Comparative performance analysis of hybrid differential evolution and pattern search technique for frequency control of the electric power system

In the current situation, operation and control of power system is a greater challenge. The most significant situation in power system control is load frequency control. In the present work, a hybrid differential evolution and pattern search (hDE-PS) method has been suggested for frequency regulation of electrical power systems. Fractional-order proportional integral derivative (FOPID) controller is implemented for design and analysis purpose. The suggested control method has been applied for two electrical power systems model, i.e., 2-area diverse source power system with/without HVDC linkage and 2-area thermal system. The performances of the suggested controller have been evaluated with PID and optimal controller. The simulation results indicate that system performances are enhanced with the suggested approach for identical structure. Robustness of the suggested approach has been analyzed by variation in random load and the system parameters. The suggested method (hDE-PS tuned FOPID) is further investigated with a 2-area thermal system. The performance of the recommended approach is analyzed by equating the results with other newly available approaches, like Genetic Algorithm (GA), Bacteria Foraging Optimization Algorithm (BFOA), Particle Swarm Optimization (PSO), hybrid BFOA and PSO (hBFOA-PSO), multi-objective Non-dominated Sorting Genetic Algorithm (NSGA)-II and Firefly Algorithm for the similar structure.

can degrade a system's performance and even cause system instability. Thus to acquire precise insight of the LFC problem, it is essential to hold the vital physical constraints in the system model. This study presents both the usage of a powerful computational intelligence technique like DE and hybrid DE-PS in order to optimize the optimal/PID/ FOPID controller parameters of a LFC system. The advantage of hybrid DE-PS is verified by equating the results with DE-optimized FOPID controller, further, to exhibit the capability of the suggested hDE-PS-optimized FOPID controller to cope with random load disturbances.
The primary contribution this paper comprises: a. To propose a hybrid DE-PS technique-based FOPID/PID/optimal output feedback controller for frequency regulation with/without HVDC link for 2-area 6-unit power system. b. The advantage of the suggested hDE-PS-optimized FOPID is presented by equating the results over conventional PID and optimal controller for the identical structure. c. Robustness of the suggested method is examined by taking into account the change in parameters and random step load patterns. d. To exhibit the advantage of hDE-PS-tuned FOID over recently existing technique, i.e., GA [7], BFOA [7], DE [8], hBFOA-PSO [9], NSGA-II [10], FA [11] optimized various controller like PI &PID for an identical test system.

System design
In the current paper, two electrical power system models are engaged to examine the ability of the suggested controller for LFC, which are commonly used in the literature.
The test system-I presented in Fig. 1 is 2-area 6-unit system with/without HVDC link [24][25][26][27]. The system comprises a source like thermal hydro-gas. The test system parameters are taken from reference [27]. The readers are advised to refer [27] for the definition and meaning of symbols used in Fig. 1. The generation power rating of every area is 2000 MW, and nominal loading is 1000 MW. The load involvement of thermal, hydro and gas systems is 600 MW, 250 MW and 150 MW, respectively. For more practical power system, time delay (TD) element is incorporated in the test model. In the present paper, the value of TD is taken as 50 ms [39]. In test system-II, a non-reheat-type thermal system [7][8][9][10][11] is considered as presented in Fig. 2. The detailed data of studied systems are available in [7][8][9][10][11].

Modeling of HVDC linkage
HVDC link (parallel AC-DC) is connected directly with the AC tie-line interconnected power system for improvement of system performance. The structure of two-area arrangement through AC-DC links is presented in Fig. 3 [24,25,27]. The modification of output in area-1 of AC tie and HVDC link is as follows: where K DC (HVDC link gain) and T DC (HVDC link-time constant).

Optimal controller design
The first step in the development of the design procedure of the controller is the linear representation of the structure. The linear form of the structure is designated in the state space form, as follows: where A (n*n) is state matrix, B (n*m) is a control matrix for n number of state variables and m number of inputs, and Ŵ is a disturbance matrix. Different variables have been defined as: State variables: Control inputs:  Fig. 1 can be expressed from which the input matrices A are found to be of the order 26 × 26, the matrix B is of the order of 26 × 2, and the matrix Ŵ is of the order of 26 × 2. The output is given by (4).
For matrix 'D' is considered as zero. Therefore, the output is represented as where 'C' matrix is the order of (2 × 26) describe the output matrix.
The values of matrices can be calculated with the help of [27]. The 26 states are x 1 , Hence, finally, the equation for control input can be written as: where 'K' is a (2 × 26) matrix known as feedback matrix gain and is represented by:  25 . The deviations of control inputs (u 1 and u 2 ) about the steady-state values are minimized. Based on the realistic control specifications requisite of LFC scheme, it is perceived from literature that the best system performance is acquired with minimum values of settling times, peak overshoots, and maximum value of damping ratio in frequency and tie-line power deviations when ITAE is employed as objective function [22]. Therefore, ITAE is elected as a cost function in the current study to determine the parameters of the controller and given by: For optimal control problem, the objective function is written as:

