.05 p. u. step change in DTm has been considered here. Performance comparison of the closed-loop con- trol system with that of the reference model is shown in Fig. 3 which shows that the designed control system is closely following the response of the reference model chosen. The performance of the proposed control system has been compared with that of the control systems proposed by Refs. [4,6,25,28e30], in Table 1 where as some of these controllers ([6,25,28 and 29]]) are further compared graphically in Figs. 4 and 5. It is observed from the tables and the figures that the proposed method gives the highest damping factor and is comparable with Fig. 3. Performance comparison of bode plots and step responses of the reference model and the design system (Example 1). Table 1 Performance comparison among various control systems for 5% change in DTm (Example 1). Method Controller tSðsÞ ypð  10À4Þ tpðsÞ x TVCO ISE ð  10À7Þ Huang Chen [28] À 23:67 À 0:1109 s 1.391 À10.25 0.1581 0.282 2.221 2.041 Huang et al. [6] À 16:305 À 0:3608 s 3.901 À10.17 0.1558 0.164 2.786 6.348 Lee Wu [29] À 11:37 À 0:2288 s 2.671 À10.08 0.1530 0.141 2.079 2.096 Bhattacharya [25] À 18:08 À 0:2483 s 2.168 10.18 0.1562 0.159 0.979 1.876 Feliachi et al. [30] À 23:63 À 0:1113 s 1.388 À10.24 0.1585 0.281 2.411 2.085 Hsu et al. [34] 20ð1þ0:017sþ0:02s2 Þ ð1þ0:05sÞ ð1þ0:05sÞ 2.692 À9.192 0.1336 0.111 8.603 2.017 Hsu Hsu (Optimal control) [4] K ¼ ½12:61 0:157 0:0039 À0:935 4:66 À304:3Š 0.7682 ¡8.182 0.1280 0.361 8.604 1.061 Hsu Hsu (Sub-optimal control) [4] À 19:24 À 0:219 s 2.033 À10.19 0.1558 0.222 2.210 1.871 Hsu Hsu (Root-locus) [4] À 29 À 0:23 s 1.562 À10.36 0.1618 0.259 2.345 2.119 Proposed À 26 À 0:163 s 1.148 À10.22 0.1580 0.666 0.579 2.038 Fig. 4. Comparison of step responses due to 0.05 p. u. change in DTm (Example 1). Fig. 5. Comparison of controller outputs due to 0.05 p. u. change in DTm (Example 1). A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121114
  6. 6. the other methods in terms of settling time ðtsÞ, time for peak overshoot ðtpÞ, peak overshoot ðypÞ, total variance of controller output (TVCO) which is a measure of smoothness of the controller output and integral square error (ISE). 3.2. Example 2: synchronous generator with IEEE type-ST1 excitation system A synchronous generator with IEEE-Type ST1 exciter connected to an infinite bus through transmission line is considered here. The relevant system parameters, system matrices and the linearized state-space representation of this test system are available in Ref. [31]. In this system, G1 ¼ KA=ð1 þ sTAÞ. The following two cases with different operating conditions have been considered as follows: 3.2.1. Case a: nominal operating condition with active power ¼ 136 MW and reactive power ¼ 83.2 MVAr The transfer function of the nominal system with DTm as input and Du as output is computed as: ToðsÞ ¼ 0:211s3 þ 4:336s2 þ 21:5s s4 þ 20:55s3 þ 192s2 þ 1828s þ 7985 (12) The open-loop poles are À0:3349±j9:422; À12:946 and À 6:936. The z and un corresponding to the dominant poles ðÀ0:3349±j9:422Þ are 0.0355 and 9.43 rad/s, respectively. For the design of achievable reference model, a pair of conjugate complex poles with z ¼ 0:295 and un ¼ 8:45 rad=s is considered with a zero fixed at origin of the s-plane. With this, the transfer function of the reference model is chosen as: Mref ðsÞ ¼ 0:00269s ½1 þ sð0:035 þ 0:1128jފ½1 þ sð0:035 À 0:1128jފ ¼ 0:00269s 0:0139s2 þ 0:0697s þ 1 (13) For the chosen reference model, the