Abstract: The Automatic Generation Control (AGC) of interconnected power systems has been considered as one of the most challenging problems for about the last four decades. Following the pioneering research of Elgerd and Fosha in 1970, a heap of research are appeared from time to time on optimal AGC of power systems considering various structural as well as technical aspects of power systems. This study is an attempt to propose a design of optimal AGC gains of a multi-area power system. The design of optimal AGC gains is based on Genetic Algorithm (GA) and Simulated Annealing (SA). It is a synergetic combination of GA and SA through a penalty function. The penalty function is derived based on the transient response specifications of dynamic response of the interconnected power system. A multi-area interconnected power system model is considered for the investigations. The power system model consists of three identical control areas consisting of reheat thermal plants. The investigations have been carried out with the designed Automatic Generation Controllers (AGC) based on GA-SA considering 2% load perturbation in one of the areas. The system dynamic responses obtained for various system states are compared with those achieved using optimal controllers derived using Linear Quadratic Regulator (LQR) theory. The simulation results show that the proposed GA-SA based optimal load frequency controllers are superior in all respects in comparison to the optimal AGC based on LQR theory.

INTRODUCTION

Due to numerous technical and economical reasons, the interconnections between individual utilities and areas as well have increased. The size and capacity of the power systems have been enlarged and with the strengthening of interconnections, abnormal phenomena have been frequent observed such as excessively large tie-line power deviations and/or sustained power oscillations under sudden system load changes. This fact suggests the necessity of more advanced control strategies to be incorporated for better control. Therefore, the efforts are always on to make the operation and control of the interconnected power systems more reliable, economic and cost effective. Intelligent systems based on computers have been employed for on-line monitoring and control of modern large-scale power systems in generation, transmission and distribution, thereby overcoming the complexities and drawbacks of the conventional control schemes. The design of system controllers based on the classical control theory in general do not yields optimal system performance where as the controller design using optimal control theory and its optimization makes the system control more meaningful in the sense of its optimality (Elgerd, 1982; Abdennour, 2002; Kothari et al., 1989; Huddar and Kulkarni, 2009; Ibraheem and Kothari, 2005). The optimal AGC regulators based on Linear Quadratic Regulator (LQR) design technique have been found to offer very promising results in comparison to those AGC regulators designed using classical control schemes. However, optimal AGC regulators are considered to be impractical due to the reasons (Mathur and Ghosh, 2006). Still they have been preferred for the applications against the AGC regulators based on classical control concepts.

The recent advancement in optimal control theory and availability of high speed digital computers coupled with enormous capability of handling large amount of data motivated the power system engineers/researchers to devise advanced AGC strategies. It is the fact that every physical system is pervasively imprecise, uncertain and hard to be categorical about. It is also a fact that precision and certainty carry a cost. To deal such systems effectively and most efficiently, a flexible approach called evolutionary algorithms has emerged on research scenario and has been very widely in use over the last two decades. One of the newer and relatively simple approaches used for optimization is GA. Besides other merits, it has a very important quality of being derivative free optimization technique. Therefore, it does not require a detailed model of the system to be optimized. It becomes desirable to consider the optimal AGC regulator design based on linear full state feedback control concept and GAs. The power engineers and researchers have considered the GA as an intelligent tool to design optimal AGC regulators due to its inherent features to get global minima for an optimization problem (Abdel-Magid and Dawoud, 1996; Aditya and Das, 2003; Daneshfar and Bevrani, 2009; Chidambaram and Paramasivam, 2009). The limitations of the GA-based method, however are that it is difficult to obtain a high quality solution because the GAs lack in fine local tuning capabilities and it also takes a large computation cost (Yip and Pao, 1995). In this study, researchers present a hybrid approach to overcome the limitations and also propose to hybridize the Simulated Annealing (SA) which is based on a local searching algorithm to compensate the fine local tuning capabilities of GA. The hybrid GA-SA technique yields more optimal gain values than the GA method (Ghoshal, 2004). The optimal automatic generation controller designs based on the approach reported for power system applications have been reported in the literature. In the present research, the hybrid form of genetic algorithm and simulated annealing techniques is proposed for better control in the multi-area power system. These techniques included system frequency and power flow deviations as a variable parameter for the new controller. The GA-SA based controller gives good transient response, insensitivity to parameter variations and external disturbances.

The effect of parameter variations on system dynamic performance has been tested comprehensively. The significant advantages of the proposed method over the optimality (LQR) theory include the following direct representation of constraints on transient response specifications treatment of constraints as hard constraintsability to handle non-linear, time varying systems also. The controller is tested on a multi-area i.e., three equal area thermal power system with reheats turbines with AC tie-lines for this test case. A comparison of test results with those from LQR method shows that the proposed controller performs better than the LQR theory based controller. The LFC design based on the LQR based strategy does not offer good system dynamic performance in the wake of 2% load perturbations in the one of the area interconnected power systems.

