Rank-Sum-Weight Method Based Systematic Determination of Weights for Controller Tuning for Automatic Generation Control

The design and performance evaluation of a grey wolf optimizer (GWO) aided rank-sum-weight method based proportional-integral-derivative regulator with derivative filter for automatic generation control of two-area interconnected power systems are presented in this research. The derivative gain filter is used to lessen the impacts of noise in the input signal. Sub-objectives based on integral of time multiplied square error (ITSE) of frequency deviations, tie-line power deviation, and area-control errors (ACEs) are used to formulate the objective function for adjusting regulator settings. A single overall objective function is formed by combining these sub-objectives. ITSEs of two areas, ITSEs of tie-line power deviation, and ITSEs of ACEs of two areas comprise up the overall objective function. In the control design, the weights in the overall objective function are used to evaluate relative significance of each sub-objective. In contrast to previous techniques, where weights are either considered equal by ignoring the relative relevance of sub-objectives or selected randomly, the weights in this article are obtained using the rank-sum-weight method systematically. Using the GWO algorithm, the overall objective function is minimized. For six different circumstances including different load disturbances in interconnected areas, the effectiveness of the proposed GWO aided rank-sum-weight method based controller is examined. The performance of the GWO-tuned controller is also compared to those of other controllers tuned using the differential evolution, elephant herding optimization, Nelder-Mead simplex, membrane computing, and Luus-Jaakola algorithms. Time domain specifications are tabulated for each of the six circumstances. The findings are also plotted to demonstrate the frequency and tie line power fluctuations. A statistical analysis is also performed in order to assess the overall efficacy of the suggested controller.


I. INTRODUCTION
Whenever there is a change in either system frequency or tieline loading, or a combination of both, the regulation of output electrical power from electric generators in a specific region is defined as the automatic generation control (AGC). Electric power is exchanged across the areas within established boundaries in order to sustain scheduled system frequency. Because of the following factors, AGC [1], [2] is required: 1) For generating enough electricity to fulfil all load needs.
The associate editor coordinating the review of this manuscript and approving it for publication was Zhenzhou Tang .
Performance indices such as integral of absolute error (IAE) [28], [39], an integral of squared error (ISE) [32], [44], an integral of time multiplied absolute error (ITAE) [45], [46], an integral of time multiplied square error (ITSE) [45], [46] are used to formulate objective functions for tuning controller settings. Frequency deviations, tie-line power variations, and area control errors (ACEs) of interconnected areas are all examined using these performance indicators. The researchers in [27] examined ISE, in order to attain controller parameters for a two-area interconnected power system. According to the findings of this research, individual ISE of ACEs from two areas are combined to form the overall objective function. In [31], the use of ISE to get controller parameters for a three-area linked power system is examined. As per the results of this study, individual ISE of ACEs from the three areas under consideration are combined to produce the overall objective function. The researchers in [32] also procured ISE to perform controller parameter tuning. The findings from this study indicate that individual ACEs from both areas of the power system are used to obtain a overall objective function.
In [11], to achieve controller parameters of a three-area interconnected power system, ISEs of frequency deviation in area-1 and area-2, as well as tie-line power deviation, are evaluated. These ISEs of frequency variations, as well as tie-line power deviation in a three-area interconnected power system, are combined to produce the overall objective function. The researchers in [12] considered ITAE to acquire controller parameters for a five-area interconnected power system. For the purpose of formulating the objective function, the ACEs of a five-area interconnected system are taken into account. In [28], for the formulation of the objective function, more than one performance indices are considered. The objective function is formulated using a combination of ISE, ITAE, ITSE, integral of squared time multiplied absolute error (ISTAE) and integral of squared time multiplied squared error (ISTSE) of frequency deviations of area-1 and area-2, as well as tie-line power. These performance metrics, evaluated in the objective function, are either incorporated directly without considering relative relevance or are given equal weights in the literature [17], [27], [31], [32], [34], [42], [47], [48]. To construct a better objective function for controller realization, a systematic process should be used to determine relative weights of distinct performance criteria.
In this article, a PID regulator for AGC of a two-area linked power system is constructed utilising the grey wolf optimization algorithm assisted with a systematic technique namely rank-sum-weight method. ITAEs of deviation in frequencies of area-1 and area-2, ITAEs of deviation in tie-line power of interconnected areas, and ITAEs of ACEs of area-1 and area-2 are used to formulate the objective function. These performance indices (i.e. ITAEs) are sub-objective functions that together constitute the overall objective function. The weighted summation of all these five ITAEs form the overall objective function. Weights associated with multiple sub-objective functions are usually chosen arbitrarily or given equal value in literature [17], [48], ignoring their relative significance. The significant contributions of this work include: • The weights of the sub-objective functions examined in this study are derived systematically using ranksum-weight method [49]. The weight calculations in rank-sum-weight method are based on relative relevance among the sub-objectives evaluated.
• The grey wolf optimization (GWO) algorithm is used to tune the PID controller parameters to maintain frequency deviation, tie-line power deviation, and ACEs within preset limits by minimising the overall objective function.
• Furthermore, six distinct test circumstances are explored for evaluating the proposed GWO-PID controller's performance. The statistical analysis provided aids the effectiveness and efficacy of the suggested GWO-PID controller. The following is a breakdown of this contribution. Section II describes the power system architecture that has been examined. The controller structure is also provided in Section II. In Section III, the problem formulation is provided. Section IV explains the rank-sum-weight method and how it is used to AGC. In Section V, the grey wolf optimization algorithm is explained. In Section VI, the performance of the proposed controller is evaluated for six different test circumstances. Finally, in Section VII, the conclusion is stated, followed by future scope.