settling time ðtsÞis 1.739 s, the peak overshoot ðypÞ is 1:533  10À4s and time for peak over- shoot ðtpÞ is 0.1658 s. With a frequency point of matching at u0 ¼ 0:08 rad=s, the PI controller is designed as: CðsÞ ¼ À7:8604 À 0:332 s (14) With this controller, the transfer function of the closed-loop is obtained as: TcðsÞ ¼ 0:211s3 þ 4:336s2 þ 21:51s s4 þ 20:55s3 þ 192s2 þ 1481s þ 3392 (15) The z and un corresponding to the dominant pole ðÀ2:14±j8:68Þ are 0.24 and 8.94 rad/s, respectively, which shows a significant improvement in the damping factor. 3.2.2. Case b: unstable operating point with active power ¼ 112 MW and reactive power ¼ À32MVAr The open-loop transfer function of the nominal system with same input and output is as follows: ToðsÞ ¼ 2:11s3 þ 4:336s2 þ 10:91s s4 þ 20:55s3 þ 125:3s2 þ 1472s þ 4631 (16) In this case, z and un corresponding to the dominant poles are À0.0308 and 8.45 rad/s, respectively. A pair of complex conju- gate poles having z ¼ 0:250 and un ¼ 7:25 rad=s with zero at origin is considered for the reference model as given by: Mref ðsÞ ¼ 0:00235s ½1 þ sð0:035 þ 0:134jފ½1 þ sð0:035 À 0:134jފ ¼ 0:00235s 0:0191s2 þ 0:0689s þ 1 (17) For the chosen reference model, the settling time ðtsÞis 2.142 s, the peak overshoot ðypÞis 1:028  10À4s and time for peak over- shoot ðtpÞ is 0.1791 s. A frequency point of matching u0 ¼ 0:14 rad=s is chosen to obtain the PI controller represented as: CðsÞ ¼ À9:11 À 0:186 s (18) The transfer function of the close-loop control system becomes: TcðsÞ ¼ 0:211s3 þ 4:336s2 þ 10:91s s4 þ 20:55s3 þ 125:3s2 þ 953:7s þ 1305 (19) For the designed system, the z and un corresponding to the dominant pole are 0.197 and 7.04 rad/s, respectively, that shows a noteworthy improvement in the damping factor. The performance comparison with the controller in Ref. [32] for nominal and unstable operating conditions for 0.01 p. u. step change in DTm is shown in Table 2 and Fig. 6. It may be observed from this table that the proposed controller shows the better re- sults in terms of settling time and comparable results for other performance indices for both the cases. 3.3. Example 3: two machine infinite bus system A two bus two hydro plants (each having 10 machines) is considered as the test system [6]. The state vector and output vector of the system are considered as h Du1 Dd1 Deq 0 1 DeFD1 Du2 Dd2 Deq 0 2 DeFD2 i and ½ Du1 Dd1 Du2 Dd2 ŠT . 3.3.1. Design of the local controllers The system is decomposed into two sub-systems as given below [3]: Machine 1: With X1 ¼ h Du1 Dd1 De0 q1 DeFD1 i as state vec- tor, the open-loop transfer function of the nominal system with DTm as input and Du as output is determined as: ToðsÞ ¼ 0:122s3 þ 6:153s2 þ 583s s4 þ 50:7s3 þ 4821s2 þ 2585s þ 1:293  105 (20) The z and un corresponding to the dominant poles are 0.0245 and 5.20 rad/s, respectively. The reference model with a pair of complex conjugate poles (with z ¼ 0:825 and un ¼ 4:60rad=s) and a zero at origin is chosen as: Mref ðsÞ ¼ 0:0038s ½1 þ sð0:18 þ 0:1236jފ½1 þ sð0:18 À 0:1236jފ ¼ 0:0038s 0:047s2 þ 0:36s þ 1 (21) Table 2 Performance comparison for 0.01 p.u. change in DTm (Example 2). Case Method tSðsÞ ypð  10À4Þ tPðsÞ Case a Ellithy et al., 2014 [31] 5.261 2.152 1.161 Proposed 1.793 2.445 0.191 Case b Ellithy et al., 2014 [31] 10.61 1.955 1.345 Proposed 2.996 2.848 0.224 A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 115