MATERIALS AND METHODS

System investigated: The power system transfer function model is shown in Fig. 1 and the state equations of the power system are written in time domain as:

(1)

(2)

The structure of vectors X, U and may be developed as follows: The structure of the state, control and disturbance vectors are shown as:

Fig. 1:	Transfer function model of reheat type multi-area thermal power system

System matrices: The system matrices, A, B and Γ described by Eq. 1 and 2 can be obtained with the structures of state, control and disturbance vectors and the transfer function block diagram representation of Fig. 1. The selection of power system configuration is done based on the development trend of electricity power scenario in India. It is assumed that for economic, technical and other reasons the power systems will continue to operate in interconnected fashion. However, the future structure of power systems may have larger geographical boundaries all over the country.

The corresponding coefficient matrices are derived using the numerical values of system variables as given in Appendix. System model and these matrices are obtained as follows:

The state cost weighting matrix Q and control cost weighting matrix R have been selected as an identity matrix of compatible dimensions, respectively.

The hybrid GA-SA technique: The hybrid GA-SA based controller is designed by Hybrid Stochastic Search (HSS) Technique (Deb, 2005). The HSS heuristic incorporates Simulated Annealing (SA) in the selection process of Evolutionary Programming (EP) (Das and Patvardhan, 2002).

Table 1:	Population structure of HSS

The solution string comprises of all feedback gains and is encoded as a string of real numbers. The population structure of HSS is shown in Table 1.

The HSS heuristic employs blend crossover and a mutation operator suitable for real number representation to provide it a better search capability. The objective is to minimize the Area Control Error (ACE) augmented with penalty terms corresponding to transient response specifications in the controller output, frequency and power flows in the transmission lines. The heuristic is quite general and various aspects like nonlinear, discontinuous functions and constraints are easily incorporated as per requirement. The heuristic can be easily understood from the following pseudo-code:

For each family, accept the best child as the parent for next generation if:

Where:

Where:

where, C is the children generated by all the families.

A briefly detailed explanation of each step is as follows: this study presents the implementation details for HGASA application. The solution string comprises of all feedback gains. A real string representation is chosen. Population size of 100 is chosen as a compromise between exhaustive search and computational burden. Following are the implementation details of the algorithm:

Crossover: For each parent i, mates are selected from the other parents at random and crossover is applied to generate m (i) children. To start with, m (i) is fixed same for all the parents. This number is changed as the search progresses as explained in the subsequent steps. The Blend Crossover operator (BLX-α) based on the theory of interval schemata is employed in this study. BLX-α operates by randomly selecting a point in the range (p1-α (p2-p1), p2 + α (p2-p1)) where p1 and p2 are the two parent points and p1<p2. In the test problem, BLX-0.5 performed better than the BLX operators with any other α value and has therefore been used.

Fitness evaluation: The objective here is to minimize the deviation in the frequency of the three areas and the deviation in the transmission line power flows. The fitness function is taken as the summation of the absolute values of the three at every discrete time instant in the simulation. An optional penalty term is added to take care of the transient response specifications viz. transient response specifications on system frequency, power flows, settling time, over shoots, etc.

Selection: The best child (the child with minimum objective value) out of the children generated from the same parent is found. The best child then competes with its parent to survive in the next generation. If the best child is better than it's parent, it is accepted as a parent in the next generation. If the best child is worse than it's parent then there is the Boltzmann criterion that the child be accepted as described.

The selection process is inspired from the SA mechanism and allows a bad move to be selected in the optimization process with a probability. This is in contrast to the conventional evolutionary algorithms where only good moves are accepted throughout the optimization process which may cause the algorithm to get stuck in local minima.

Boltzmann criterion: As in SA, the selection of temperatures is such that initially the probability of acceptance of a bad move i.e., accepting the best child as the parent for next generation when it is worse than the current parent is high (approximately 1) but as the temperatures are successively lowered this probability is decreased till at the end, the probability of accepting a bad move is negligible (approximately 0). Such a strategy enables the technique to seek the global optimum without getting stuck in any local optimum (Das and Patvardhan, 2002).

The initial and final temperatures are calculated as follows: A bad move is accepted according to the Boltzmann Criterion. Initially, the probability of accepting a bad move is approximately one, i.e.,

and finally,

therefore,

Where:

Cooling schedule: Temperature is cooled down with the iterations as under:

Where:

where superscript k represents the iteration count.

Acceptance number: The number of children generated in the next generation is proportional to a parameter called the acceptance number. This number provides a measure of the goodness of the solutions in the vicinity of the current parent. The number is computed by sampling the search space around the current parent and counting the number of good samples out of the total samples as per step 7-10 of the pseudo-code. This strategy enables the algorithm to focus search on the better regions of the search space.

