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Load–frequency control of interconnected power system with governor backlash nonlinearities
Electrical Power and Energy Systems 33 (2011) 1542–1549
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Load–frequency control of interconnected power system with governor backlash nonlinearities Tain-Sou Tsay ⇑ Department of Aeronautical Engineering, National Formosa University, 64, Wen-Hua Road, Huwei, YunLin 63208, Taiwan
a r t i c l e
i n f o
Article history: Received 10 May 2010 Received in revised form 12 February 2011 Accepted 3 June 2011 Available online 4 August 2011 Keywords: Interconnected power system Nonlinear multivariable feedback system Limit cycle
a b s t r a c t In this paper, the stability-equation method is applied to the analysis and design of an interconnected power system with governor backlash nonlinearities. The considered system is a nonlinear multivariable feedback control system. The governor nonlinearities tend to produce a sustaining oscillation in area frequency and tie-line power transient responses. Most conventional linear design techniques are usually unable to find the sustaining oscillation in design phase and need simulation verifications to check the validations after designs. However, the proposed method can consider effects of nonlinearities in the design phase. Some nonlinear design techniques need parameters optimization method by Lyapunov theorem or Integral of Square Errors (ISE) criteria. They are effective. However, they need large computation efforts. The proposed method can choice frequency bias parameters and integrator gains of supplementary controllers for avoiding the oscillation or reducing the amplitude of the oscillation to be acceptable. Simulation verifications show that the proposed method can provide a simple and effective way for the considered system. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction For reasons of economy and system reliability, neighboring power systems are interconnected, forming an augmented system referred to as ‘‘power pool’’. The various areas are interconnected through tie-lines. The net power flow on the tic-lines connecting a system to the external system is frequently scheduled by a priori contract basis. The tie-lines are used for contractual energy exchange between areas and provide inter-area support in case of abnormal conditions. Area load changes and abnormal conditions, such as outages of generation, lead to mismatches in frequency and scheduled power interchanges between areas. These mismatches have to be corrected through a Load–Frequency Control (LFC). The load–frequency control is based on an error signal called Area Control Error (ACE) which is a linear combination of net-interchange and frequency errors. The conventional control strategy used in industry is to take the integral of ACE as the control signal [1–14]. The control purpose is to reduce the frequency and tie-line power errors to zero in the steady state. Many decentralized control strategies; e.g., variable structure controls [5–7], PI/PID and Fuzzy controls [9–14], have been employed in the design of LFC for interconnected power systems. Among the various types of decentralized LFC, the most widely employed is the simple conventional control. The conventional con⇑ Tel.: +886 5 631 5537; fax: +886 5 631 2415. E-mail address: [email protected] 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.06.005
trol for LFC is still popular with the power industries because of their simplicity, easy realization, low cost, robust and decentralized nature of the control strategy. It has also been shown that there is no significant advantage in using the more complex controllers over the conventional controllers [9–11]. The conventional proportional plus integral control strategy, which is widely used in power industry, is to take the integral of ACE as the control signal. It is well known that many LFC scheme does not yield adequate control performance with consideration of the system nonlinearities such as governor deadband or generation rate constraint [1– 4]. Governor deadband (GDB) nonlinearities tend to produce an unexpected sustaining oscillation in area frequency and tie-line power transient responses. Avoiding sustaining oscillation or reducing the amplitude of the sustaining oscillation is expected. Most conventional linear design techniques [5–14] need digital simulations to check effects of nonlinearities after controller designed. The nonlinearities considered in the design phase are expected. Some nonlinear design techniques need parameters optimization method by Lyapunov theorem [1–3] or Integral of Square Errors (ISE) criteria [4]. They are effective. However, they need large computation efforts. Simple and effective ways to evaluate effects of nonlinearities are expected. This is the motivation of this paper. In this paper, the stability-equation method [15–17] is used to analyze and design the considered system. The considered system is a nonlinear multivariable feedback control system. Harmonicbalance equations and the characteristic equation [18–21] are
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
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Nomenclature ACEi Ai Bi D Ki Ri Tgi Tti Tri Kri Tpi Kpi DFi
area control error of area i amplitude of sinusoidal input of Ni frequency bias parameter for area control error of area i governor deadband integrator gain of supplementary controller of area i governor speed regulation of area i time constant of speed governor control system of area i time constant of steam turbine of area i reheat time constant of area i reheat coefficient of steam turbine of area i generator time constant of area i generator gain constant of area i frequency deviation of area i
evaluated for finding the unstable, limit-cycle and asymptotically stable regions in the parameter plane [16]. It will be see that there is no asymptotically stable region for some specified values of controlling and system parameters. This implied that sustain oscillation is always exist; i.e., the stable limit cycle is exist. In this condition, the acceptable amplitude of the sustaining oscillation is expected (e.g., tie-line power). The proposed method can provide information of amplitudes of sustain oscillations. In another way, the proposed method can be used to find the asymptotically stable region in the parameter-plane by selecting other specified values of controlling or system parameters. Furthermore, this paper can be applied the analyses and refining of the systems controlled by conventional linear control techniques; e.g., adjusting parameters of PI/PID controllers to refining the system performance. This paper is organized as follows. First, the considered interconnected power system is described. Then, the proposed method is presented. Finally the proposed method is applied to analyze and design the considered system. Simulation verification results show that the proposed method can provide a simple way for choosing proper values of parameters to avoid sustaining oscillation or reducing amplitudes of sustaining oscillations. 2. The two-area interconnected power system The two-area interconnected power system consists of two single areas connected through a power line called the tie line. Each area feeds its user pool, and the tie line allows electric power to flow between areas. Since two areas are tied together, a load perturbation in one area affects the output frequencies of both areas as well as the power flow on the tie line. The control system of each area needs information about the transient situation of both areas to bring the local frequency back to its steady state value. Information about the other area is found in the output frequency fluctuation of that area and in the tie line power fluctuation. Thus, the tie line power is sensed, and the resulting tie line power signal is fed back into both areas. A block diagram of the two area interconnected power system is given in Fig. 1 in which governors, reheat turbines and generators are considered. Transfer functions of reheat turbines are given by
DPgi ðsÞ 1 þ K ri T ri s ¼ ; DX Ei ðsÞ ð1 þ sT ri Þð1 þ sT ti Þ
i ¼ 1; 2
ð1Þ
and differential equations of the overall system [1–4] are given as: 1 DF_ 1 ¼ T 1 p1 DF 1 þ K p1 T p1 ðDP g1 DP tie DP d1 Þ
ð2Þ
h i 1 1 1 DP_ g1 ¼ T 1 r1 DP g1 þ DP r1 T r1 K r1 T r1 þ K r1 T t1 DX E1
ð3Þ
Dpgi Dpci Dpdi Dpri Dptie DXEi T12 a12 Pri Ni
mechanical power deviation of area i change in speed changer position real power increase in area i mechanical power deviation during steam reheat in area i tie-line power deviation out of area i governor valve displacement of area i synchronizing power coefficient of tie-line connected between areas 1 and 2 Pr1/Pr2 MW capacity of area i governor deadband nonlinearity of area i
1 DP_ r1 ¼ T 1 t1 DX E1 T t1 DP r1
ð4Þ
1 1 1 DX_ E1 ¼ R1 1 N 1 ðA1 ÞT g1 DF 1 T g1 DX E1 þ T g1 N 1 ðA1 ÞDP c1
ð5Þ
ACE1 ¼ B1 DF 1 þ DPtie
ð6Þ
DPc1 ¼ K 1
Z
ACE1 dt
ð7Þ
1 DF_ 2 ¼ T 1 p2 DF 1 þ K p2 T p2 ðDP g2 a12 DP tie DP d2 Þ
ð8Þ
h i 1 1 1 DP_ g2 ¼ T 1 r2 DP g2 þ DP r2 T r2 K r2 T r2 þ K r2 T t2 DX E2
ð9Þ
1 DP_ r2 ¼ T 1 t2 DX E2 T t2 DP r2
ð10Þ
1 1 1 DX_ E2 ¼ R1 2 N 2 ðA2 ÞT g2 DF 2 T g2 DX E2 þ T g2 N 2 ðA2 ÞDP c2
ð11Þ
DP_ tie ¼ 2pT 12 ðDF 1 DF 2 Þ
ð12Þ
ACE2 ¼ B2 DF 2 þ a12 DP tie
ð13Þ
DPc2 ¼ K 2
Z
ACE2 dt
ð14Þ
where N1(A1) and N2(A2) represent quasi-linear gains or describing functions [15–18] of governor deadband (backlash) nonlinearities [1–4]; A1 and A2 represent amplitudes of sinusoidal inputs of nonlinearities N1 and N2, respectively. The corresponding transfer functions of governors and generators are shown in Fig. 1. In conventional systems, an integral controller sets the turbine reference power of each area. Since a perturbation in either area affects the frequency in both areas and a perturbation in one area is perceived by the other through a change in tie line power, the controller of each area should not only; take the frequency variation as input but also the tie line power variations. Since an integral controller has just one input, these two contributions, namely local frequency variation and tie line power variation, must be combined into a single signal so that it can be used as an input of the controller. The easiest way of doing this is to combine them linearly, i.e. the input of the integrator in area 1 is DPtie + B1DF1, and the input of the integrator in area 2 is a12DPtie + B2DF2. The input of each controller is called the Area Control Error (ACE). The nonlinearities destabilize the system and tend to produce a sustaining oscillation in area and tie-line power transient responses [1–4]. There are frequency bias parameters Bi(i = 1, 2) and integrator gains Ki(i = 1, 2) of the supplementary controllers
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T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
Fig. 1. Transfer function block diagram of two-area interconnected power system.