  7. 7. For the chosen reference model, the settling time ðtsÞ is 1.095 s, the peak overshoot ðypÞ is 3:670  10À4s and time for peak over- shoot ðtpÞ is 0.2317 s. The obtained PI controller with frequency point of matching of u0 ¼ 3 rad=s is: CðsÞ ¼ À91:593 À 1:11 s (22) The transfer function of the closed-loop system becomes: TcðsÞ ¼ 0:122s3 þ 6:153s2 þ 583s s4 þ 50:7s3 þ 4821s2 þ 82380s þ 5:434  105 (23) The z and un corresponding to the dominant poles are 0.830 and 11.5 rad/s, respectively that show appreciable improvement in damping factor. Machine 2: With X2 ¼ h Dd2 Du2 De0 q2 DeFD2 i as state vec- tor, the open-loop transfer function of the nominal system with DTm as input and Du as output is computed as: ToðsÞ ¼ 0:1237s3 þ 6:223s2 þ 248:2s s4 þ 50:55s3 þ 2086s2 þ 3869s þ 1:401  105 (24) The z and un corresponding to the dominant poles are 0.0105 and 8.35 rad/s, respectively. For designing the reference model, a pair of complex conjugate poles with z ¼ 0:7 and un ¼ 11:5 rad=s is chosen with a zero at origin as given by: Mref ðsÞ ¼ 0:0018s ½1 þ sð0:061 þ 0:062jފ½1 þ sð0:061 À 0:062jފ ¼ 0:0018s 0:0076s2 þ 0:122s þ 1 (25) For the chosen reference model, the settling time ðtsÞ is 0.616 s, the peak overshoot ðypÞ is 4:793  10À4s and time for peak over- shoot ðtpÞ is 0.097 s. Considering, the frequency point of matching at u ¼ 2:12 rad=s, the designed PI controller is obtained is obtained as: CðsÞ ¼ À65:683 À 1:2586 s (26) With this, the closed-loop control system becomes: TcðsÞ ¼ 0:122s3 þ 6:153s2 þ 583s s4 þ 50:7s3 þ 4821s2 þ 39590s þ 3:981  105 (27) The z and un corresponding to the dominant poles are 0.759 and 19 rad/s, respectively that show a significant improvement in damping factor. 3.3.2. Design of the centralized controller For the design of the centralized controller, a reduced order power system, as stated in Ref. [8], is considered here to achieve approximate optimum performance. For this purpose, a reduced order model is obtained by eliminating the states that have the least contribution to input/output behavior by using the Hankel singular values. The reduced order state model ðAr; Br; CrÞ is ob- tained as: Ar ¼ 2 6 6 4 À0:244 À0:073 0 0:0731 377 0 0 0 0 0:1843 À0:2473 À0:1847 0 0 377 0 3 7 7 5 Br ¼ À0:2212 0 0:1017 0 0:07233 0 À0:3008 0 ; Cr ¼ 1 1 0 0 0 0 1 1 9 = ; (28) The gain matrix for global control using eqn. (28) for minimizing the interaction between the sub-systems as in Ref. [8] is computed as: G ¼ 0 0:1458 0 À0:1459 0 À0:5634 0 0:5647 (29) It is to note, the bold faced diagonal blocks of the gain matrix G are to be replaced by the local controller gains that are already designed to arrive at the overall gain matrix. Accordingly, the overall gain matrix, including local and global controllers, is obtained as: K ¼ À91:593 À1:11 0 À0:1459 0 À0:5634 À65:683 À1:2586 (30) The simulation of the complete system has been shown in Fig. 7. The performance comparison of the proposed controller with the controllers stated in Refs. [3,5,7 and 30]] are shown in Fig. 7. From Table 3 and Fig. 7, it is observed that the proposed controllers give the better overall performances in all cases of this example than the other controllers. 3.4. Example 4: three bus infinite bus system A three bus two hydro plants is considered here with h Dd1 Du1 De 0 q1 DeFD1 Dd2 Du2 De 0 q2 DeFD2 i as state vector and ½ Dd1 Du1 Dd2 Du2 Dd3 Du3 Š as output vector [32]. Fig. 6. Comparison of step responses for 0.01 p. u. change in DTm (Example 2): (a) for case a, (b) for case b. A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121116
