a variable structure-based controller with fuzzy tuning

International Journal of Power and Energy Systems, Vol. 0, No. 0, 2001

A VARIABLE STRUCTURE-BASED CONTROLLER WITH FUZZY TUNING FOR LOAD-FREQUENCY CONTROL Q.P. Ha∗ and H. Trinh∗∗

Abstract

ear feedback optimal control [2, 3], adaptive control [46], and variable structure control [710]. In optimal control, either the feasibility for implementation still must be established or systematic methodology is not recommended for the proper choice of equivalent dynamics for the closedloop system. The adaptive control strategies usually require satisfaction of the perfect model-following conditions or explicit parameter identification. The variable structure systems with a sliding mode have been well-known for their robust properties. The complexity of sliding mode controllers and the associated chattering problem may be the reason these controllers were not fully appreciated in LFC applications. Furthermore, controllers based on the state equation of the linearized model may require estimates of inaccessible state variables. Observers can be designed for this, but it would involve the additional cost of data telemetering. Using only the area control error, conventional integral-proportional controllers with switching proportional and integral gains are proposed in [11, 12]. However, issues relating to the robustness against parameter variations have not been clearly addressed by this method. Recently, fuzzy logic has proven to be a prospective tool for dealing with uncertainties in dynamic systems. In this direction, fuzzy tuning schemes have been shown to remarkably enhance system robustness in conventional proportional integral control [13]. Inspired by [13], tuning schemes are proposed in this paper to enhance the control performance of proportional-integral controllers for singlearea and multi-area interconnected systems with nonreheat and reheat generating units. Simulation results indicate that the control scheme is able to provide good performance in single-area as well as multi-area interconnected systems with non-reheat or reheat turbines. Moreover, the responses of the system employing the proposed controllers are shown as being rather insensitive to parameter changes and speed governor deadband, with or without generation rate constraints. The explicit expressions for tuning require low computational cost and thus, are easy to implement.

This paper presents a variable structure-based approach to the load frequency control problem in electric power generation systems. This approach combines the salient features of both variable structure and fuzzy systems to achieve high-performance and robustness. Fuzzy tuning is proposed for this purpose to continuously adjust the controller parameters according to the area control error. Simulation results indicate that the control scheme is able to substantially enhance control performance in single-area and multi-area interconnected systems with non-reheat or reheat turbines. Furthermore, it is shown that the responses of the system employing the proposed controllers are quite insensitive to parameter changes and speed governor deadband, with or without generation rate constraints. The control strategy requires low computational cost and is amenable for practical implementation.

Key Words Load-frequency control, variable structure-based control, fuzzy tuning

1. Introduction In electric power generation, system disturbances caused by load fluctuations result in changes to the desired frequency value. Load-frequency control (LFC), or automatic generation control, is a very important issue in power system operation and control for supplying sufficient and both good quality and reliable electric power. The conventional control strategy for the LFC problem is to take the integral of the control error as the control signal. An integral controller provides zero steady-state frequency deviation but exhibits poor dynamic performance [1], especially in the presence of other destabilizing effects such as parameter variations and nonlinearities. In order to improve the transient response, advanced control techniques have been proposed, which include lin∗

Department of Mechanical & Mechatronic Engineering, J07, the University of Sydney, NSW 2006, Australia; e-mail: [email protected] ∗∗ chool of Engineering, James Cook University, Townsville QLD 4811, Australia; e-mail: [email protected] (paper no. 203-2148)

1

and singletons for µKP N L , µKP N S , µKP P L , and µKP P S [14], the value for kP is found to be:

2. Controller Design In this section, a new control scheme for refining the dynamic properties of the conventional integral controller is proposed, employing both integral and proportional control with fuzzy tuning (IPFT). Fuzzy logic schemes are applied here to continuously change both the integrator gain and additional proportional gain in such a way that the control error is smoothly reduced, regardless of any load disturbances and parameter variations, while its zero steady-state is also guaranteed. Let us begin with the control law: u(t) = kP e + kI

Z

edt

kP = kP,max

(1)

