GA-Based Integral Sliding Mode Control for

AGC

Dianwei Qian1, Xiangjie Liu1, Miaomiao Ma1, and Chang Xu2

1 School of Control and Computer Engineering, North China Electric PowerUniversity, Beijing 102206, P.R. China

dianwei.qian@ncepu.edu.cn2 College of Energy and Electricity, Hohai University, Nanjing, 210098, P.R. China

Abstract. This paper addresses an integral sliding mode control ap-proach for automatic generation control (AGC) of a single area powersystem. Genetic algorithm (GA) is employed to search the parametersof the sliding surface. The proposed design is investigated for AGC of asingle area power system, made up of reheated thermal and gas powergenerations. Compared with the GA-based proportion-integral (PI) con-trol, simulation results show the feasibility of the presented method.

1 Introduction

Automatic Generation Control (AGC) is one of the most important issues in theoperation and design of contemporary power systems [1]. The primary objectivesof AGC are to adjust the power output of the electrical generator within aprescribed area in response to changes in system frequency, tie-line loading (forinterconnected areas), so as to maintain the scheduled system frequency andinterchange with the other areas with predetermined limits [2].

A large number of approaches concerning the AGC problem have been pre-sented in the last two decades, e.g., optimal control [3], variable structure control[4], adaptive control [5], robust control [6], intelligent control [7]. In the refer-eed literature, the AGC problem can be cataloged as single area with a thermalor hydro power source and interconnected double areas with thermal-thermalpower sources. In this paper, we focus on a single area with reheated thermaland gas power sources, which is rarely refereed in the above references. With theincrease in size and complexity of power systems, there may exist large numberof various sources of generations in a prescribed area, which make our researchinteresting in practical accounts [8].

Integral sliding mode control (ISMC) [9] is a robust feedback control method,possessing the property that the order of its motion equation is equal to the orderof the original system. Such technology could avoid the chartering phenomenon ofthe conventional sliding mode and preserve the robustness and accuracy providedby the sliding mode. But we have to select the parameters of the sliding surface ofISMC after trial and error during the design process of ISMC. Genetic algorithm(GA) is a searching strategy inspired by natural evolution behavior, pointing out

Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part II, LNCS 6146, pp. 260–267, 2010.c© Springer-Verlag Berlin Heidelberg 2010

GA-Based ISMC for AGC 261

a gateway to free the this time-consuming business. Thus, the combination ofISMC and GA provides a good candidate to solve the AGC problem concerningthe single area with multiple sources.

2 Power System Models

The power system for the AGC problem under consideration is expressed onlyto relatively small changes so that it can be adequately represented by the linearmodel in Fig. 1. Figure 1 represents the block diagram of a single area sys-tem with multi-source power generations. The total generation is from reheatedthermal and gas power generating units equipped with speed governors, whichrepresents the thermal and gas power generating units lumped together in thisprescribed area, respectively.

Fig. 1. Linear model of a single area power system

The symbols in Fig. 1 are explained as a and c are constants of valve posi-tioner, b is time constant of valve positioner, X is gas turbine speed governorlead time constant, Y is gas turbine speed governor lag time constant, Tgf isfuel time constant, Tgcr is combustion reaction time delay, Tgcd is compressordischarge volume time constant, Kg is gas power generation contribution, Rg isspeed governor regulation parameter of the gas unit, Tthg is steam turbine speedgovernor time constant, Kthr is coefficient of re-heater steam turbine, Tthr is re-heater time constant, Ttht is steam turbine time constant, Kth is thermal powergeneration contribution, KPS is power system gain constant, TPS is power sys-tem time constant, Rth is speed governor regulation parameter of the reheatedthermal unit, ΔPCth is change in thermal turbine speed-changer position, ΔPCg

is change in gas turbine speed-changer position, ΔPGth is thermal power devi-ation, ΔPGg is gas power deviation, ΔPG is total power deviation, ΔPd is loaddisturbance, ΔF is frequency deviation, ΔPCth is the AGC control signal of thereheated thermal unit, ΔPCg is the AGC control signal of the gas unit.