  8. 8. Fig. 7. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other or both machines (Example 3). A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 117
  9. 9. 3.4.1. Design of the local controllers The system is decomposed into three machines as discussed below: Machine 1: With h Dd1 Du1 De0 q1 DeFD1 i as state vector, the transfer function of the open-loop system with DTm1 as input and D u1 as output is: T0ðsÞ ¼ 0:2165s3 þ 4:529s2 þ 17:18s s4 þ 21:6s3 þ 137:7s2 þ 1164s þ 4226 (31) The z and un corresponding to the dominant poles are 0.0189 and 7.42 rad/s, respectively. For the above transfer function model, a pair of conjugate complex poles is considered with z ¼ 0:325 and un ¼ 7:15 rad=s with a zero at origin. The chosen reference model is given as: Mref ðsÞ ¼ 0:0031s ½1 þ sð0:049 þ 0:132jފ½1 þ sð0:049 À 0:132jފ ¼ 0:0031s 0:00195s2 þ 0:098s þ 1 (32) For the chosen reference model, the settling time ðtsÞ is 1.738 s, the peak overshoot ðypÞ is 6:852  10À4s and time for peak over- shoot ðtpÞ is 0.1761 s. Choosing the frequency point of matching at u0 ¼ 6:4 rad=s, the designed PI controller obtained as: CðsÞ ¼ À29:733 þ 0:5953 s (33) The transfer function of the closed-loop is obtained as. TCðsÞ ¼ 0:2165s3 þ 4:529s2 þ 17:18s s4 þ 21:06s3 þ 137:7s2 þ 1475s þ 1891 (34) The z and un corresponding to the dominant poles are 0.114 and 8.64 rad/s, respectively that shows the increase in damping factor. Machine 2: With X2 ¼ h Dd2 Du2 De0 q2 DeFD2 i as state vec- tor, the transfer function of the open-loop system with DTm2 as input and Du2 as output is: ToðsÞ ¼ 0:1471s3 þ 2:972s2 þ 3804s s4 þ 20:06s3 þ 81:38s2 þ 1125s þ 1677 (35) The z and un corresponding to the dominant poles are À0.0142 and 7.45 rad/s, respectively. A pair of conjugate complex poles is considered with z ¼ 0:295 and un ¼ 4:75 rad=s for selection of the reference model as given by: Mref ðsÞ ¼ 0:0055s ½1 þ sð0:062 þ 0:201jފ½1 þ sð0:062 À 0:201jފ ¼ 0:0055s 0:0443s2 þ 0:129s þ 1 (36) For the chosen reference model, the settling time ðtsÞ is 3.107 s, the peak overshoot ðypÞ is 8:804  10À4s and time for peak over- shoot ðtpÞ is 0.296 s. A frequency point of matching u0 ¼ 4:7 rad/s is chosen for obtaining the PI controller and as is computed as: CðsÞ ¼ À22:436 þ 0:4699 s (37) The closed-loop control system becomes: TcðsÞ ¼ 0:1471s3 þ 2:972s2 þ 3:804s s4 þ 20:18s3 þ 81:38s2 þ 1286s þ 401:2 (38) The z and un corresponding to the dominant poles are 0.0315 and 8.08 rad/s, respectively that clearly indicate the improvement of damping factor. Machine 3: With X3 ¼ h Dd3 Du3 De 0 q3 DeFD3 i as state vector, the transfer function, taking DTm3 as input and Du3 as output is determined for the system considered. TOðsÞ ¼ 0:108s3 þ 2:181s2 þ 6:3s s4 þ 20:21s3 þ 79:79s2 þ 415:1s þ 1388 (39) The z and un corresponding to the dominant poles are À0.0358 and 4.61 rad/s, respectively. For selection of the reference model, a Table 3 Performance comparison (Example 3). Method Abdel-Magid and Aly [3] Flechai et al. [30] T. Huang et al. [5] C. Huang et al. [7] Proposed Transient Response in Du1 due to 0.05 p.u. change in DTm1 tSðsÞ 1.332 2.688 0.9229 0.5868 0.6113 ypð  10À4Þ 8.790 10.44 7.036 5.744 4.672 tpðsÞ 0.1194 0.1910 0.1403 0.1170 0.1082 Transient Response in Du2 due to 0.05 p.u. change in DTm1 tSðsÞ 1.466 3.128 0.6884 0.7087 0.5857 ypð  10À4Þ 1.754 12.86 6.348 6.216 4.138 tpðsÞ 0.2288 0.1910 0.1079 4.138 0.0812 Transient Response in Du1 due to 0.05 p.u. change in DTm2 tSðsÞ 1.488 3.121 