µES =

2 , µEL = 1 − µES [exp(e/σe ) + exp(−e/σe )]

singletons for µKIL and µKIS and using the technique described in [13], the value for kI is found to be: kI =

−2kI,max [exp(−e/σe ) + exp(e/σe )]

(3)

where kI,max is the maximal value of kI , chosen to satisfy the system stability. Remark 1: The controller is of a variable structure type in the sense that its parameters can be adaptively adjusted using fuzzy tuning schemes based on information from the control error. It thus avoids the cumbersome aspect of estimating all the inaccessible system state variables when realizing other control strategies such as linear feedback optimal or sliding mode control.

Figure 1. Block diagram of the IPFT controller.

3. Application to Load-frequency Control of a Single-area System

Control engineering heuristics suggests that if is large, then kP should increase and kI should decrease and if ACE is sufficiently small, then kP should diminish to zero while kI remains large. As indicated below, the system stability requires that kP be limited and lie between a negative and a positive value, while normally, kP = 0 when ACE = 0. Note further that physically, the control output should be in the opposite direction to the ACE. Therefore, the following rules are proposed for tuning the proportional gain kP [14]: • If ACE is positive large (EPL), then kP is negative large (KPNL), • if ACE is positive small (EPS), then kP is negative small (KPNS), or • if ACE is negative large (ENL), then kP is positive large (KPPL), and • if ACE is negative small (EPS), then kP is positive small (KPPS). By introducing the following membership functions:

µEN L =

(2)

where kP,max is the maximal value of kP , chosen to satisfy the system stability and σe is some positive constant. The fuzzy rules for tuning kI can be formalized as [14]: • If the ACE is large (ES), then kI is small (KIS) and • if the ACE is small (EL), then kI is large (KIL). By defining the following membership functions:

where e is the area control error (ACE), and kP and kI are the control gains. Control practice indicates that a large kI will result in a high overshoot, but when kI diminishes to zero the system will display a steady-state error. A feasible approach to this static and dynamic accuracy conflict is to switch to the control output u(t) between the proportional and integral control action according to the comparison of ACE with a constant value ε [11]. This technique however, may lead to an oscillatory response, and the settings for ε, kP , and kI must be carefully rechosen when the system is subject to a new condition. This may be explained by the fact that kP and kI are chosen to be constants. Motivated by [11, 13], a continuous variable-structured integralproportional controller is proposed. The controller, shown in Fig. 1, is capable of self-adjusting its parameters according to the error information.

µEP L =

[exp(−e/σe ) − exp(e/σe )] [exp(e/σe ) + exp(−e/σe )]

Assuming the power system is exposed to small changes in load during its normal operation, the linear model will adequately represent its dynamics. Let us consider the same linearized model of a single control area power system with numerical parameters given in [3, 8, 9]. The block diagram is shown in Fig. 2 where the values for the system nominal parameters are, respectively, R = 2.4 Hz/p.u. MW, Tg = 0.08 s, Tt = 0.3 s, Tp = 20 s, and Kp = 120 Hz/p.u. MW, and turbine power of 2000 MW. Figure 2. Block diagram of the single control area. From the closed-loop characteristic equation, the stability conditions for the system with the nonreheat turbine using proportional-integral control are found as:   − (Tg +Tt +Tp )(1/Tg +1/Tt +1/Tp )−(1+Kp /R) < k < p Kp B  −a (1+Kp /R+Kp Bkp ) < k < 0 Kp B(Tg Tt +Tt Tp +Tp Tg )

[exp(e/σe ) − exp(−e/σe )] , µEP S = 1 − µEP L [exp(e/σe ) + exp(−e/σe )]

1+Kp /R Kp B

I

(4)

where:

[exp(−e/σe ) − exp(e/σe )] , µEN S = 1 − µEN L [exp(e/σe ) + exp(−e/σe )]

a= 2

(Tg + Tt + Tp )(1/Tg + 1/Tt + 1/Tp ) − (1 + Kp /R + Kp Bkp ) (1/Tg + 1/Tt + 1/Tp )