It is obvious that the plants for the AGC in the area consist of three parts:

– Reheated thermal turbine and its governor with dynamics Gthg(s) = 1Tthgs+1

and Gtht(s) = 1+sKthrTthr(1+sTthr)(1+sTtht)

262 D. Qian et al.

– Valve positioner, fuel system, gas turbine and its generator with dynamicsGvp(s) = a

bs+c , Gfc(s) = Xs+1Y s+1 , Ggt(s) = 1

1+Tgcds and Ggg(s) = 1−Tgcrs1+Tgf s

– Power systems with dynamics Gps(s) = KPS

TPSs+1

Assume there is no mismatch between generation and load under normal oper-ating conditions. The total generation is determined by

PG = PGth + PGg (1)

where PGth = KthPG, PGg = KgPG, and Kth + Kg = 1. The values of Kth andKg depend upon the total load and also involve economic load dispatch. Forsmall perturbation, (1) can be written as

ΔPG = ΔPGth + ΔPGg (2)

In Fig. 1, both the AGC control signals ΔPCth and ΔPCg will be produced bytwo integral sliding mode controllers, designed in the following section.

3 GA-Based Integral Sliding Mode Controller

3.1 Design of Integral Sliding Mode Controller

In the integral sliding mode method, a sliding surface should be constructed bythe system state variables in the state space so that the state space models of theabove single area with the two generation sources should be transformed fromtheir transfer functions.

Without loss of generality, the state space expression of the AGC problem ofa single area with a generation source can be depicted as

X = AX + Bu + f(X) (3)

where X is the n-dimensional state vector, u is the control scalar, produced bythe integral sliding mode controller, A is the n × n state matrix, and B is then × 1 input vector, f(X) is the nonlinear perturbation vecotor with a knownupper bound vector f0 > 0. (3) is a simple model, associated with the AGCproblem of a single area with multiple generation sources in the paper. In Fig.1, we can get such the state space model as (3) by separating one generationunit from the other. For example, let Kth = 0, we could get the transfer functionmodel of a single area with a gas generation unit, then the state space model ofsuch the system can be gotten.

For the control design, we define the control input u as

u = uic + urp (4)

where uic is the predetermined ideal control, denoting the state trajectory of thesystem X = AX+Buic, e.g. uic may be obtained through linear static feedbackcontrol uic = −kTX, urp is discontinuous to reject the perturbation.

GA-Based ISMC for AGC 263

Then, the sliding surface s is defined as

s = s0(X) + z (5)

where s0 may be designed as the linear combination of the system states s0 =CTX (similar to conventional sliding surface), z induces the integral term andmay be determined as z = −CT (AX + Buic), z(0) = −CTX(0).

Theorem 1. Consider the single area power system with a generation unit as(3). If the control law and the integral sliding surface are defined as (4) and (5),then the sliding motion of the nominal system of (3) will occur at t = 0.

Proof. Let f(X) = 0 in (3), its nominal system could be written as

X = AX + Bu (6)

From (5), we can get the sliding surface as

s = s0(X) + z = CT X +∫ t

0

[−CT (AX + Buic)]dt (7)

When the sliding mode occurs, we have

s = CT X − CT (AX + Buic) = 0 (8)

which means the motion equation of the sliding surface coincides with the nom-inal system dynamics. Further, at t = 0, we have

s(0) = CTX(0) + z(0) = CT X(0) − CT X(0) = 0

Thus, the sliding mode of the nominal system will occur at t = 0. ��Theorem 2. Consider the single area power system with a generation unit as(3), define the control law (4), the integral sliding surface (5) and the discontin-uous term urp = −ρs − σsign(s), (ρ > 0, � > 0). If CT Bσ ≥ CT f0 is satisfied,then the integral sliding surface is asymptotically stable.

Proof. Define the Lyapunov function as V = s2

2 . Differentiating V with respectto time t, we obtain

V = ss (9)Substituting (3), (4) and (5) into (9), we have

V = ss = s(s0 + z)

= s[CT X − CT (AX + Buic)]

= s{CT [AX + Bu + f(X)] − CT (AX + Buic)}= s{CT [AX + B(uic + urp) + f(X)] − CT (AX + Buic)}= s{CT Burp + CT f(X)}= s{CT B[−ρs − σsign(s)] + CT f(X)}= −CT Bρs2 − CT Bσ| s | + CT f(X)s

≤ −CT Bρs2 − CT Bσ| s | + CT f0s < 0

(10)

Thus, the sliding surface s with integral term is asymptotically stable. ��

264 D. Qian et al.

3.2 Parameter Tuning by Genetic Algorithm

Genetic algorithm (GA) is a searching strategy inspired by natural evolutionbehavior. Each individual consisting of a set of parameters to be tuned can berepresented by a chromosome. A simple GA includes individual selection, mu-tation and crossover steps. The selection from the whole population is based oneach individual’s fitness. A roulette selection strategy is adopted in the followingcomparison. The mutation causes a complete opposite change on gene bit ran-domly. The crossover exchanges part of the information between two individuals.After genetic operation, new individuals are generated to form a new popula-tion. The fitness mapping is a key problem for the genetic learning process. Thereciprocal of the integral squared error (ISE) of the system states is selected asthe individual fitness