RESULTS AND DISCUSSION

Computational exeperience: The investigation has been performed on MATLAB 7.4 version computing system. The case study-wise optimal feedback gain matrices is computed and is shown in Table 2. The response plots for various state variables are obtained for 2% load disturbance in area₁. These response plots are shown in Fig. 2-10.

The results obtained have been compared with those from LQR theory an optimal control algorithm. The test case assumes 2% load disturbance in area₁. Figure 2-7 show the dynamic response of frequency deviations (ΔF₁-ΔF₃) changes in the power generations (ΔPg₁-ΔPg₃) and Fig. 8-10 presents area control errors deviations (∫ACE₁-∫ACE₃) with optimal controller and the proposed optimal state feedback controller (GA-SA). Response of LQR is sluggish and associated with large number of oscillations, large settling time and large magnitudes of overshoots when load disturbance is 2% is considered in area₁. The settling time of Hybrid GA-SA algorithm is approximately <6 sec. However, improvement with the system is about 50%.

Comparison of this method with LQR theory shows that inspite of its wider applicability the search power of the heuristic is immense and it succeeds in finding better solutions than the existing method for any case. The comparison shows that the proposed controller gives better performance than the conventional optimal control.

Fig. 2:	Response of Δ F1 for 2% load perturbation in area1

Fig. 3:	Response of Δ F2 for 2% load perturbation in area1

Fig. 4:	Response of Δ F3 for 2% load perturbation in area1

Table 2:	Optimal State Feedback gains for LQR and HGASA based Regulators

Fig. 5:	Response of Δ Pg1 for 2% load perturbation in area1

Fig. 6:	Response of Δ Pg2 for 2% load perturbation in area1

Fig. 7:	Response of Δ Pg3 for 2% load perturbation in area1

Fig. 8:	Response of ∫ACE1 for 2% load perturbation in area1

For the case, the parameters of HSS are kept same and are given:

Number of parents (N) = 10
Number of children (NC) = 100
Maximum number of iterations (MAXIT) = 50
Probability of crossover = 0.6
Probability of mutation = 0.01

Fig. 9:	Response of ∫ACE2 for 2% load perturbation in area1

Fig. 10:	Response of ∫ACE3 for 2% load perturbation in area1

CONCLUSION

A new Hybrid GA-SA approach has been proposed for optimizing the state feedback gains for AGC in this study. A penalty function based technique is used for handling the constraints imposed on the transient response of the system. Test results presented on a multi-area power system with an A.C. link and are compared with those obtained from an optimal controller based on Linear Quadratic Regulator theory. Test results describe the efficacy of the proposed approach in optimizing the state variable feedback gains to give a fairly better controller performance.

NOMENCLATURE

ΔFi	=	Incremental change in frequency
i	=	Subscript referring to area (i = 1, 2 and 3)
ΔPgi	=	Incremental change in generator power output
ΔPdi	=	Incremental change in load demand (p.u.MW/Hz)
Δxgi	=	Incremental change in governor valve position
Tpi	=	Electric system time constants
Ri	=	Speed regulation parameter, Hz/p.u.MW
Ki	=	Integral gain constant
ΔPtie I	=	Incremental change in tie-line power (MW)
Tgi	=	Speed governor time constant of area ‘i’, s
Kri, Tri	=	Reheat coefficient’s and reheat time’s
Bi	=	Frequency bias constant (p.u.MW/Hz)
ACEi	=	Area control error’s
Tti	=	Turbine time constants
A, B, C, Γ	=	System matrices of a dynamic model in state space form
Dpci	=	Incremental change in speed changer position
Kgi	=	Speed governor gain
Kti	=	Reheat thermal turbine gain constant
T12, T13	=	Synchronizing coefficient of ac tie-link of area (i)
	=	State, control, output and disturbance vector of a model in state space form

APPENDIX

Nominal parameters of the system investigated: B₁ = B₂ = B₃ = 0.425, R₁ = R₂ = R3 = 2.4 Hz/ p.u. MW,T_g1 = T_g2 = T_g3 = 0.08 sec, K_r1 = K_r2 = K_{r3 =} 0.5 sec, T_r1 = T_{r2 =} T_r3= 10 sec, T_t1 = T_t2 = T_t3= 0.3 sec, K_p1 = K_p2 = K_p3 = 120 Hz/ p.u. MW, T_p1 = T_p2 = T_p3 = 20 sec, a₁₂ = -1, ΔPd₁ = 0.02 p.u.MW, Nominal frequency = 50Hz, 2πT₁₂ = 0.545 p.u.MW .