In this case, we find that −5.3471 < kp < 1 and −2.252 < kI < 0 (for kP = 0). Remark 2: For the case with reheat turbines, a reheater’s transfer function, of the form (1+Tr Kr s)/(1+Tr s), is inserted between the governor and turbine blocks in Fig. 2. With IPFT controller parameters chosen as kP,max = 0.2, kI,max = 1, and σe = 0.2 for a step load change of ∆PL = 0.02 p.u. MW, the frequency deviation for the nonreheat-type system is shown in Fig. 3(a). The responses, obtained by the sliding mode controller (SMC) in [8], and the variable-structured switching proportionalintegral controller (VSSPI) in [11] are also shown in the same figure for the purposes of comparison. Fig. 3(b) shows the responses for the reheat-type system (Tr = 10s and Kr = 0.5). It is observed that the response with the IPFT controller is less oscillatory and has a smaller frequency swing.

performance than with VSSPI [11]. Note that the system employing SMC [8] cannot provide a stable response as the state trajectory does not remain in the sliding surface under generation rate constraints and speed governor deadband. Figure 5. Frequency deviation (a) and output power change (b) with nonreheat turbine (2D = 0.06%, Pmax = 0.1 p.u. MW/min, and Pmax = 0.03 p.u. MW). Figure 6. Frequency deviation (a) and and output power change (b) with a reheat turbine (2D = 0.06%, Pmax = 0.03 p.u. MW/min, and Pmax = 0.01 p.u. MW). 4. Application to Load-frequency Control of Interconnected Systems As the IPFT control technique requires only the information of the area-control error, it can be directly applied to the automatic generation control of multi-area interconnected systems.

Figure 3. Frequency deviation with a (a) nonreheat and a (b) reheat turbine.

4.1 A Two-area Interconnected System Assume that the system parameters R, Tg , Tt , Tp , and Kp are subject to simultaneous changes of +25% from their nominal values. Fig. 4(a) depicts the frequency deviation versus time for the nonreheat-type system using IPFT, SMC, and VSSPI controllers. The responses for the reheattype system are presented in Fig. 4(b). It is inherent that the responses using the proposed IPFT controller exhibit insensitivity to load and parameter perturbations for the power system with nonreheat or reheat turbines. This merit is explained by the controller’s capability to selfadjust its gains in accordance to proper tuning schemes.

Consider in the following the two-area thermal power system reported in [11]. It is assumed that the power demand is subject to small changes so that LFC and reactivepower/voltage control problems are decoupled and the system is adequately represented by a linearized model. The LFC block diagram for equal areas with reheat steam turbines is shown in Fig. 7 where a1 2 = −1. The values of the system parameters for each area are the same as discussed above. The responses obtained by employing the proposed IPFT controller (kP,max = 0.001, kI,max = 0.04, and σe = 0.005) will be compared with those obtained by employing the variable structure-type switching proportional integral controller (VSSPI) [11] with kP = 0.001, kI = 0.04, and ε = 0.01, and the conventional integral controller (CI) with kI = 0.04. For a step load change of ∆PL1 = 0.01 p.u. MW, the frequency and output power deviations of area 1 and the tie-line power deviation with nonreheat turbines are shown in Fig. 8 and with reheat turbines in Fig. 9. Comparing the responses using the VSSPI and CI controllers in both cases, the responses with the IPFT controller exhibit better performance with respect to static and dynamic accuracy. The parameter variations problem is considered next by assuming a +25% change in R, Tg , Tt , Tp , and Kp from their nominal values. Fig. 10 shows the responses of the two-area interconnected power system with reheat turbines. The results obtained demonstrate the robustness of the proposed control scheme against parameter variations. In fact, as concluded in [3], in two-area systems with reheat turbines and fixed controller parameters, the optimum controller setting, cost index, and stability margin are sensitive to the reheat coefficient, governor time constant, turbine time constant, and inertia constant. In this paper, the values for the controller parameters, namely the proportional and integral gains, can be adjusted, continuously depending on the value of AC. The control performance n the presence of system parameter variations therefore has improved substantially.