J =1

∞∫0

{∑i=ni=0 x2

i (t)}dt

(11)

Here xi is the ith element of the n−dimensional state vector X.A good individual corresponds to a small objective value or a big fitness. As

the genetic operation goes on, the individual maximum fitness and the popula-tion average fitness are increased steadily. In our simulations, we find that thecontroller parameters can be searched out by using such simple genetic algorithm,but they vary greatly with different crossover probability, mutation probability,and population size. Controller parameters often converge to different results indifferent experiments, which may not be an optimized solution even may lead toa false solution.

For such the case of the AGC problem in this paper, some modifications areproposed on the basis of the simple genetic algorithm. Large crossover probabilityand small mutation probability will ensure population diversities and preventpremature convergence of maximum individual fitness so that crossover fractionand mutation fraction are set to 0.95 and 0.05, respectively. Elitist individualreservation is applied to ensure the maximum fitness to keep on increasing andprevent fluctuation of maximum fitness caused by large crossover probabilities.From (5), the design of the sliding surface with integral term can be summarizedas the process of finding a suitable vector CT . Further, we have known that uic

can be designed as the state feedback controller uic = −CTX so that we canconclude the constraint as the eigenvalues of (A − BCT ) < 0 on the aspect ofthe system stability. For accelerating our search, we preset σ and ρ before theoptimization by GA.

4 Simulation Results

In this section, we shall demonstrate the application of the presented GA-basedintegral sliding mode control for the AGC problem of a single area with reheatedthermal and gas generation units as shown in Fig. 1. The values of the parametersof this power systems are determined [8] as follows.

GA-Based ISMC for AGC 265

– Reheated thermal generation unit: Tthg = 0.08 s, Ttht = 0.3 s, Tthr = 10 s,Kth = 0.3, Rth = 2.4 Hz/puMW

– Gas generation unit: X = 0.6, Y = 1.0, a = 1, b = 0.05, c = 1, Tgf = 0.23 s,Tgcr = 0.01 s, Tgcd = 0.2 s, Rg = 2.4 Hz/puMW

– Power system: KPS = 68.57 Hz/puMW, TPS = 11.43 s for the operatingload 1750 MW ; KPS = 75 Hz/puMW, TPS = 12.5 s for the operating load1600 MW

For verifying the robustness of the integral sliding mode controller, we employthe parameters at the operation load 1750 MW as the design point and theparameters at the operation load 1600 MW as the checking point.

Let the system output �fth = [0 0 74.99 24.99] × [xth1 xth

2 xth3 xth

4 ]T , wherexth

i (i = 1, 2, 3, 4) is the state variables. For the reheated thermal generationunit, the corresponding values of the state matrix Ath and the input vector Bth

can be obtained as

Ath =

⎡⎢⎢⎣−16.02 −44.66 −39.19 −10.78

1 0 0 00 1 0 00 0 1 0

⎤⎥⎥⎦ Bth =

⎡⎢⎢⎣

1000

⎤⎥⎥⎦

Similarly, the values of the state matrix Ag and the input vector Bg for the gasgeneration unit can be obtained as

Ag =

⎡⎢⎢⎢⎢⎣

−30.43 −240.70 −657.66 −1132.37 −1124.761 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0

⎤⎥⎥⎥⎥⎦ Bg =

⎡⎢⎢⎢⎢⎣

10000

⎤⎥⎥⎥⎥⎦

where the system output �fg = [0 0 −15.65 1538.99 2608.22]×[xg1 xg

2 xg3 xg

4 xg5]

T ,xg

i (i = 1, 2, 3, 4, 5) is the state variables.Due to the AGC problem in Fig. 1 with relatively small changes, we set σ = 2

and ρ = 0.3, which is enough to resist the perturbation up to 30% according to(10), far from the small changes. Utilizing the modified GA with the constraintAth−BthCT

th < 0, and adopting the two-point crossover method and the uniformmutation function, we can get the optimized CT

th during the continuous four ex-periments shown in Table 1 and average each element of CT

th as the parametersof the integral sliding mode controller of the reheated thermal generation unit.Similarly, Table 2 shows the optimized CT

g during the continuous four experi-ments by the modified GA with the constraint Ag −BgC

Tg < 0 and average each

element of CTg as the parameters of the integral sliding mode controller of the

gas generation unit. As shown in both the tables, the controller parameters areable to converge to similar results with the modified GA.