1.179 0.9961 0.6450 ypð  10À4Þ 6.525 6.379 1.897 1.215 1.3053 tpðsÞ 0.2286 0.4456 0.2266 0.2797 0.1677 Transient Response in Du2 due to 0.05 p.u. change in DTm2 tSðsÞ 1.339 2.995 1.1817 0.5868 0.7201 ypð  10À4Þ 2.026 11.58 1.006 5.744 1.8232 tpðsÞ 0.2785 0.2266 0.2267 0.1177 0.1407 Transient Response in Du1 due to 0.05 p.u. change in both DTm1 and DTm2 tSðsÞ 1.322 2.793 0.7708 1.1792 0.5947 ypð  10À4Þ 8.796 13.46 6.7524 1.8973 5.072 tpðsÞ 0.1193 0.1910 0.1325 0.2266 0.0974 Transient Response in Du2 due to 0.05 p.u. change in both DTm1 and DTm2 tSðsÞ 1.128 2.078 1.159 0.8701 0.5057 ypð  10À4Þ 9.251 20.40 10.57 7.071 6.432 tpðsÞ 0.1293 0.3356 0.1835 0.1472 0.1244 A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121118
  10. 10. pair of conjugate complex poles are considered with z ¼ 0:215 and un ¼ 4:484 rad/s and a zero at origin as given by: Mref ðsÞ ¼ 0:0046s ½1 þ sð0:044 þ 0:2017jފ½1 þ sð0:044 À 0:2017jފ ¼ 0:0046s 0:0426s2 þ 0:088s þ 1 (40) For the chosen reference model, the settling time ðtsÞ is 3.807 s, the peak overshoot ðypÞ is 8:253  10À4s and time for peak over- shoot ðtpÞ is 0.2676 s. With frequency point of matching at u0 ¼ 1 rad=s, the PI controller is obtained as: C3ðsÞ ¼ À28:649 þ 0:0433 s (41) The closed loop system is obtained as: TCðsÞ ¼ 0:108s3 þ 2:181s2 þ 6:3s s4 þ 20:21s3 þ 79:79s2 þ 673:1s þ 1237 (42) The z and un corresponding to the dominant poles are 0.041 and 5.76 rad/s, respectively that shows the effective improvement in the damping of the system. It is to note that the damping factor of the nominal system is negative as the system is unstable. 3.4.2. Design of the centralized controller For the design of the centralized controller, first the reduced order power system model is obtained as: Ar ¼ 2 6 6 6 6 6 6 4 0 377 0 0 0 0 À0:1418 0:0225 0:0196 0:0056 0:0399 0:2342 0 0 0 377 0 0 0:0098 À0:0019 À0:1724 0:2034 0:0714 0:0312 0 0 0 0 0 377 0:0031 À0:0025 0:0165 À0:0087 À0:0642 0:165 3 7 7 7 7 7 7 5 Br ¼ 2 6 6 6 6 6 6 4 0 0 0 À0:1379 0:0132 0:0747 0 0 0 À0:0318 À0:2781 0:0601 0 0 0 À0:0214 0:0128 À0:1527 3 7 7 7 7 7 7 5 ;Cr ¼ 0 1 0 1 0 1 1 0 1 0 1 0 9 = ; (43) Following the procedure illustrated in the earlier example the gain matrix for centralized controller using Eqn. (43) is obtained as: Now, the bold faced diagonal blocks of the gain matrix G are replaced by the local controller gains that are already designed in Section 3.4.1 to obtain the overall gain matrix: The performance comparison of the proposed controller with some of the controllers stated in Refs. [5,22,33] under the various operating conditions has been shown in Table 4. The simulation results employing all these controllers have been shown in Fig. 8. By comparing Fig. 8 and Table 4 simultaneously, it is clear that the proposed controller provides the better performances than the others controllers taken from the literature. Table 4 Performance comparison (Example 4). Method Huang et al. [5] Solimon et al. [22] Optimal [33] Proposed Transient Response in Du1due to 0.05 p.u. change in DTm1 tSðsÞ 3.261 7.511 1.834 1.959 ypð  10À4Þ 12.688 14.317 12.59 13.933 tpðsÞ 0.1848 0.2080 0.1888 0.2747 Transient Response in Du1 due to 0.05 p.u. change in DTm2 tSðsÞ 3.910 8.266 3.369 2.498 ypð  10À4Þ 2.554 11.05 0.8050 0.7055 tpðsÞ 0.4681 1.491 0.6421 0.5219 Transient Response in Du1 due to 0.05 p.u. change in DTm3 tSðsÞ 2.864 7.907 3.3212 2.9262 ypð  10À4Þ 5.171 8.021 0.2485 4.086 tpðsÞ 0.4434 0.4161 0.3273 0.4945 Transient Response in Du2 due to 0.05 