to be chosen for desirable performance. In general, the sustaining oscillation is not desirable. In this paper, the stability-equation method [15–17] is applied to the analysis and design the considered system in parameter planes, where unstable, limit-cycle, and asymptotically stable regions can be clearly defined. The constant-Ai loci in the limit-cycle region show amplitudes (Ai) of sinusoidal inputs of governor deadband nonlinearities with specified values of Bi and Ki(i = 1, 2). Therefore, one can avoid the oscillation or reduce the amplitude of oscillation by choosing proper values of adjustable parameters Bi and Ki(i = 1, 2). It will be seen that if there is no asymptotically stable region in the parameter plane, the system will always have a sustaining oscillation for all the values of Bi and Ki(i = 1, 2) [6,7]. In this case, the design goal is to keep the amplitude of oscillation as small as possible [8]. Note that, one can improve the damping characteristics of each subsystem to stabilize the overall system, then an asymptotically stable region may exist. Although the controllers considered in this paper are simple, the proposed method can be applied easily to systems with more complicated type of supplementary controllers [5–14]; e.g., parameters of PD/PID controllers can be used analyzed parameters to refine the system performance.
3. The proposed method Fig. 2 shows the equivalent system block diagram of a two-area interconnected power system in which A1 and A2 represent amplitudes of sinusoidal inputs of nonlinearities N1 and N2, respectively; and h21 is the phase shift angle of the sinusoidal input of N2 related to N1. For zero power increment inputs (DPd1 = DPd2 = 0) of each area, harmonic-balance equations [18–21] of the system are
A1
A2 ejh21
¼
E11 ðjxÞ E12 ðjxÞ
F 11 ðjxÞ F 12 ðjxÞ
E21 ðjxÞ E22 ðjxÞ F 21 ðjxÞ F 22 ðjxÞ 0 N1 ðA1 Þ A1 0
H11 ðjxÞ H12 ðjxÞ H21 ðjxÞ H22 ðjxÞ
N2 ðA2 Þ
N1 ðA1 Þ 0
A2 ejh21 0 N2 ðA2 Þ
A1 jh21
A2 e
ð15aÞ
2 X k¼1
Eik ðjxÞF kj ðjxÞ;
i; j ¼ 1; 2
where
h i E11 ðjxÞ ¼ G3 ðjxÞ C 1 ðjxÞB1 þ C 1 ðjxÞG5 ðjxÞ R1 1
ð17Þ
E12 ðjxÞ ¼ C 1 ðjxÞG4 ðjxÞG5 ðjxÞ
ð18Þ
E21 ðjxÞ ¼ a12 C 2 ðjxÞG3 ðjxÞG5 ðjxÞ
ð19Þ
h i E22 ðjxÞ ¼ G4 ðjxÞ C 2 ðjxÞB2 a12 C 2 ðjxÞG5 ðjÞ R1 2
ð20Þ
DðjxÞ ¼ 1 þ G3 ðjxÞG5 ðjxÞ a12 G4 ðjxÞG5 ðjxÞ
ð21Þ
F 11 ðjxÞ ¼ G1 ðjxÞ½1 a12 G4 ðjxÞG5 ðjxÞ=DðjxÞ
ð22Þ
F 12 ðjxÞ ¼ G2 ðjxÞG4 ðjxÞG5 ðjxÞ=DðjxÞ
ð23Þ
F 21 ðjxÞ ¼ a12 G1 ðjxÞG3 ðjxÞG5 ðjxÞ=DðjxÞ
ð24Þ
F 22 ðjxÞ ¼ G2 ðjxÞ½1 þ G3 ðjxÞG5 ðjxÞ=DðjxÞ
ð25Þ
Solving Eq. (15b), one has
ejh21 ¼ ð15bÞ
A1 ½1 N1 ðA1 ÞH11 ðjxÞ A2 N2 ðA2 ÞH12 ðjxÞ
ð26Þ
A1 N1 ðA1 ÞH21 ðjxÞ A2 ½1 N2 ðA2 ÞH22 ðjxÞ
ð27Þ
And
where s = jx; N1(A1) and N2(A2) represent describing functions of deadbad nonlinearities N1 and N2 [15,18] with amplitudes A1 and A2, respectively; and
Hi;j ðjxÞ ¼
Fig. 2. Equivalent system block diagram of Fig. 1.
ð16Þ
ejh21 ¼
Equating Eqs. (26) and (27), one has the characteristic equation the considered system:
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
P0r ðA1 ; A2 ; xÞ þ K 1 Q 0r ðA1 ; A2 ; xÞ þ K 2 W 0r ðA1 ; A2 ; xÞ
1 þ N 1 ðA1 ÞH11 ðjxÞ þ N2 ðA2 ÞH22 ðjxÞ
þ K 1 K 2 T 0r ðA1 ; A2 ; xÞ ¼ 0
þ N1 ðA1 ÞN2 ðA2 Þ½H11 ðjxÞH22 ðjxÞ H12 ðjxÞH21 ðjxÞ ¼0
ð28Þ
Multiplying the Lowest Common Multiple (LCM) of denominators of Hij(jx) and determinate of Hij(jx) in Eq. (28), Eq. (28) can rewritten as
PðjxÞ þ N1 ðA1 ÞQ ðjxÞ þ N2 ðA2 ÞWðjxÞ þ N1 ðA1 ÞN 2 ðA2 ÞTðjxÞ ¼ 0 ð29Þ Inverting the substitution of s by jx in Eq. (29), then the characteristic equation of the considered system with complex coefficients will be get; i.e.,
PðsÞ þ N1 ðA1 ÞQ ðsÞ þ N 2 ðA2 ÞWðsÞ þ N 1 ðA1 ÞN2 ðA2 ÞTðsÞ ¼ 0
ð30Þ
where P(s), Q(s), W(s), T(s) are rational polynomials of Laplace operator s. Note that the phase shift angle h21 is not represented in the characteristic equation. Describing functions of deadband nonlinearities Ni(Ai) are complex numbers [15,18]. They are
Ni ðAi Þ ¼
1
p
(
1=2 ) 2D 2D D D þ2 1 1 1 þ sin Ai Ai Ai Ai 2 4D D j Nir ðAi Þ þ jN ii ðAi Þ 1 ð31Þ Ai pAi
p
1
where D is the dead band, and Ai is the amplitude of the sinusoidal input. Eq. (29) can be rewritten as
Pr ðxÞ þ jP i ðxÞ þ ½N1r ðA1 Þ þ jN 1i ðA1 Þ½Q r ðxÞ þ jQ i ðxÞ þ ½N2r ðA2 Þ þ jN 2i ðA2 Þ ½W r ðxÞ þ jW i ðxÞ þ ½N1r ðA1 Þ þ jN 1i ðA1 Þ½N 2r ðA2 Þ þ jN 2i ðA2 Þ½T r ðxÞ þ jT i ðxÞ ¼ 0
ð32Þ
where PðjxÞ Pr ðxÞ þ jPi ðxÞ; Q ðjxÞ Q r ðxÞ þ jQ i ðxÞ, WðjxÞ W r ðxÞ þ jW i ðxÞ and TðjxÞ T r ðxÞ þ jT i ðxÞ. Taking the real and imaginary parts of Eq. (32), then two stability-equations for analyses and designs [15–17] are found. They are
Pr ðxÞ þ ½N1r ðA1 ÞQ r ðxÞ N1i ðA1 ÞQ i ðxÞ þ ½N2r ðA2 ÞW r ðxÞ N2i ðA2 ÞW i ðxÞ þ ½N1r ðA1 ÞN2r ðA2 Þ jN 1i ðA1 ÞN2i ðA2 ÞT r ðxÞ ½N1r ðA1 ÞN2i ðA2 Þ þ N 1i ðA1 ÞN2r ðA2 ÞT i ðxÞ ¼ 0
ð33Þ
and
Pi ðxÞ þ ½N1r ðA1 ÞQ i ðxÞ þ N 1i ðA1 ÞQ r ðxÞ þ ½N2r ðA2 ÞW i ðxÞ þ N2i ðA2 ÞW r ðxÞ þ ½N1r ðA1 ÞN2r ðA2 Þ jN 1i ðA1 ÞN2i ðA2 ÞT i ðxÞ þ ½N1r ðA1 ÞN2i ðA2 Þ þ N 1i ðA1 ÞN2r ðA2 ÞT r ðxÞ ¼ 0
1545
ð34Þ
ð36Þ
and
P0i ðA1 ; A2 ; xÞ þ K 1 Q 0i ðA1 ; A2 ; xÞ þ K 2 W 0i ðA1 ; A2 ; xÞ þ K 1 K 2 T 00i ðA1 ; A2 ; xÞ ¼ 0
ð37Þ
Eqs. (36) and (37) are functions of (A1 ; A2 and x), and five parameters (K 1 ; K 2 ; A1 ; A2 and x) are needed to be found. There are an infinite number of solutions (A1 ; A2 and x) satisfy Eqs. (36) and (37) simultaneously for specified values of K1 and K2 [15–17]. For solving this problem, the following constraint equation must be added; i.e., f26g