Employing the searched CTth & CT

g , the simulation results in Fig. 2 showthe frequency for 1% step load disturbance at the operating load 1750MW ,where kth = 1 and Kg = 0 in Fig. 2(a) and kth = 71.43% and Kg = 28.57% in

266 D. Qian et al.

Table 1. CTth Table 2. CT

g

Jth cth1 cth

2 cth3 cth

4 Jg cg1 cg

2 cg3 cg

4 cg5

5.4754 0.4627 48.416344.779019.5862 7.0887 0.1632 48.347243.63402.4417 7.5909

5.0243 0.1776 48.928049.327524.2908 7.0421 0.2724 49.595342.79752.3488 3.7315

5.0053 0.1461 49.296246.567117.3229 7.0770 0.1971 47.886842.27312.0943 6.3027

5.5255 0.5245 49.685946.927920.1061 7.0994 0.1252 48.036242.82880.6516 4.4351

Average0.3277 49.081646.900420.3265 Average0.1895 48.466442.88341.8841 5.5151

Fig. 2(b). To verify the robustness of the GA-Based integral sliding mode controlmethod for the AGC problem of the single area with multi-sources, Fig.3 showsthe frequency for 1% step load disturbance at the operating load 1600MW withthe same AGC controller parameters, where kth = 1 and Kg = 0 in Fig. 3(a)and kth = 71.43% and Kg = 28.57% in Fig. 3(b). Compared with the GA-basedPI controllers, it is obvious that the ISMC method can decrease the overshootin Fig. 2 and Fig. 3.

0 5 10 15 20−0.005

0

0.005

0.01

0.015

0.02

0.025

time ( s) (a)

Δ f

0 5 10 15 20−0.005

0

0.005

0.01

0.015

0.02

0.025

time ( s) (b)

Δ f

GA−based PI Kth

=1

GA−based ISMC Kg=0

GA−based PI Kth

=71.43%

GA−based ISMC Kg=28.57%

Fig. 2. Frequency deviation of the single area at the setting load point 1750MW

0 5 10 15 20−0.005

0

0.005

0.01

0.015

0.02

0.025

time ( s)(a)

Δ f

0 5 10 15 20−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

time ( s)(b)

Δ f

GA−based PI Kth

=1

GA−based IMSC Kg=0

GA−based PI Kth

=71.43%

Ga−based ISMC Kg=28.57%

Fig. 3. Frequency deviation of the single area at the checking load point 1600MW

5 Conclusions

This paper designs an integral sliding mode control approach for the AGC prob-lem of a single area power system with multiple-source generation unit. The

GA-Based ISMC for AGC 267

stability analysis of the integral sliding surface is proven as well. For searchingtwo groups of the sliding surface parameters of both the AGC controllers, GAis employed. Simulation results show the controller’s feasibility.

Acknowledgements

Thisworkwas supportedby theNSFCProjects (No.60904008, 60974051), theFun-damental ResearchFunds for the Central Universities (No. 09MG19, 09QG29), theNational 863 Program (No. 2007AA05Z445).

References

1. Saadat, H.: Power system analysis. McGraw-Hill, New York (1999)2. Kundur, P.: Power system stability and control. McGraw-Hill, New York (1994)3. Mariano, S., Pombo, J., Calado, M., Ferreira, L.: Optimal output control: Load

frequency control of a large power system. In: Proceeding of International Conferenceon Power Engineering, Energy and Electrical Drives, vol. 2, pp. 369–374 (2009)

4. Al-Hamouz, Z.M., Al-Duwaish, H.N.: A new load frequency variable structurecontroller using genetic algorithms. Electric Power Systems Research 55, 1–6 (2000)

5. Shoults, R.R., Jativa Ibarra, J.A.: Multi-area adaptive LFC developed for acomprehensive AGC simulator. IEEE Transaction on Power Systems 8, 541–547(1993)

6. Yu, X.F., Tomsovic, K.: Application of linear matrix inequalities for load frequencycontrol with communication delays. IEEE Transactions on Power Systems 19,1508–1515 (2004)

7. Cam, E.: Application of fuzzy logic for load frequency control of hydroelectricalpower plants. Energy Conversion and Management 48, 1281–1288 (2007)

8. Ramakrishna, K.S.S.: Automatic generation control of single area power systemwith multi-source power generation. Proceedings of the Institution of MechanicalEngineers, Part A: Journal of Power and Energy 222, 1–11 (2008)

9. Utkin, V., Shi, J.X.: Integral sliding mode in systems operating under uncertaintyconditions. In: Proceedings of the 35th IEEE conference on Decision and Control,vol. 4, pp. 4591–4596 (1996)