II. STUDIED POWER SYSTEM ARCHITECTURE
A power system is made up of a network of transmission lines that interconnect a large number of generators. Electrical power is delivered to consumers at rated voltage and frequency via these transmission lines. There are several energy sources for generating electrical power which include fossil fuels like natural gas, oil, and coal; and hydro, wind, solar, geothermal, nuclear, tidal, wave, biomass, etc. The energy gained from these sources is first transformed into mechanical energy. This mechanical energy is then converted to electrical energy with the help of generators. There are two types of power systems based on the number of generation and distribution networks: 1) Single-area power systems and 2) Multi-area power systems. The increased load demand in a single-area power system is fulfilled either by borrowing rotating kinetic energy from machines in the power system or increasing generation whereas the increased load demand in a multi-area power system is fulfilled by power interchanges across interconnected areas through tie-lines.

A. CONFIGURATION OF MODEL UNDER INVESTIGATION
The Fig. 1 depicts the two-area power system investigated in this work. The examined two-area power system is considered from Ali and Abd-Elazim [50] along with the system parameters. It is a veristic linked system made up of two non-reheat thermal power plants with a total capacity of 2000 MW and a nominal load of 1000 MW each. A network of two-area power system is depicted in the Fig. 1 where f a1 , and f a2 are variations in system frequencies, ACE a1 , and ACE a2 are area control errors, µ a1 , and µ a2 are control inputs, β a1 , and β a2 are frequency bias factors, R a1 , and R a2 are governor speed regulation constants, τ gr1 , and τ gr2 are governor time constants, τ te1 , and τ te2 are turbine time constants, τ a1 , and τ a2 are power system time constants, κ a1 , and κ a2 are power system gains, P gr1 , and P gr2 are power deviations of governors, P te1 , and P te2 are power deviations of non-reheat steam turbines, P la1 , and P la2 are change in load demands of power system in area-1, and area-2 respectively. P tl is the change in tie-line power of interconnected power system.

B. CONFIGURATION OF CONTROLLER
As it is simple to comprehend and implement, PID controller is the most widely used feedback control mechanism. The controller structure used in this research is depicted in Fig. 2.
κ pg , κ ig , and κ dg are three controller parameters in the controller structure which indicate proportional gain, integral gain, and derivative gain, respectively. These parameters must be tuned finely to fit the dynamics of the process under control. A filter η with derivative gain is used to reduce noise in the signal. The PID controller transfer function [50] with filter in derivative gain is given by The area control errors, ACE a1 and ACE a2 of area-1 and area-2, respectively, are the controllers' inputs. The area control errors (ACEs) of area-1 and area-2 [50] are calculated as follows:

III. PROBLEM FORMULATION
The aim of automatic generation control is to optimize the reliable and persistent power flow in a multi-area interconnected power system such that frequency fluctuations are reduced while the system's constraints are met. The formulation of an objective function is critical for finding the best solution to this problem. The tuning of a controller is based on well-defined objectives that meet the demands and limits of the system. The objective function as well as the various constraints are listed below.

A. OBJECTIVE FUNCTION FORMULATION
The effectiveness of an optimization is considerably influenced by the objective function selected. An integral error (IE), an integral of absolute error (IAE), an integral of squared error (ISE), an integral of time multiplied absolute error (ITAE) or an integral of time multiplied square error (ITSE) may be chosen for formulation of objective function. While tuning controller parameters, ITAE error minimization of frequency deviations, tie-line power deviation, and area control errors (ACEs) are among the design objectives covered in this article. The following are the various minimization objectives considered.
where, τ s denotes the overall simulation time. The first objective function, 1 studied is a combination of ITAE of deviation in system frequencies of area-1 and area-2. The second objective function studied is 2 , which defines the ITAE of power deviation in tie-line. The third objective function studied is 3 , which describes the ITAE of deviation in ACEs VOLUME 10, 2022 of area-1 and area-2. The total objective function, taking into account all three objectives, is as follows: where, the weights of the objectives 1 , 2 , and 3 are ε 1 , ε 2 , and ε 3 respectively. These weights show the relative significance of the sub-objectives in the overall objective. Most of the time, the relative importance of objectives is overlooked or viewed as a set of arbitrary values. The overall aim established utilising the sub-objectives is deemed to be of similar relative importance, as indicated in literature [17], [48]. The overall objective function in (7) is realised by combining (4), (5), and (6) and can be formulated as follows.