State-space model of Test system-II
The state-space representation of input, control and disturbance vectors for the system under study is: State vector: Control vector: Disturbance vector: The matrices of state space are found by using the equations of state space. The detailed equations of state space are given in reference [27].

FOPID controller
Conventional PID controllers may not offered required system performance if it is connected with nonlinearity parameters. Extensive development has been noticed in the growth of intelligent controllers applied to different power systems, but still, it remains a demanding area and a general problem for researchers. The fuzzy logicbased controller needs more fuzzy variables for better accuracy. This will exponentially increase the rues. The PID controller has been effectively used in many applications. The acceptance of the PID is due to the ease of the design processes and acceptable performance. The fractional-order controller design methods are in principle founded on additions of the traditional PID control theory, with an importance on the greater flexibility in the tuning approach ensuing improved control performances as related to classical control. Fractional calculus has become very beneficial in recent times because of its applications in many applied sciences. Persuaded from the positive results of these developments, a FOPID structure is suggested for LFC of power system. FOPID controller has been suggested in the current paper which includes a fractionalorder and PID configuration. Conventional PID controllers are normally not effectual as of their linear arrangement, mainly, if higher-order plants are concerned or if time delay systems and uncertainties are there. On the contrary, the FOPID can handle nonlinearity and uncertainties. The FOPID can be intended to match the necessary performance of the control system. From the literature, it is seen that application of FOPID enhanced the performance of PID/PI. The proposed FOPID controller gets advantages of outstanding ability of a PID in addition to the feedback control mechanism in removing the steady-state error in addition to predicting and controlling future error.
The main advantages of FOPID are that if the parameter of a power system varies, a fractional-order (FO) PID is less responsive over a conventional PID [33,37]. Additionally, the FO has two additional variables to optimize. Its offers further degrees of freedom to the dynamic properties of FO structure. The FOPID configuration assumed in each generating element is shown in Fig. 4. The inputs to the controllers are the respective ACEs, and outputs of the controllers are the reference power setting of generating units P c . In Fig. 4, K P , K I, K D, λ and µ are to be optimized. The expression of FOPID is given by Eq. (15).

Outline of hybrid Differential Evolution and Pattern Search (hDE-PS) method
Finding a global optimal solution is a difficult assignment in several applications due to the fact that data and models are generally nonlinear and subject to diverse sources of error. By the way, hybrid optimization algorithms have achieved attractiveness as it has become apparent that there cannot be a universal optimization strategy which is globally more beneficial than any other. In the current work, an effort has been prepared to apply a hybrid DE method and PS technique to tune the controller parameters.
In this method, DE is utilized for overall exploration & the Pattern Search technique [38] engaged for local search. The initial stage is explorative, using a conventional DE to recognize capable regions of the explore space. The superb result set up by DE is subsequently refined with PS technique through a succeeding exploitative stage. To set up the advantage of suggested hDE-PS method, the results are evaluated by the individual DE method.
Differential Evolution (DE) technique is a straightforward, capable, but effective technique and applied to numerous design problems [40]. It gives remarkable performance for dynamic, multi-objective, constraint problems. Four main steps of differential evaluation are, namely initialization, mutation, crossover and selection. Boot starts with creating an initial population vector of N P . The initial population is proposed to give rise to consecutive generations. Selection of the initial population is made initial arbitrarily. For each generation entities of the existing population are called target vector. Beginning step toward generating new solutions is called a mutation. The crossover of the population is done, which is resultant from mutation and the original population, where a recent vector known as the anticipated vector is created. C R as the crossover factor is a constant value between 0 and 1. If the normal vector attains better fitness rates that of target vector, it substitutes the target vector in the successive generation. The recent population is substituted by the new population, and a new loop will be generated. Some fundamental issues need to be determined for the implementation of DE. They are initialization, DE scaling factor (f), crossover probability (C R ) and population size (N P ). The range of scaling factor is (0, 2). Amount of perturbation is controlled by this range in the process of mutation. Crossover probability (C R ) is usually selected between the interval (0, 2). The DE technique is described in [8] in a detailed manner.
The PS method is an easy concept, simple to realize and computationally competent [22,38]. The PS method calculates a series of spots that could or could not come up