Figure 4. Frequency deviation with a parameter changes of +25% from nominal values with a (a) nonreheat, and a (b) reheat turbine. Remark 3: As pointed out in [15], the purposes and objectives of LFC are limited by the physical elements involved in the process. Due to delays associated with physically limited response rates of energy conversion, it is desired that LFC acts slowly and deliberately over tens of seconds or a few minutes. From a utility operations perspective, no particular control purpose is served by speeding up the LFC action. Therefore, the gains kp and kI should be chosen so that the LFC response time will take place at least one to two minutes. In the simulation, kP,max = 0.001, kI,max = 0.05, and σe = 0.002 are chosen. Fig. 5 shows the frequency changes and generator output power with response to a step load change of ∆PL = 0.005 p.u. MW when the speed governor is subject to backlash of deadband 2D = 0.06% and generation limits Pmax = 0.1 p.u.MW/min and Pmax = 0.03 p.u. MW are imposed for the nonreheat-type system. With generation limits Pmax = 0.03 p.u.MW/min and Pmax = 0.01 p.u. MW for the system with the reheat turbine, the frequency deviation and power generation output are shown in Fig. 6. It can be observed that the responses with our proposed control scheme give a better 3

Figure 7. A two-area thermal power system.

that additional links between the companies are negligible. Interchanges over the multi-area ties, represented by the synchronising coefficients Tij , are scheduled to specific levels so that unwanted power flow on these tie-lines is undesirable.

Figure 8. System response with a nonreheat turbine: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation. Figure 9. System response with reheat a turbine, no constraint: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation.

Figure 12. Simplified diagram of a four-area interconnected system. In the simulation, the system and controller parameters described above are used for each company and area 1 is the considered area with a step load change of 1% p.u. MW. The frequency, output power, and tie-line power deviations of area 1 versus time are shown in Fig. 13. Comparing the responses obtained by employing the proposed IPFT, VSSPI, and CI controllers reveals the advantage of the proposed controller in terms of smaller swings and dynamic error. Now let us assume that the power system with the proposed IPFT controllers is subject to speed governor deadbands of 0.05% and generation rate constraints Pmax = 0.1 p.u. MW/min and Pmax = 0.03 p.u. MW for nonreheat turbines, and Pmax = 0.03 p.u. MW/min and Pmax = 0.01 p.u. MW for reheat turbines are imposed. The responses to a load change of 1% p.u. MW with nominal parameters are shown in Fig. 14(a) and with simultaneous changes of +20% over the nominal values of Ri , Tgi , Tti , Tpi , and Kpi (i = 1, 2, 3, 4) are shown in Fig. 14(b). The results obtained demonstrate that the proposed approach is highly robust against parameter and load variations, even with the presence of nonlinearities existing in the system or imposed by designers.

Figure 10. System response with a reheat turbine at parameter changes of +25% from nominal values, no constraint: (a) frequency deviation, (b) power deviation, and (c) tie-line power deviation. Now assume that the power system is subject to speed governor deadbands of 0.05% and generation rate constraints Pmax = 0.1 p.u. MW/min and Pmax = 0.03 p.u. MW for the system with nonreheat turbine, and Pmax = 0.03 p.u. MW and Pmax = 0.01 p.u. MW for the system when reheat turbine are imposed. For a step load change of ∆PL1 = 0.005 p.u. MW, the responses obtained with the IPFT controller are presented in Fig. 11. It is shown that, in the presence of speed governor deadbands with imposed generation rate constraints, the effect on the interconnected areas of a disturbance in one area can be considerably reduced in the system with nonreheat and reheat turbines using the proposed control scheme. Note that, due to the constraints, the first swing in the frequency deviation response need not be limited [15]. Figure 11. System response with governor deadband and GRC with a: (a) non-reheat and a (b) reheat turbine.

Figure 13. System response, no constraint: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation of area 1.

4.2 A Four-area Interconnected System

Figure 14. System response with governor deadbands and generation rate constraints at (a) nominal parameters and (b) parameter changes of +20% from nominal values.