f27g
h21 h21 ¼ 0
ð38Þ
where hf26g and hf27g represent phase angles found by Eqs. (26) and 21 21 (27), respectively; i.e., f26g h21 hf27g 21 ¼ \
A1 ½1 N1 ðA1 ÞH11 ðjxÞ A1 N1 ðA1 ÞH21 ðjxÞ \ ¼0 A2 N2 ðA2 ÞH12 ðjxÞ A2 ½1 N2 ðA2 ÞH22 ðjxÞ ð39Þ
Then, there are three simultaneously equations (i.e., Eqs. (36)– (38)) for solving three parameters (A1 ; A2 and x) of the considered system with specified values of (K1, K2). Fig. 3 shows the typical solving processes of a stable limit cycle in which the loci represent roots of Eqs. (36) and (37). They represent an infinite number possible solutions (A1 ; A2 and x) of limit-cycle. But, the unique solution represented by Q 1 (1:179 1103 ; 1:179 103 ; 2:5294) is found from Eq. (39). This is called stable limit cycle. The other solutions are called unstable limit cycles. For controller designs purpose, another parameters finding procedure for (A2 , K 1 and K 2 ) of Eqs. (36) and (37) are evaluated for specified values of (A1 ; x), and plotted in the K1 vs. K2 plane; i.e., parameter plane. The constant-A1 locus [16], which is plotted in the K1 vs. K2 plane, is constituted by simultaneous solutions for a specified value of amplitude (A1) and a number of frequencies (x). For a number of constant-A1 loci, the unstable, limit-cycle and asymptotically stable regions of the considered system can be clearly defined in the K1 vs. K2 plane. Those three regions are classified by constant-A1 ð1Þ and A1 (0) loci. For example, the constant-A1 ð1Þ locus represents the stability boundary between the limit-cycle region and the unstable region; the constant-A1 ð0Þ locus represents the boundary between the limit-cycle region and the asymptotically stable region. If parameters (K 1 , K 2 ) are chosen in the limit-cycle region, then sustaining oscillation will occur. The amplitude (A1 ) and frequency
Eqs. (33) and (34) are rational polynomial of the frequency x. Roots of Eq. (33) are (xei, i = 1, . . . , nr) and t roots of Eq. (34) are (xoj,j = 1, . . . ,ni). nr and ni are maximal order of x of Eqs. (33) and (34); respectively. The stability criteria [15–17] of considered nonlinear feedback control systems developed are (a) system is stable: xei and xoj are all real and alternative in sequence; (b) sustaining oscillating: xei and xoj are all real and alternative in sequence except that one root pair is equal to each other (i.e., xei = xoj = x); (c) system is unstable: xei and xoj are not all real or and not alternative in sequence. Another representation of the Eq. (28) for controllers design is
P0 ðA1 ; A2 ; jxÞ þ K 1 Q 0 ðA1 ; A2 ; jxÞ þ K 2 W 0 ðA1 ; A2 ; jxÞ þ K 1 K 2 T 0 ðA1 ; A2 ; jxÞ ¼ 0
ð35Þ
for controllers Ci(s) = Ki/s(i = 1, 2), Similar to the evaluation of Eqs. (33) and (34), taking the real and the imaginary parts of Eq. (35), one has another two stability-equations. They are
Fig. 3. Limit cycle analyses in A1 vs. A2 plane for finding a stable limit cycle.
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T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
(x) of the oscillation can be predicted from the constant-A1 loci. If parameters are chosen in the asymptotically stable region, the system will be stable. Note that the locus of constant-A1 ðD=2Þ will be used to show the boundary between the asymptotically stable and the limit cycle region because the considered system is open-loop stable with deadband D. A typical value of D is 0.06% [3]. Note also that above evaluations are ready to be applied to multi-area interconnected power systems. For example, there are two stabilityequations and three phase angle shift constraint equations (i.e., for h21 ; h31 ; h41 ) are evaluated solutions (Ai ; i ¼ 1; 2; 3; 4; x) for a four-area interconnected power system. In the following section, high precision predictions of limit-cycle characteristics will be used to illustrate the merit of the proposed method for controller parameter designs and deadband nonlinearities in the system cannot be neglected. 4. Stability analyses results For the system considered in Section 2, assume that the values of parameters are as follows [1,2]: Area capacity Pr1 = Pr2 = 2000 MW Nominal area loading=1000 MW f ¼ 60Hz K p1 ¼ K p2 ¼ 120 Hz/p.u. MW T p1 ¼ T p2 ¼ 20 s R1 ¼ R2 ¼ 2:4 Hz/p.u. MW T 12 ¼ 0:0707 s
T g1 ¼ T g2 ¼ 0:2 s T t1 ¼ T t2 ¼ 0:3 s K r1 ¼ K r2 ¼ 1=3 Dpd1 ¼ 0:01 p.u. MW Dpd2 ¼ 0:00 B1 ¼ B2 ¼ 0:425 T r1 ¼ T r2 ¼ 20 s
K1 and K2 are two adjustable parameters of the supplemental controllers. The stability analyses of the system without deadband nonlinearities is analyzed first; i.e., the describing functions are replaced by unities. The shaded region of Fig. 4 shows the asymptotically stable region of the considered system. And then the system with deadband nonlinearities are analyzed and shown on K1 vs. K2 plane. Fig. 4 shows the stable region is replaced by the limit-cycle region in which shows the constant-A1 curves. From the constantA1 curves shown in Fig. 4, one can see that small values of Ki are expected for small amplitude of oscillating signal. The smallest values of A1 occurs at K1 = K2 = 0.0. This implies that the governor deadband nonlinearities tend to produce a sustaining oscillation(limit-cycle). Fig. 5 shows a sustaining oscillation simulated result for K1 = K2 = 0.0 and B1 = B2 = 0.425.
Fig. 4. Parameter analysis of Eqs. (36)–(38) for Bi = 0.425 .