2-Area diverse source system (Test system-I)
Simulation of test system-I, which is a 2-area multi-source system with/without HVDC link exhibited in Fig. 1 [24][25][26][27], is done using FOPID/PID/Optimal output feedback controller. Each area includes hydro, gas and thermal generation units. The test system is performed with a sudden rise in a load of 1% in area-1 (�P D1 = 0.01 p.u.MW) at t = 0 s in the first area, and optimized controller parameters are obtained with the DE/hDE-PS techniques. Tables 1 and 2 provide optimal gains and PID/FOPID controller for (1) AC link and (2) parallel AC/DC links, respectively. For comparison, the results of other controllers like optimal and PID controller are also considered. For identical power structure, the lowest error value is acquired by DE optimized FOPID (ITAE = 0.8723) equated to PID (ITAE = 2.7718) and optimal (ITAE = 9.3341) controller which is observable from Table 3. It is noticed that the ITAE value with FOPID is reduced by 90.65% and 68.53%  Table 3. This exhibits the advantage of the suggested FOPID over PID and optimal controller. It is also clear from Table 3  It is useful to state at this point that in the above assessment, same model, controller (FOPID) and ITAE objective function are considered. It can be concluded suggested hDE-PS approach offers improved result equated to DE method as minimum error value is attained. Least settling times, O S and U S in ΔF and ΔP Tie , are also obtained with a suggested approach compared to others.
The transient response of systems is presented in Fig. 5a-c for P D1 = 0.01 p.u.MW at t = 0 s. For evaluation, the results with suggested approach (hDE-PS: FOPID controller), DE-optimized FOPID, PID and optimal controllers are also presented in Fig. 5a-c. Since the diverse generations are associated with a rigid network in each area, the frequency variations in an area remain the same in that area. Considerable enhancement is observed through the suggested approach over to other approaches.
To exhibit the efficacy of the suggested approach (hDE-PS: FOPID), simultaneously load disturbances are considered (�P D1 = 0.01 p.u.MW &�P D2 = 0.02p.u.MW ) . In this case, a sudden rise in a load of 1% is applied in area-1 & 2% in area-2. The system  responses for the suggested approach and other methods are displayed in Fig. 5d-f. It is noticeable from Fig. 5d-f that the recommended controller illustrates significantly improved performance than the other approaches. Hence, the designed hDE-PS: FOPID controllers are robust and it acts reasonably regardless of the position of disturbance.

Extension to 2-area diverse source system through parallel AC/DC linkages
To exhibit the capability of the suggested technique the work is additionally comprehensive to a 2-area multi-source system through parallel AC/DC links which is presented in Fig. 3 [24,25,27]. The representation of the test system is displayed in Fig. 1. The test system is carried out with a sudden rise in a load of 1% used in area-1 (�P D1 = 0.01p.u.MW ) at t = 0 s, and optimized controller parameters with HVDC link utilizing ITAE objective function are available in Table 2. The system performance solutions are given in Table 3. It can be observed from Table 3    Further, it is noticeable from Table 3 Table 3. This shows the supremacy of the HVDC link. The transient response of the suggested approach (hDE-PS: FOPID) is shown in Fig. 6a-c) with HVDC link(�P D1 = 0.01p.u.MW ) . For a better illustration of results, the transient response obtained by DE-based optimal/PID/FOPID controller is offered in Fig. 6a-c. It is seen from Fig. 6a-c that the result of the suggested method is better in terms of T S , O S and U S than other methods. To show the strength of the suggested method, simultaneously load disturbances (�P D1 = 0.01p.u.MW &�P D2 = 0.02p.u.MW ) are considered. The system responses are exposed in Fig. 6d-f from which it is obvious that the FOPID controller is strong and execute satisfactorily for different location and size of the disturbance variation. The dynamics of all the energy sources in the areas are presented in Fig. 7a-c. The contribution of each generation source following a load disturbance is shown in Fig. 7. It is worthwhile to mention here that as per the requirement of LFC scheme, under normal operating conditions, each area should carry its own load and the power exchange between control areas following a load perturbation should be maintained at its prescheduled value as quickly as possible. It is clear from Fig. 7 that when a load disturbance is applied in area -1, all the plants in area-1 increase their generation to meet the load demand. To minimize the frequency deviations following a load disturbance in area-1, all the plants in area-2 supply the power during the transient phases only.