For ease of operation and control, an interconnected power system is generally considered as an amalgamation of a number of areas [12]. Fig. 12 depicts the simplified diagram of a multi-area interconnected system consisting of four operating companies with different kinds of generating units (nonreheat and reheat-turbine type) linked together in an interconnection. Each company has its own generation Gi , representing all the generating sources within the company area and its own load Li , representing the aggregate of all loads within the area (i = 1, 2, 3, 4). Any area attempting to adjust its generation to restore the frequency-to-schedule requires a block of controllable generation large enough to respond to the mismatched power in the interconnection. Moreover, the interconnection requires tie-lines, which can carry such amounts of power between this area and the others [15]. When a load change takes place in a given area, the generation of that area is expected to provide a control action to accommodate that load change. The difference between an area’s generation and the power taken internally by loads and losses is the sum of power flows on all tie-lines between this area and others. The inter-company ties are shown in a simplified form, where each company has a tie-line representing all of its links with each of its neighbour areas. It is assumed

5. Conclusion This paper has presented a variable structure-based approach, incorporating tuning schemes to load-frequency control in electric power generation systems. Integralproportional controllers with fuzzy tuning are applied to the load frequency control problem of single-area and multiarea systems. The control technique is of a variable structure type in the sense that the controller parameters are continuously adjusted according to the area-control error. The validity of the control technique is verified through extensive simulations for single-area and two- and fourarea interconnected systems with nonreheat and reheat turbines, taking into account a number of practical aspects such as parameter variations, speed governor deadband, and generation rate constraints. It is demonstrated that the enhancement in load-frequency control performance is achieved through the use of fuzzy tuning schemes that can capture engineering heuristics and experience. Explicit expressions for adjusting the controller parameters ensures 4

Biographies

that the proposed technique is feasible for implementation in power generation systems.

Q.P. Ha received the B.E. degree in electrical engineering from the Ho Chi Minh City University of Technology, Vietnam, the Ph.D. degree in engineering science from the Moscow Power Institute, Russia, and the Ph.D. degree in electrical engineering from the University of Tasmania, Australia, in 1983, 1992, and 1997, respectively. Early in his career, he worked as a Lecturer in electrical engineering in Ho Chi Minh City for 8 years. From 1993 to 1994 he was a postdoctoral in the Department of Electric Drives at the Moscow Power Institute, Russia. Since September 1997, he has been a Senior Research Associate in the Department of Mechanical and Mechatronic Engineering at the University of Sydney, Australia. His research interests include robust control systems, sliding mode control, and application of artificial intelligence in engineering.

Acknowledgement The support of the Australian Research Council is gratefully acknowledged.

References [1] P. Kundur, Power system stability and control (New York: McGraw-Hill, 1994). [2] S.M. Miniesy & E.V. Bohn, Optimum load-frequency continuous control with unknown deterministic power demand, IEEE Trans. Power Apparatus and Systems, 91(5), 1972, 19101915.

[3] J. Nanda & B.L. Kaul, Automatic generation control of an interconnected power system, IEE Proceedings, 125(5), 1978, 385390. [4] I. Vajk, M. Vajta, L. Keviczky, R. Haber, J. Hetthessy, & K. Kovacs, Adaptive load-frequency control of Hungarian power system control, Automatica, 21(2), 1985, 129137.

H. Trinh obtained the B.E., M.Eng.Sc., and Ph.D. degrees in electrical engineering from the University of Melbourne, Victoria, Australia in 1990, 1992, and 1996, respectively. From March 1995 to December 1996, he was a postdoctoral Research Fellow in the Department of Electrical and Electronic Engineering at the University of Melbourne. He joined James Cook University in January 1997, where currently, he is a Lecturer. His research interests lie in the areas of modelling and control of dynamic systems.