The amplitudes of Ai are 6:04 104 p:u: and the oscillation frequency is 2.479 rad/s. This gives excellent agreement with the analyzed results. Note that the system is unstable for Ki are less than zeros. For comparison purpose, two simulations for system with/without deadband nonlinearities are given for K1 = K2 = 0.5 (point Q2 in Fig. 4). Fig. 6 shows step response for Dpd1 ¼ 0:01 p:u:; Dpd2 ¼ 0:0 with deadband nonlinearities; and Fig. 7 shows step response for Dpd1 ¼ 0:01p:u:; Dpd2 ¼ 0:0 without deadband nonlinearities(linear system). Fig. 6 shows a limit cycle, but there is not limit-cycle in Fig. 7. The deadband nonlinearities degrade the system stability and the sustaining oscillating is always exist with given system parameters. Table 1 gives the calculated and simulated results comparisons of some other nine points in the limit-cycle region. From Table 1, one can see that the calculated results are quite close to those of simulated results. This implies that the proposed method can be used to analyze the considered system accurately. Since the sustaining oscillating is always exist, therefore system analyses for modification is needed. In the above analysis, the frequency bias parameters B1 and B2 are set at 0.425 and Ki (i = 1, 2) are varying. In another way, one can also assume specified values of Ki (i = 1, 2) and perform parameter analysis varying B1 and B2. The desirable solutions are B1, B2, A1, A2 and x. Since the considered system is symmetrical, the analysis can be performed simply in the Bi (B1 = B2) vs. Ai (A1 = A2) plane. For Ki = 0, 0.5 and 0.6 (i = 1, 2), the analyzed results are shown in Fig. 8. It can be seen that the smaller the values of Bi and Ki are the smaller the amplitude of Ai (A1 = A2) will be. The dot-points shown in Fig. 8 show the simulation verifications. For illustration, the step responses (Dpd1 ¼ 0:01p:u:) for K i ¼ 0:5 and Bi ¼ 0:1 are shown in Fig. 9. From Figs. 6 and 9, it can be seen that the amplitude Ai (and DF i ; Dpci ; etc:) are reduced largely by adjusting the values of K i and Bi from 0.425 to 0.1. Therefore, small values of K i and Bi are desirable. But values of K i can not be too small because the settling time may become too long. Note that the amplitudes of the oscillating signals in Fig. 9 are kept in acceptable narrow band. From responses of Ai ; DF i , and Dpci ; ði ¼ 1; 2Þ shown in Figs. 6 and 9, one can see that increase the damping of each subsystem is needed for avoiding the sustaining oscillation. For example, the time constant T gi (i = 1, 2) of governors can be changed for less phase lag to get higher damping. Fig. 10 shows results of analysis for T gi ¼ 0:08 s [2], Bi ¼ 0:1 and D ¼ 6:0 104 in the Ai ðA1 ¼ A2 Þ vs. K i ðK 1 ¼ K 2 Þ plane, where the asymptotically stable, limit-cycle and unstable regions are clearly defined. Note that there is no asymptotically stable region (see Fig. 4) with original system
Fig. 5. The simulated limit cycle of the system for Ki = 0.0 and Bi = 0.425.
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
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Fig. 6. Transient responses of LFC for 0.01 p.u. load increased in area 1 with dead-band nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
Fig. 7. Transient responses of LFC for 0.01 p.u. load increased in area 1 without dead-band nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
Table 1 The calculated and simulated results of the system with governor deadband nonlinearities for Bi = 0.425. Parameters
Calculated
K1
K2
x
A1
A2
x
Simulated A1
A2
0.1601 0.4513 1.0297 0.3365 0.0807 1.1533 0.0455 0.7459 0.9660
1.0175 0.7142 0.5550 1.2087 1.6173 0.7403 1.8302 1.1610 0.9660
2.525 2.535 2.545 2.540 2.535 2.550 2.535 2.550 2.551
0.0015 0.0015 0.0030 0.0030 0.0050 0.0300 0.0500 0.0500 1.0120
0.00114 0.00138 0.00334 0.00247 0.00400 0.03180 0.04310 0.04710 1.0120
2.523 2.532 2.547 2.523 2.537 2.548 2.532 2.548 2.562
0.00152 0.00149 0.00316 0.00314 0.00525 0.03347 0.05357 0.05515 1.10796
0.00116 0.00137 0.00352 0.00258 0.00427 0.03543 0.04625 0.05205 1.10799
Fig. 8. Parameter analysis and simulation results for Bi(B1 = B2) varying and Ki = 0, 0.5 and 0.6.
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T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
Fig. 9. Transient responses of LFC for 0.01 p.u. load increased in area 1 for Ki = 0.5 and Bi = 0.1 with deadband nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
parameters. In Fig. 10, doted-points show the simulated verifications. They give excellent agreement with the analyzed result also. For illustration, the step responses (Dpd1 ¼ 0:01 p:u:) for K i ¼ 0:5 and Bi ¼ 0:1 are shown in Fig. 11. It can be seen that the controlled system with the selected parameters is asymptotically stable. 5. Conclusions
Fig. 10. Parameter analysis and simulation results of the damped system for Bi = 0.1.
In this paper, the stability-equation method has been applied to the analysis and design of an interconnected power system with reheat steam turbine and governor deadband nonlinearities. The asymptotically stable, limit-cycle and unstable regions can be clearly defined in parameter planes. Based on these regions, one can choose the desirable parameters easily. The study pays attention to found the asymptotically stable region and reduce the
Fig. 11. Transient responses of LFC for 0.01 p.u. load increased in area 1 for Ki = 0.5, Bi = 0.1 and Tgi = 0.08 s with deadband nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
amplitudes of sustaining oscillations. The proposed method can be applied to analyze systems controlled by PI, PID, VSC, and Fuzzy methods [5–14] also. For example, parameters of PI and PID controller can be used as parameters to be analyzed and designed. From Figs. 4, 8 and 10, one can see that the proposed method provided a simple way for choosing desirable parameters. References [1] Tripathy SC, Hop GS, Malik OP. Optimisation of load–frequency control parameters for power system with read steam turbines and governor deadband nonlinearity. PROC IEE PtC 1982;129(1):10–6. [2] Tripathy SC, Balasubramanian R, Nair PSC. Effect of superconducting magnetic energy storage on automatic generation control considering governor deadband and boiler dynamics. IEEE Trans Power Syst 1992;7(3):1266–73. [3] Lu CF, Liu CC, Wu CJ. Effect of battery energy storage system on load frequency control considering governor deadband and generation rate constraint. IEEE Trans Energy Convers 1995;10(1):555–61. [4] Chidambaram IA, Velusami S, Decentralized biased controllers for load– frequency control of interconnected power systems considering governor dead band non-linearity. In: INDICON, 2005 Annual IEEE; 2005; p. 521–5. [5] Kumar A, Malik OP, Hope GS. Variable–structure-system control applied to AGC of an interconnected power system. IEE PROC Gen Transm Distrib 1985;132(1):23–9. [6] Kumar A, Malik OP, Hope GS. Discrete variable structure controller for load frequency control of multiarea interconnected power systems. IEE PROC Gen Transm Distrib 1987;134(2):116–22. [7] Yang MS. Decentralized sliding mode stabilizer design for multi-area interconnected power systems. In: Proceedings of the 2000 IEEE international symposium on industrial electronics, vol. 1; 2000. p. 185–90. [8] Aldeen M, Marsh JF. Decentralised proportional-plus-integral design method for interconnected power systems. IEE PROC Gen Transm Distrib 1991;138(4):263–74.