Sensitivity study
The robustness of the recommended method (hDE-PS: FOPID; with HVDC link) is tested with a variation of ± 25% in system parameters [8][9][10][11]. The tuned parameters under different circumstances the recommended method (hDE-PS: FOPID; with HVDC link) are shown in Tables 4 and 5. It can be verified from Table 6 that settling time(T S ), peak overshoot (Os)/ undershoot(Us) and ITAE values differ within suitable ranges and are close to the values with nominal values. For example, the change in the frequency of area-1 with a deviation of loading is publicized in Fig. 8. From Fig. 8 and Table 6, it can be established that suggested method (hDE-PS: FOPID; with HVDC link) is robust under varied in system parameters. Hence, it proved the strength of the recommended approach.
To examine the advantage of the suggested method (hDE-PS: FOPID; with HVDC link) a random pattern load is considered to area-1. Figure 9a displays the random pattern load [19]. The size and period of the step load are arbitrary. The response for random load pattern with proposed approach is exposed in Fig. 9b-d. The suggested hybrid DE-PS-optimized FOPID controller shows better transient responses than DE-optimized FOPID controller which can be noticed from Fig. 9b-d. The comparative study validates that the suggested hDE-PS-tuned FOPID controller improves system performs Table 4 Tuned parameters with loading, T G and T T conditions Controller parameters adequately. The eigenvalues and damping ratio of the proposed approach (hDE-PS: FOPID; with HVDC link) is shown in Table 7. It is well known in control system that as long as the real part of complex eigenvalue is negative, then the system is stable. It is marked from Table 7 that all the eigenvalues lie in the left half of s-plane for the proposed approach, so preserve the stability.

Comparison with recent AGC approaches
The performance of the suggested AGC method also examined in an extensively utilized a non-reheat-type thermal system [7][8][9][10][11] is considered as presented in Fig. 2. Two      The performance of the suggested method (hDE-PS: FOPID) is equated by conventional as well as some new optimization method such as conventional ZN: PI [7], GA: PI [7], BFOA: PI [7, PSO: PI [7], hBFOA-PSO: PI [9], NSGA-II: PI [10] NSGA-II: PIDF [10], DE: PI [8], FA: PI [11] and FA: PID [11]. The ITAE values obtained with each approach are presented in Table 8. It is seen from Table 8 that the lowest error value (ITAE = 0.2527) is acquired by the suggested method (hDE-PS: FOPID) as compared to newly suggested AGC methods. The transient responses of the system are publicized in Fig. 10a-c from which it is apparent that the suggested method outperforms newly suggested automatic generation control methods.

Conclusions
In this manuscript, hDE-PS technique-based FOPID/PID/optimal controller has been suggested for control of frequency in electrical power system with/without HVDC link. Performance of the suggested approach is tested on two electrical power system models. Initially, a 2-area system with varied sources of generations like hydro, gas and thermal via parallel AC/DC link is considered. To make the system more sensible, time delay has been integrated into the model. The gains of FOPID are tuned using a hDE-PS method. The advantage of the suggested hDE-PS method over the DE technique has been verified. To confirm the supremacy of the suggested method (hDE-PS: FOPID), results are equated with DE-based optimal/PID/FOPID controller for the equal test system. It is recognized that the suggested method (hDE-PS: FOPID) provides better performance than others. Simulation results also show that with DC-link further progress the system performance with the suggested method. Further, sensitivity investigation is executed by changing the parameters and load conditions from their nominal values to show the strength of the suggested hybrid DE-PS algorithm-optimized FOPID. It was observed that the suggested method (hDE-PS: FOPID) is robust. Finally, the usefulness and robustness of the suggested scheme against random load variations were investigated. The performance of the suggested AGC method is also investigated in an extensively exercised two-area system. It is noticed that the proposed method is superior to various newly suggested approaches.