[5] C.T. Pan & C.M. Law, An adaptive controller for power system load-frequency control, IEEE Trans. Power Systems, 4(1), 1989, 122128. [6] A. Rubaii & V. Udo, Self-tuning load frequency control: Multilevel adaptive approach, IEE Proceedings -Gener. Transm. Distrib., 141(4), 1994, 285290. [7] W.C. Chan & Y.Y. Hsu, Automatic generation control of interconnected power systems using variable-structure controllers, IEE Proceedings Pt. C, 128(5), 1981, 269-279. [8] N.N. Bengiamin & W.C. Chan, Variable structure control of electric power generation, IEEE Trans. Power Apparatus and Systems, 101(2), 1982, 376380. [9] A.Y. Sivaramakrishnan, M.V. Hariharan, & M.C. Srisailam, Design of variable-structure load-frequency controller using pole assignment technique, Int. J. Control, 40(3), 1984, 487 498. [10] Q.P. Ha, A fuzzy sliding mode controller for power system loadfrequency control, Proc. IEEE Int. Conf. on Knowledge-Based Intelligent Electronic Systems, Adelaide, Australia, 1998, 149 154. [11] A. Kumar, O.P. Malik, & G.S. Hope, Variable-structure-system control applied to AGC of an interconnected power system, IEE Proceedings Pt. C, 132(1), 1985, 2329. [12] A. Kumar, O.P. Malik, & G.S. Hope, Discrete variable-structure controller for load frequency control of multiarea interconnected power system, IEE Proceedings Pt. C, 134(2), 1987, 116122. [13] Q.P. Ha, PI controllers with fuzzy tuning, Electronics Letters, 32(11), 1996, 10431044. [14] Q.P. Ha & M. Negnevitsky, Fuzzy tuning in electric power generation control, Proc. IEE Int. Conf. on Advances in Power System Control, Operation and Management, Hong Kong, 1997, 662667. [15] N. Jaeeli, L.S. Vanslyck, D.N. Ewart, L.H. Fink, & A.G. Hoffmann, Understanding automatic generation control, IEEE Trans. Power Systems, 7(3), 1992, 11061111.

5

K s x4

1 R Backlash 2D

Rate Limits Pmax 1 Tg

u

1 s Governor

Position Limits ∆Pmax 1 1+sTt x2 x3 Turbine

Load ∆PL KP 1+sTP

Power System

Control

Figure 1. Block diagram of single control area.

(a)

x1

(b)

(c) (d) Figure 2. Change in frequency versus time: (a) Integral controller, (b) Controller proposed by Bengiamin & Chan, (c) Controller proposed by Sivaramakrishnan et al., and (d) Controller proposed in this paper.

(a)

(b)

Figure 3. Control input: (a) Controller proposed by Sivaramakrishnan et al., and (b) Controller proposed in this paper.

(a) (b) Figure 4. Responses without fuzzy tuning: (a) Frequency deviation, and (b) Control input.

(a) (b) Figure 5. Frequency deviation (a) and change in output power (b) with the proposed controller when the speed governor is subject to a deadband of 2 D = 0.001 , and generation limits Pmax = 0.1 p. u.MW / min and ∆Pmax = 0.03 p. u. MW are imposed.

k (e) I s

e

u k P (e)

Frequency deviation, Hz

Figure 6. Block diagram for the IPFT controller.

SMC IPFT

VSSPI

Time, x0.1sec

Frequency deviation, Hz

(a)

SMC IPFT

VSSPI

Time, x0.1sec

(b) Figure 7. Frequency deviation with (a) nonreheat, and (b) reheat turbine.

Frequency deviation, Hz

VSSPI SMC IPFT

Time, x0.1sec

Frequency deviation, Hz

(a)

VSSPI IPFT SMC Time, x0.1sec

(b) Figure 8. Frequency deviation at parameter changes of +25% from nominal values

Frequency deviation, Hz

with (a) nonreheat, and (b) reheat turbine.

IPFT VSSPI

Time, x0.1sec

(a)

Change in output power, p.u. MW

IPFT VSSPI

Time, x0.1sec

(b) Figure 9. Frequency deviation (a) and output power change (b) with nonreheat

Frequency deviation, Hz

turbine ( 2 D = 0.06% , PDmax = 0.1 p.u.MW / min , and Pmax = 0.03 p.u.MW ).