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[9] Khodabakhshian A, Hooshmand R. A new PID controller design for automatic generation control of hydro power systems. Int J Electr Power Energy Syst 2010;32(5):375–82. [10] Alrifai MT, Hassan MF, Zribi M. Decentralized load frequency controller for a multi-area interconnected power system. Int J Electr Power Energy Syst 2011;33(2):198–209. [11] Tan W. Tuning of PID load frequency controller for power systems. Energy Convers Manage 2009;50(6):1465–72. [12] Juang CF, Lu CF. Power system load frequency control by evolutionary Fuzzy PI controller. In: 2004 Proceedings of IEEE international conference on fuzzy systems, vol. 2; 2004. p. 715–9. [13] Kocaarslan I, Cam E. Fuzzy logic controller in interconnected electrical power systems for load–frequency control. Int J Electr Power Energy Syst 2005;27(8):542–9. [14] Mathur HD, Manjunath HV. Extended fuzzy logic based integral controller for three area power system with generation rate constraint. In: ICIT 2006 IEEE international conference on industrial technology; 2006. p. 917–21. [15] Han KW. Nonlinear control system: some practical method. Academic Culture Company; 1977. [16] Tsay TS. Analyses and designs of a nonlinear mechanical scanning control system. J Franklin Inst 2006;343(1):83–93. [17] Tsay TS, Han KW. Limit cycle analysis of nonlinear multivariable feed back control systems. J Franklin Inst 1988;325(6):721–30. [18] Atherton DP, Spurgeon S. Nonlinear control systems, analytical methods. Electrical engineering encyclopedia. New York: John Wiley & Sons, Inc.; 1999. [19] Patra KC, Singh YP. Graphical method of prediction of limit cycle for multivariable nonlinear systems. IEE PROC Control Theory Appl 1996;143(5):423–8. [20] Gray JO, Taylor PM. Computer aided design of multivariable nonlinear control system using frequency domain techniques. Automatica 1979;5:281–97. [21] Gray JO, Nakhla NB. Prediction of limit cycle in multivariable nonlinear systems. PROC IEE PtD 1981;128(5):238–41.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Load–frequency control of interconnected power system with governor backlash nonlinearities Tain-Sou Tsay ⇑ Department of Aeronautical Engineering, National Formosa University, 64, Wen-Hua Road, Huwei, YunLin 63208, Taiwan
a r t i c l e
i n f o
Article history: Received 10 May 2010 Received in revised form 12 February 2011 Accepted 3 June 2011 Available online 4 August 2011 Keywords: Interconnected power system Nonlinear multivariable feedback system Limit cycle
a b s t r a c t In this paper, the stability-equation method is applied to the analysis and design of an interconnected power system with governor backlash nonlinearities. The considered system is a nonlinear multivariable feedback control system. The governor nonlinearities tend to produce a sustaining oscillation in area frequency and tie-line power transient responses. Most conventional linear design techniques are usually unable to find the sustaining oscillation in design phase and need simulation verifications to check the validations after designs. However, the proposed method can consider effects of nonlinearities in the design phase. Some nonlinear design techniques need parameters optimization method by Lyapunov theorem or Integral of Square Errors (ISE) criteria. They are effective. However, they need large computation efforts. The proposed method can choice frequency bias parameters and integrator gains of supplementary controllers for avoiding the oscillation or reducing the amplitude of the oscillation to be acceptable. Simulation verifications show that the proposed method can provide a simple and effective way for the considered system. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction For reasons of economy and system reliability, neighboring power systems are interconnected, forming an augmented system referred to as ‘‘power pool’’. The various areas are interconnected through tie-lines. The net power flow on the tic-lines connecting a system to the external system is frequently scheduled by a priori contract basis. The tie-lines are used for contractual energy exchange between areas and provide inter-area support in case of abnormal conditions. Area load changes and abnormal conditions, such as outages of generation, lead to mismatches in frequency and scheduled power interchanges between areas. These mismatches have to be corrected through a Load–Frequency Control (LFC). The load–frequency control is based on an error signal called Area Control Error (ACE) which is a linear combination of net-interchange and frequency errors. The conventional control strategy used in industry is to take the integral of ACE as the control signal [1–14]. The control purpose is to reduce the frequency and tie-line power errors to zero in the steady state. Many decentralized control strategies; e.g., variable structure controls [5–7], PI/PID and Fuzzy controls [9–14], have been employed in the design of LFC for interconnected power systems. Among the various types of decentralized LFC, the most widely employed is the simple conventional control. The conventional con⇑ Tel.: +886 5 631 5537; fax: +886 5 631 2415. E-mail address: [email protected] 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.06.005
trol for LFC is still popular with the power industries because of their simplicity, easy realization, low cost, robust and decentralized nature of the control strategy. It has also been shown that there is no significant advantage in using the more complex controllers over the conventional controllers [9–11]. The conventional proportional plus integral control strategy, which is widely used in power industry, is to take the integral of ACE as the control signal. It is well known that many LFC scheme does not yield adequate control performance with consideration of the system nonlinearities such as governor deadband or generation rate constraint [1– 4]. Governor deadband (GDB) nonlinearities tend to produce an unexpected sustaining oscillation in area frequency and tie-line power transient responses. Avoiding sustaining oscillation or reducing the amplitude of the sustaining oscillation is expected. Most conventional linear design techniques [5–14] need digital simulations to check effects of nonlinearities after controller designed. The nonlinearities considered in the design phase are expected. Some nonlinear design techniques need parameters optimization method by Lyapunov theorem [1–3] or Integral of Square Errors (ISE) criteria [4]. They are effective. However, they need large computation efforts. Simple and effective ways to evaluate effects of nonlinearities are expected. This is the motivation of this paper. In this paper, the stability-equation method [15–17] is used to analyze and design the considered system. The considered system is a nonlinear multivariable feedback control system. Harmonicbalance equations and the characteristic equation [18–21] are
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
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Nomenclature ACEi Ai Bi D Ki Ri Tgi Tti Tri Kri Tpi Kpi DFi
area control error of area i amplitude of sinusoidal input of Ni frequency bias parameter for area control error of area i governor deadband integrator gain of supplementary controller of area i governor speed regulation of area i time constant of speed governor control system of area i time constant of steam turbine of area i reheat time constant of area i reheat coefficient of steam turbine of area i generator time constant of area i generator gain constant of area i frequency deviation of area i
evaluated for finding the unstable, limit-cycle and asymptotically stable regions in the parameter plane [16]. It will be see that there is no asymptotically stable region for some specified values of controlling and system parameters. This implied that sustain oscillation is always exist; i.e., the stable limit cycle is exist. In this condition, the acceptable amplitude of the sustaining oscillation is expected (e.g., tie-line power). The proposed method can provide information of amplitudes of sustain oscillations. In another way, the proposed method can be used to find the asymptotically stable region in the parameter-plane by selecting other specified values of controlling or system parameters. Furthermore, this paper can be applied the analyses and refining of the systems controlled by conventional linear control techniques; e.g., adjusting parameters of PI/PID controllers to refining the system performance. This paper is organized as follows. First, the considered interconnected power system is described. Then, the proposed method is presented. Finally the proposed method is applied to analyze and design the considered system. Simulation verification results show that the proposed method can provide a simple way for choosing proper values of parameters to avoid sustaining oscillation or reducing amplitudes of sustaining oscillations. 2. The two-area interconnected power system The two-area interconnected power system consists of two single areas connected through a power line called the tie line. Each area feeds its user pool, and the tie line allows electric power to flow between areas. Since two areas are tied together, a load perturbation in one area affects the output frequencies of both areas as well as the power flow on the tie line. The control system of each area needs information about the transient situation of both areas to bring the local frequency back to its steady state value. Information about the other area is found in the output frequency fluctuation of that area and in the tie line power fluctuation. Thus, the tie line power is sensed, and the resulting tie line power signal is fed back into both areas. A block diagram of the two area interconnected power system is given in Fig. 1 in which governors, reheat turbines and generators are considered. Transfer functions of reheat turbines are given by
DPgi ðsÞ 1 þ K ri T ri s ¼ ; DX Ei ðsÞ ð1 þ sT ri Þð1 þ sT ti Þ
i ¼ 1; 2
ð1Þ
and differential equations of the overall system [1–4] are given as: 1 DF_ 1 ¼ T 1 p1 DF 1 þ K p1 T p1 ðDP g1 DP tie DP d1 Þ
ð2Þ
h i 1 1 1 DP_ g1 ¼ T 1 r1 DP g1 þ DP r1 T r1 K r1 T r1 þ K r1 T t1 DX E1
ð3Þ
Dpgi Dpci Dpdi Dpri Dptie DXEi T12 a12 Pri Ni
mechanical power deviation of area i change in speed changer position real power increase in area i mechanical power deviation during steam reheat in area i tie-line power deviation out of area i governor valve displacement of area i synchronizing power coefficient of tie-line connected between areas 1 and 2 Pr1/Pr2 MW capacity of area i governor deadband nonlinearity of area i
1 DP_ r1 ¼ T 1 t1 DX E1 T t1 DP r1
ð4Þ
1 1 1 DX_ E1 ¼ R1 1 N 1 ðA1 ÞT g1 DF 1 T g1 DX E1 þ T g1 N 1 ðA1 ÞDP c1
ð5Þ
ACE1 ¼ B1 DF 1 þ DPtie
ð6Þ
DPc1 ¼ K 1
Z
ACE1 dt
ð7Þ
1 DF_ 2 ¼ T 1 p2 DF 1 þ K p2 T p2 ðDP g2 a12 DP tie DP d2 Þ
ð8Þ
h i 1 1 1 DP_ g2 ¼ T 1 r2 DP g2 þ DP r2 T r2 K r2 T r2 þ K r2 T t2 DX E2
ð9Þ
1 DP_ r2 ¼ T 1 t2 DX E2 T t2 DP r2
ð10Þ
1 1 1 DX_ E2 ¼ R1 2 N 2 ðA2 ÞT g2 DF 2 T g2 DX E2 þ T g2 N 2 ðA2 ÞDP c2
ð11Þ
DP_ tie ¼ 2pT 12 ðDF 1 DF 2 Þ
ð12Þ
ACE2 ¼ B2 DF 2 þ a12 DP tie
ð13Þ
DPc2 ¼ K 2
Z
ACE2 dt
ð14Þ
where N1(A1) and N2(A2) represent quasi-linear gains or describing functions [15–18] of governor deadband (backlash) nonlinearities [1–4]; A1 and A2 represent amplitudes of sinusoidal inputs of nonlinearities N1 and N2, respectively. The corresponding transfer functions of governors and generators are shown in Fig. 1. In conventional systems, an integral controller sets the turbine reference power of each area. Since a perturbation in either area affects the frequency in both areas and a perturbation in one area is perceived by the other through a change in tie line power, the controller of each area should not only; take the frequency variation as input but also the tie line power variations. Since an integral controller has just one input, these two contributions, namely local frequency variation and tie line power variation, must be combined into a single signal so that it can be used as an input of the controller. The easiest way of doing this is to combine them linearly, i.e. the input of the integrator in area 1 is DPtie + B1DF1, and the input of the integrator in area 2 is a12DPtie + B2DF2. The input of each controller is called the Area Control Error (ACE). The nonlinearities destabilize the system and tend to produce a sustaining oscillation in area and tie-line power transient responses [1–4]. There are frequency bias parameters Bi(i = 1, 2) and integrator gains Ki(i = 1, 2) of the supplementary controllers
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T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
Fig. 1. Transfer function block diagram of two-area interconnected power system.