p.u. change in DTm1 tSðsÞ 4.067 8.296 3.356 2.487 ypð  10À4Þ 4.353 3.359 1.255 3.709 tpðsÞ 0.6651 2.080 0.6295 0.5219 Transient Response in Du2 due to 0.05 p.u. change in DTm2 tSðsÞ 2.566 7.813 2.186 1.601 ypð  10À4Þ 9.239 9.874 8.824 8.093 tpðsÞ 0.2093 0.2080 1.889 0.2671 Transient Response in Du2 due to 0.05 p.u. change in DTm3 tSðsÞ 2.618 7.588 2.535 2.717 ypð  10À4Þ 6.339 5.183 1.308 2.624 tpðsÞ 0.4680 0.4854 0.4406 0.4121 Transient Response in Du3 due to 0.05 p.u. change in DTm1 tSðsÞ 3.534 7.807 4.029 2.923 ypð  10À4Þ 2.352 1.985 0.1520 2.243 tpðsÞ 0.7152 0.9014 0.9064 0.4669 Transient Response in Du3 due to 0.05 p.u. change in DTm2 tSðsÞ 3.051 7.759 2.838 2.647 ypð  10À4Þ 1.863 4.317 1.108 0.3653 tpðsÞ 0.4081 0.5162 0.4532 0.3846 Transient Response in Du3 due to 0.05 p.u. change in DTm3 tSðsÞ 1.944 5.966 1.257 2.397 ypð  10À4Þ 7.554 6.289 7.197 6.5024 tpðsÞ 0.2340 0.1734 0.2140 0.2472 G ¼ 2 4 0:92212 0:00971 À0:10465 À0:007508 À0:07846 À1:5628 À0:17662 0:009278 0:62303 0:01369 À0:15743 0:11626 À0:16465 0:016288 À0:04113 0:059415 0:41826 0:22959 3 5 (44) K ¼ 2 4 0:5953 ­29:733 À0:10465 À0:007508 À0:07846 À1:5628 À0:17662 0:009278 0:4699 ­22:436 À0:15743 0:11626 À0:16465 0:016288 À0:04113 0:059415 0:0433 ­28:649 3 5 (45) A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 119
  11. 11. Fig. 8. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other machines (Example 4). A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121120
  12. 12. 4. Conclusion An enhancement of the small-signal stability of the power sys- tem in terms of improving the damping factor using a two-level control scheme has been proposed in this paper. In the local level the proposed design method is based on frequency domain model matching that avoids elaborate frequency response analysis and outperforms extensive mathematical calculations. This matching method proposed here is independent of the order and structure of the system and, hence, does not require reduction of the system before designing the controllers. Suitable reference models for the purpose of model matching are developed that are easily achiev- able by the system dynamics fulfilling the desired response. Close matching of the designed systems with the reference models is obtained following the design process. In the global level, the proposed controller works on minimizing the effect of inter- connection of the sub-systems leading to improvement in overall responses. The proposed design method has been applied on four examples taken from the literature with four different configurations of po- wer system such as OMIB with IEEE Type-DC1 excitation system, OMIB with IEEE Type-ST1 excitation system, two-machine infinite bus system and three-machine infinite bus system. The simulation results are elaborately illustrated through figures and tables, which evidently show the efficacy of the proposed scheme in improving the damping of the low frequency oscillations appreciably. The local controllers or its blending with the centralized (global) controllers (as the case may be) shows favorable performance when compared with some controllers prevalent in the literature. The extension of the proposed control strategy for large power system and consid- eration of time-delay in communication channels along with practical implementation of the proposed controllers may be considered as relevant future work. References [1] Pai MA, Gupta DPS, Padiyar KR. 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