IPFT VSSPI

Time, x0.1sec

Change in Output Power, p.u. MW

(a)

IPFT VSSPI

Time, x0.1sec

(b)

Figure 10. Frequency deviation (a) and output power change (b) with reheat turbine ( 2 D = 0.06% , PDmax = 0.03 p.u.MW / min and Pmax = 0.01 p.u.MW ).

1 R1

B1 ACE

+

IPFT1

+

Load ∆PL1 1

+

Controller

1+sTg 1

1+sTr 1Kr1 1+sTr 1

Governor

Reheater

1 1+sTt 1

+

KP1 1+sTP 1

Turbine

Power System

∆Ptie1

2πT12 s

a12 ACE

+

IPFT2

+ B2

+

a12

+

Controller

1 R2

1 1+sTg 2

1+sTr2Kr2

Governor

Reheater

1+sTr 2

1

KP2

1+sTt2

1+sTP2

Turbine

Power System Load

∆PL 2

Frequency deviation ∆ f1, Hz

Figure 11. A two-area thermal power system.

IPFT VSSPI CI

Time, x0.1sec

Output power deviation ∆ Pg1, p.u.MW

(a)

IPFT VSSPI CI

Time, x0.1sec

Tie-line power deviation, p.u.MW

(b)

IPFT VSSPI CI

Time, x0.1sec

(c)

Frequency deviation ∆ f1, Hz

Figure 12. System response with nonreheat turbine: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation.

IPFT VSSPI CI

Time, x0.1sec

Output power deviation ∆ Pg1, p.u.MW

(a)

IPFT VSSPI CI

Time, x0.1sec

(b)

Tie-line power deviation, p.u.MW

IPFT VSSPI CI

Time, x0.1sec

Frequency deviation ∆ f1, Hz

(c) Figure 13. System response with reheat turbine, no constraint: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation.

IPFT VSSPI CI

Time, x0.1sec

Tie-line power deviation, p.u.MW

(a)

IPFT VSSPI CI

Time, x0.1sec

(b)

Output power deviation ∆ Pg1, p.u.MW

IPFT VSSPI CI

Time, x0.1sec

(c) Figure 14. System response with reheat turbine at parameter changes of +25% from nominal values, no constraint,: (a) frequency deviation, (b) power deviation, and (c) tie-line power deviation.

Output power deviation ∆ Pg1, p.u.MW

Tie-line power deviation, p.u.MW Frequency deviation ∆ f1, Hz

Time, x0.01sec

(a) Output power deviation ∆ Pg1, p.u.MW

Tie-line power deviation, p.u.MW Frequency deviation ∆ f1, Hz

Time, x0.1sec

(b) Figure 15. System response with governor deadband and GRC: (a) non-reheat, and (b) reheat turbine.

L2

G2

Company 2

nonreheat turbine

T12=T21

L1

L4 T14=T41

Company 1

T23=T32

T13=T31 G3

Company 3

Company 4

G4

G1 reheat turbine

reheat turbine

nonreheat turbine

L3

Frequency deviation ∆ f1, Hz

Figure 16. Simplified diagram of a four-area interconnected system.

IPFT VSSPI CI

Time, x0.1sec

Output power deviation ∆ Pg1, p.u.MW

(a)

IPFT VSSPI CI

Time, x0.1sec

(b)

Tie-line power deviation ∆ Ptie1, p.u.MW

IPFT VSSPI CI

Time, x0.1sec

(c) Figure 17. System response, no constraint: (a) frequency deviation, (b) output power deviation, and (c) tie-line power deviation of area 1.

Output power deviation ∆ Pg1, p.u.MW

Frequency deviation ∆ f1, Hz Tie-line power deviation, ∆ Ptie1, p.u.MW

Time, x0.1sec

(a)

Output power deviation ∆ Pg1, p.u.MW Frequency deviation ∆ f1, Hz Tie-line power deviation ∆ Ptie1, p.u.MW

Time, x0.1sec

(b) Figure 18. System response with governor deadbands and generation rate constraints at (a) nominal parameters, and (b) parameter changes of +20% from nominal values.