to be chosen for desirable performance. In general, the sustaining oscillation is not desirable. In this paper, the stability-equation method [15–17] is applied to the analysis and design the considered system in parameter planes, where unstable, limit-cycle, and asymptotically stable regions can be clearly defined. The constant-Ai loci in the limit-cycle region show amplitudes (Ai) of sinusoidal inputs of governor deadband nonlinearities with specified values of Bi and Ki(i = 1, 2). Therefore, one can avoid the oscillation or reduce the amplitude of oscillation by choosing proper values of adjustable parameters Bi and Ki(i = 1, 2). It will be seen that if there is no asymptotically stable region in the parameter plane, the system will always have a sustaining oscillation for all the values of Bi and Ki(i = 1, 2) [6,7]. In this case, the design goal is to keep the amplitude of oscillation as small as possible [8]. Note that, one can improve the damping characteristics of each subsystem to stabilize the overall system, then an asymptotically stable region may exist. Although the controllers considered in this paper are simple, the proposed method can be applied easily to systems with more complicated type of supplementary controllers [5–14]; e.g., parameters of PD/PID controllers can be used analyzed parameters to refine the system performance.
3. The proposed method Fig. 2 shows the equivalent system block diagram of a two-area interconnected power system in which A1 and A2 represent amplitudes of sinusoidal inputs of nonlinearities N1 and N2, respectively; and h21 is the phase shift angle of the sinusoidal input of N2 related to N1. For zero power increment inputs (DPd1 = DPd2 = 0) of each area, harmonic-balance equations [18–21] of the system are
A1
A2 ejh21
¼
E11 ðjxÞ E12 ðjxÞ
F 11 ðjxÞ F 12 ðjxÞ
E21 ðjxÞ E22 ðjxÞ F 21 ðjxÞ F 22 ðjxÞ 0 N1 ðA1 Þ A1 0
H11 ðjxÞ H12 ðjxÞ H21 ðjxÞ H22 ðjxÞ
N2 ðA2 Þ
N1 ðA1 Þ 0
A2 ejh21 0 N2 ðA2 Þ
A1 jh21
A2 e
ð15aÞ
2 X k¼1
Eik ðjxÞF kj ðjxÞ;
i; j ¼ 1; 2
where
h i E11 ðjxÞ ¼ G3 ðjxÞ C 1 ðjxÞB1 þ C 1 ðjxÞG5 ðjxÞ R1 1
ð17Þ
E12 ðjxÞ ¼ C 1 ðjxÞG4 ðjxÞG5 ðjxÞ
ð18Þ
E21 ðjxÞ ¼ a12 C 2 ðjxÞG3 ðjxÞG5 ðjxÞ
ð19Þ
h i E22 ðjxÞ ¼ G4 ðjxÞ C 2 ðjxÞB2 a12 C 2 ðjxÞG5 ðjÞ R1 2
ð20Þ
DðjxÞ ¼ 1 þ G3 ðjxÞG5 ðjxÞ a12 G4 ðjxÞG5 ðjxÞ
ð21Þ
F 11 ðjxÞ ¼ G1 ðjxÞ½1 a12 G4 ðjxÞG5 ðjxÞ=DðjxÞ
ð22Þ
F 12 ðjxÞ ¼ G2 ðjxÞG4 ðjxÞG5 ðjxÞ=DðjxÞ
ð23Þ
F 21 ðjxÞ ¼ a12 G1 ðjxÞG3 ðjxÞG5 ðjxÞ=DðjxÞ
ð24Þ
F 22 ðjxÞ ¼ G2 ðjxÞ½1 þ G3 ðjxÞG5 ðjxÞ=DðjxÞ
ð25Þ
Solving Eq. (15b), one has
ejh21 ¼ ð15bÞ
A1 ½1 N1 ðA1 ÞH11 ðjxÞ A2 N2 ðA2 ÞH12 ðjxÞ
ð26Þ
A1 N1 ðA1 ÞH21 ðjxÞ A2 ½1 N2 ðA2 ÞH22 ðjxÞ
ð27Þ
And
where s = jx; N1(A1) and N2(A2) represent describing functions of deadbad nonlinearities N1 and N2 [15,18] with amplitudes A1 and A2, respectively; and
Hi;j ðjxÞ ¼
Fig. 2. Equivalent system block diagram of Fig. 1.
ð16Þ
ejh21 ¼
Equating Eqs. (26) and (27), one has the characteristic equation the considered system:
T.-S. Tsay / Electrical Power and Energy Systems 33 (2011) 1542–1549
P0r ðA1 ; A2 ; xÞ þ K 1 Q 0r ðA1 ; A2 ; xÞ þ K 2 W 0r ðA1 ; A2 ; xÞ
1 þ N 1 ðA1 ÞH11 ðjxÞ þ N2 ðA2 ÞH22 ðjxÞ
þ K 1 K 2 T 0r ðA1 ; A2 ; xÞ ¼ 0
þ N1 ðA1 ÞN2 ðA2 Þ½H11 ðjxÞH22 ðjxÞ H12 ðjxÞH21 ðjxÞ ¼0
ð28Þ
Multiplying the Lowest Common Multiple (LCM) of denominators of Hij(jx) and determinate of Hij(jx) in Eq. (28), Eq. (28) can rewritten as
PðjxÞ þ N1 ðA1 ÞQ ðjxÞ þ N2 ðA2 ÞWðjxÞ þ N1 ðA1 ÞN 2 ðA2 ÞTðjxÞ ¼ 0 ð29Þ Inverting the substitution of s by jx in Eq. (29), then the characteristic equation of the considered system with complex coefficients will be get; i.e.,
PðsÞ þ N1 ðA1 ÞQ ðsÞ þ N 2 ðA2 ÞWðsÞ þ N 1 ðA1 ÞN2 ðA2 ÞTðsÞ ¼ 0
ð30Þ
where P(s), Q(s), W(s), T(s) are rational polynomials of Laplace operator s. Note that the phase shift angle h21 is not represented in the characteristic equation. Describing functions of deadband nonlinearities Ni(Ai) are complex numbers [15,18]. They are
Ni ðAi Þ ¼
1
p
(
1=2 ) 2D 2D D D þ2 1 1 1 þ sin Ai Ai Ai Ai 2 4D D j Nir ðAi Þ þ jN ii ðAi Þ 1 ð31Þ Ai pAi
p
1
where D is the dead band, and Ai is the amplitude of the sinusoidal input. Eq. (29) can be rewritten as
Pr ðxÞ þ jP i ðxÞ þ ½N1r ðA1 Þ þ jN 1i ðA1 Þ½Q r ðxÞ þ jQ i ðxÞ þ ½N2r ðA2 Þ þ jN 2i ðA2 Þ ½W r ðxÞ þ jW i ðxÞ þ ½N1r ðA1 Þ þ jN 1i ðA1 Þ½N 2r ðA2 Þ þ jN 2i ðA2 Þ½T r ðxÞ þ jT i ðxÞ ¼ 0
ð32Þ
where PðjxÞ Pr ðxÞ þ jPi ðxÞ; Q ðjxÞ Q r ðxÞ þ jQ i ðxÞ, WðjxÞ W r ðxÞ þ jW i ðxÞ and TðjxÞ T r ðxÞ þ jT i ðxÞ. Taking the real and imaginary parts of Eq. (32), then two stability-equations for analyses and designs [15–17] are found. They are
Pr ðxÞ þ ½N1r ðA1 ÞQ r ðxÞ N1i ðA1 ÞQ i ðxÞ þ ½N2r ðA2 ÞW r ðxÞ N2i ðA2 ÞW i ðxÞ þ ½N1r ðA1 ÞN2r ðA2 Þ jN 1i ðA1 ÞN2i ðA2 ÞT r ðxÞ ½N1r ðA1 ÞN2i ðA2 Þ þ N 1i ðA1 ÞN2r ðA2 ÞT i ðxÞ ¼ 0
ð33Þ
and
Pi ðxÞ þ ½N1r ðA1 ÞQ i ðxÞ þ N 1i ðA1 ÞQ r ðxÞ þ ½N2r ðA2 ÞW i ðxÞ þ N2i ðA2 ÞW r ðxÞ þ ½N1r ðA1 ÞN2r ðA2 Þ jN 1i ðA1 ÞN2i ðA2 ÞT i ðxÞ þ ½N1r ðA1 ÞN2i ðA2 Þ þ N 1i ðA1 ÞN2r ðA2 ÞT r ðxÞ ¼ 0
1545
ð34Þ
ð36Þ
and
P0i ðA1 ; A2 ; xÞ þ K 1 Q 0i ðA1 ; A2 ; xÞ þ K 2 W 0i ðA1 ; A2 ; xÞ þ K 1 K 2 T 00i ðA1 ; A2 ; xÞ ¼ 0
ð37Þ
Eqs. (36) and (37) are functions of (A1 ; A2 and x), and five parameters (K 1 ; K 2 ; A1 ; A2 and x) are needed to be found. There are an infinite number of solutions (A1 ; A2 and x) satisfy Eqs. (36) and (37) simultaneously for specified values of K1 and K2 [15–17]. For solving this problem, the following constraint equation must be added; i.e., f26g
f27g
h21 h21 ¼ 0
ð38Þ
where hf26g and hf27g represent phase angles found by Eqs. (26) and 21 21 (27), respectively; i.e., f26g h21 hf27g 21 ¼ \
A1 ½1 N1 ðA1 ÞH11 ðjxÞ A1 N1 ðA1 ÞH21 ðjxÞ \ ¼0 A2 N2 ðA2 ÞH12 ðjxÞ A2 ½1 N2 ðA2 ÞH22 ðjxÞ ð39Þ
Then, there are three simultaneously equations (i.e., Eqs. (36)– (38)) for solving three parameters (A1 ; A2 and x) of the considered system with specified values of (K1, K2). Fig. 3 shows the typical solving processes of a stable limit cycle in which the loci represent roots of Eqs. (36) and (37). They represent an infinite number possible solutions (A1 ; A2 and x) of limit-cycle. But, the unique solution represented by Q 1 (1:179 1103 ; 1:179 103 ; 2:5294) is found from Eq. (39). This is called stable limit cycle. The other solutions are called unstable limit cycles. For controller designs purpose, another parameters finding procedure for (A2 , K 1 and K 2 ) of Eqs. (36) and (37) are evaluated for specified values of (A1 ; x), and plotted in the K1 vs. K2 plane; i.e., parameter plane. The constant-A1 locus [16], which is plotted in the K1 vs. K2 plane, is constituted by simultaneous solutions for a specified value of amplitude (A1) and a number of frequencies (x). For a number of constant-A1 loci, the unstable, limit-cycle and asymptotically stable regions of the considered system can be clearly defined in the K1 vs. K2 plane. Those three regions are classified by constant-A1 ð1Þ and A1 (0) loci. For example, the constant-A1 ð1Þ locus represents the stability boundary between the limit-cycle region and the unstable region; the constant-A1 ð0Þ locus represents the boundary between the limit-cycle region and the asymptotically stable region. If parameters (K 1 , K 2 ) are chosen in the limit-cycle region, then sustaining oscillation will occur. The amplitude (A1 ) and frequency
Eqs. (33) and (34) are rational polynomial of the frequency x. Roots of Eq. (33) are (xei, i = 1, . . . , nr) and t roots of Eq. (34) are (xoj,j = 1, . . . ,ni). nr and ni are maximal order of x of Eqs. (33) and (34); respectively. The stability criteria [15–17] of considered nonlinear feedback control systems developed are (a) system is stable: xei and xoj are all real and alternative in sequence; (b) sustaining oscillating: xei and xoj are all real and alternative in sequence except that one root pair is equal to each other (i.e., xei = xoj = x); (c) system is unstable: xei and xoj are not all real or and not alternative in sequence. Another representation of the Eq. (28) for controllers design is
P0 ðA1 ; A2 ; jxÞ þ K 1 Q 0 ðA1 ; A2 ; jxÞ þ K 2 W 0 ðA1 ; A2 ; jxÞ þ K 1 K 2 T 0 ðA1 ; A2 ; jxÞ ¼ 0
ð35Þ
for controllers Ci(s) = Ki/s(i = 1, 2), Similar to the evaluation of Eqs. (33) and (34), taking the real and the imaginary parts of Eq. (35), one has another two stability-equations. They are
Fig. 3. Limit cycle analyses in A1 vs. A2 plane for finding a stable limit cycle.
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(x) of the oscillation can be predicted from the constant-A1 loci. If parameters are chosen in the asymptotically stable region, the system will be stable. Note that the locus of constant-A1 ðD=2Þ will be used to show the boundary between the asymptotically stable and the limit cycle region because the considered system is open-loop stable with deadband D. A typical value of D is 0.06% [3]. Note also that above evaluations are ready to be applied to multi-area interconnected power systems. For example, there are two stabilityequations and three phase angle shift constraint equations (i.e., for h21 ; h31 ; h41 ) are evaluated solutions (Ai ; i ¼ 1; 2; 3; 4; x) for a four-area interconnected power system. In the following section, high precision predictions of limit-cycle characteristics will be used to illustrate the merit of the proposed method for controller parameter designs and deadband nonlinearities in the system cannot be neglected. 4. Stability analyses results For the system considered in Section 2, assume that the values of parameters are as follows [1,2]: Area capacity Pr1 = Pr2 = 2000 MW Nominal area loading=1000 MW f ¼ 60Hz K p1 ¼ K p2 ¼ 120 Hz/p.u. MW T p1 ¼ T p2 ¼ 20 s R1 ¼ R2 ¼ 2:4 Hz/p.u. MW T 12 ¼ 0:0707 s
T g1 ¼ T g2 ¼ 0:2 s T t1 ¼ T t2 ¼ 0:3 s K r1 ¼ K r2 ¼ 1=3 Dpd1 ¼ 0:01 p.u. MW Dpd2 ¼ 0:00 B1 ¼ B2 ¼ 0:425 T r1 ¼ T r2 ¼ 20 s
K1 and K2 are two adjustable parameters of the supplemental controllers. The stability analyses of the system without deadband nonlinearities is analyzed first; i.e., the describing functions are replaced by unities. The shaded region of Fig. 4 shows the asymptotically stable region of the considered system. And then the system with deadband nonlinearities are analyzed and shown on K1 vs. K2 plane. Fig. 4 shows the stable region is replaced by the limit-cycle region in which shows the constant-A1 curves. From the constantA1 curves shown in Fig. 4, one can see that small values of Ki are expected for small amplitude of oscillating signal. The smallest values of A1 occurs at K1 = K2 = 0.0. This implies that the governor deadband nonlinearities tend to produce a sustaining oscillation(limit-cycle). Fig. 5 shows a sustaining oscillation simulated result for K1 = K2 = 0.0 and B1 = B2 = 0.425.
Fig. 4. Parameter analysis of Eqs. (36)–(38) for Bi = 0.425 .
The amplitudes of Ai are 6:04 104 p:u: and the oscillation frequency is 2.479 rad/s. This gives excellent agreement with the analyzed results. Note that the system is unstable for Ki are less than zeros. For comparison purpose, two simulations for system with/without deadband nonlinearities are given for K1 = K2 = 0.5 (point Q2 in Fig. 4). Fig. 6 shows step response for Dpd1 ¼ 0:01 p:u:; Dpd2 ¼ 0:0 with deadband nonlinearities; and Fig. 7 shows step response for Dpd1 ¼ 0:01p:u:; Dpd2 ¼ 0:0 without deadband nonlinearities(linear system). Fig. 6 shows a limit cycle, but there is not limit-cycle in Fig. 7. The deadband nonlinearities degrade the system stability and the sustaining oscillating is always exist with given system parameters. Table 1 gives the calculated and simulated results comparisons of some other nine points in the limit-cycle region. From Table 1, one can see that the calculated results are quite close to those of simulated results. This implies that the proposed method can be used to analyze the considered system accurately. Since the sustaining oscillating is always exist, therefore system analyses for modification is needed. In the above analysis, the frequency bias parameters B1 and B2 are set at 0.425 and Ki (i = 1, 2) are varying. In another way, one can also assume specified values of Ki (i = 1, 2) and perform parameter analysis varying B1 and B2. The desirable solutions are B1, B2, A1, A2 and x. Since the considered system is symmetrical, the analysis can be performed simply in the Bi (B1 = B2) vs. Ai (A1 = A2) plane. For Ki = 0, 0.5 and 0.6 (i = 1, 2), the analyzed results are shown in Fig. 8. It can be seen that the smaller the values of Bi and Ki are the smaller the amplitude of Ai (A1 = A2) will be. The dot-points shown in Fig. 8 show the simulation verifications. For illustration, the step responses (Dpd1 ¼ 0:01p:u:) for K i ¼ 0:5 and Bi ¼ 0:1 are shown in Fig. 9. From Figs. 6 and 9, it can be seen that the amplitude Ai (and DF i ; Dpci ; etc:) are reduced largely by adjusting the values of K i and Bi from 0.425 to 0.1. Therefore, small values of K i and Bi are desirable. But values of K i can not be too small because the settling time may become too long. Note that the amplitudes of the oscillating signals in Fig. 9 are kept in acceptable narrow band. From responses of Ai ; DF i , and Dpci ; ði ¼ 1; 2Þ shown in Figs. 6 and 9, one can see that increase the damping of each subsystem is needed for avoiding the sustaining oscillation. For example, the time constant T gi (i = 1, 2) of governors can be changed for less phase lag to get higher damping. Fig. 10 shows results of analysis for T gi ¼ 0:08 s [2], Bi ¼ 0:1 and D ¼ 6:0 104 in the Ai ðA1 ¼ A2 Þ vs. K i ðK 1 ¼ K 2 Þ plane, where the asymptotically stable, limit-cycle and unstable regions are clearly defined. Note that there is no asymptotically stable region (see Fig. 4) with original system
Fig. 5. The simulated limit cycle of the system for Ki = 0.0 and Bi = 0.425.
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Fig. 6. Transient responses of LFC for 0.01 p.u. load increased in area 1 with dead-band nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
Fig. 7. Transient responses of LFC for 0.01 p.u. load increased in area 1 without dead-band nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
Table 1 The calculated and simulated results of the system with governor deadband nonlinearities for Bi = 0.425. Parameters
Calculated
K1
K2
x
A1
A2
x
Simulated A1
A2
0.1601 0.4513 1.0297 0.3365 0.0807 1.1533 0.0455 0.7459 0.9660
1.0175 0.7142 0.5550 1.2087 1.6173 0.7403 1.8302 1.1610 0.9660
2.525 2.535 2.545 2.540 2.535 2.550 2.535 2.550 2.551
0.0015 0.0015 0.0030 0.0030 0.0050 0.0300 0.0500 0.0500 1.0120
0.00114 0.00138 0.00334 0.00247 0.00400 0.03180 0.04310 0.04710 1.0120
2.523 2.532 2.547 2.523 2.537 2.548 2.532 2.548 2.562
0.00152 0.00149 0.00316 0.00314 0.00525 0.03347 0.05357 0.05515 1.10796
0.00116 0.00137 0.00352 0.00258 0.00427 0.03543 0.04625 0.05205 1.10799
Fig. 8. Parameter analysis and simulation results for Bi(B1 = B2) varying and Ki = 0, 0.5 and 0.6.
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Fig. 9. Transient responses of LFC for 0.01 p.u. load increased in area 1 for Ki = 0.5 and Bi = 0.1 with deadband nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
parameters. In Fig. 10, doted-points show the simulated verifications. They give excellent agreement with the analyzed result also. For illustration, the step responses (Dpd1 ¼ 0:01 p:u:) for K i ¼ 0:5 and Bi ¼ 0:1 are shown in Fig. 11. It can be seen that the controlled system with the selected parameters is asymptotically stable. 5. Conclusions
Fig. 10. Parameter analysis and simulation results of the damped system for Bi = 0.1.
In this paper, the stability-equation method has been applied to the analysis and design of an interconnected power system with reheat steam turbine and governor deadband nonlinearities. The asymptotically stable, limit-cycle and unstable regions can be clearly defined in parameter planes. Based on these regions, one can choose the desirable parameters easily. The study pays attention to found the asymptotically stable region and reduce the
Fig. 11. Transient responses of LFC for 0.01 p.u. load increased in area 1 for Ki = 0.5, Bi = 0.1 and Tgi = 0.08 s with deadband nonlinearities; (a) inputs of governor; (b) frequency deviations; (c) speed changer deviations; (d) tie-line power deviation.
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amplitudes of sustaining oscillations. The proposed method can be applied to analyze systems controlled by PI, PID, VSC, and Fuzzy methods [5–14] also. For example, parameters of PI and PID controller can be used as parameters to be analyzed and designed. From Figs. 4, 8 and 10, one can see that the proposed method provided a simple way for choosing desirable parameters. References [1] Tripathy SC, Hop GS, Malik OP. Optimisation of load–frequency control parameters for power system with read steam turbines and governor deadband nonlinearity. PROC IEE PtC 1982;129(1):10–6. [2] Tripathy SC, Balasubramanian R, Nair PSC. Effect of superconducting magnetic energy storage on automatic generation control considering governor deadband and boiler dynamics. IEEE Trans Power Syst 1992;7(3):1266–73. [3] Lu CF, Liu CC, Wu CJ. Effect of battery energy storage system on load frequency control considering governor deadband and generation rate constraint. IEEE Trans Energy Convers 1995;10(1):555–61. [4] Chidambaram IA, Velusami S, Decentralized biased controllers for load– frequency control of interconnected power systems considering governor dead band non-linearity. In: INDICON, 2005 Annual IEEE; 2005; p. 521–5. [5] Kumar A, Malik OP, Hope GS. Variable–structure-system control applied to AGC of an interconnected power system. IEE PROC Gen Transm Distrib 1985;132(1):23–9. [6] Kumar A, Malik OP, Hope GS. Discrete variable structure controller for load frequency control of multiarea interconnected power systems. IEE PROC Gen Transm Distrib 1987;134(2):116–22. [7] Yang MS. Decentralized sliding mode stabilizer design for multi-area interconnected power systems. In: Proceedings of the 2000 IEEE international symposium on industrial electronics, vol. 1; 2000. p. 185–90. [8] Aldeen M, Marsh JF. Decentralised proportional-plus-integral design method for interconnected power systems. IEE PROC Gen Transm Distrib 1991;138(4):263–74.
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