IV. RANK-SUM-WEIGHT METHOD AND ITS IMPLEMENTATION TO AGC
Rank-sum-weight method is a multi-attribute-decisionmaking methodology proposed by Stillwell et al. [49]. The rank-sum-weight method is used for weighing of attributes with respect to each other for decision making problem.
Firstly, attributes are identified. Then, the rank ordering of the attributes in the attribute set are decided. From this information, the weight [49] for the given attribute is determined by where, χ denotes the total number of attributes, ϑ r indicates the rank position of r th attribute and ε r gives the normalized weight of r th attribute. In this contribution, three sub-objectives formulated are considered as attributes (i.e. χ = 3). The attributes considered are 1 , 2 , and 3 as given in (4), (5) and (6), respectively. Once, the attributes are considered, the rank ordering of these sub-objectives is accomplished as given in Table 1. The ranking order of sub-objectives is considered as 1 , 2 and 3 in order. This ranking order of sub-objectives is considered because the effect of change in load will be very high on ITAE of deviations in frequencies of both the areas followed by ITAE of deviations in tie-line power and ITAE of ACEs of both the areas. Now, the normalized weight of each sub-objective is obtained from (13) as These weights depict the weighing factor of sub-objective in the overall objective function. Thus, the overall objective function (7) with 1 , 2 , and 3 and their weights ε 1 , ε 2 , and ε 3 respectively, turns out to be ( 1 , 2 , 3 ) = 0.5 1 + 0.3333 2 + 0.1666 3 (15) From (4), (5), and (6), the overall objective function obtained in (15) is rewritten as In order to obtain controller gains, the weighted objective function, given in (16), must be minimized. In this article, grey wolf optimizer is used to minimize (16), subject to constraints specified in (9), (10), (11), and (12).

V. GREY WOLF OPTIMIZATION ALGORITHM
In 2014, Mirjalili et al. [51] presented the grey wolf optimization (GWO) algorithm. It is based on the behaviour of grey wolves when they are exploring and hunting for prey. Grey wolves are members of the canidae family and are scientifically known as canis lupus. They are predators who prefer to live in a group namely pack. The social dominance order among these grey wolves is rigid. Grey wolves' behaviour is characterised by social hierarchy, encircling prey, hunting, exploitation, and exploration. In dependance on these behavioural patterns, grey wolves are divided into four categories: ρ, σ , ν and υ. The first level is ρ, and the wolf at this level is in charge of making judgments. The pack is bound to the judgments made by ρ. Wolves of this rank do not have to be the strongest members of the pack, but they are the greatest in terms of pack management. In this level of pack, the wolves demonstrate that group discipline and structure are more essential than collective power. The second level is σ , where the wolves aid the ρ level in decision-making and other endeavours. In the absence of a ρ level, the pack is officiated by the σ level. Wolves at the σ level promote ρ's choice all over the pack, even while providing feedback to ρ level on the judgment and its execution. Scouts, sentinels, hunters, and caretakers make up the third level of the ranking system (i.e. ν). Scouts are responsible for being vigilant when watching the territory's borders and informing the pack if there is a threat while sentinels are responsible for the pack's safety. During the search for prey, hunters are responsible for assisting ρ and σ levels. The pack's lowest ranking is the υ level. The wolves at this υ level are the last to devour food and are subject to submission by higher ranked wolves.
The new position of wolves [51] is calculated mathematically by aggregating the locations of ρ, σ and ν wolves in the search space.
where, Itr+1 represents new position of wolves in search space, ρ denotes new position of first level wolves (ρ), σ represents new position of second level wolves (σ ) and ν denotes new position of third level wolves (ν). The new positions of these three levels of wolves are calculated as where, ρ,Itr , σ,Itr and ν,Itr represent current positions of ρ, σ and ν level wolves respectively; ρ , σ , ν denote the encircling behaviour of different levels of wolves. The encircling behaviour is given as   where, represents position of prey in search space and ϒ, T are coefficient vectors which are calculated as where, the component ι declines linearly from 2 to 0 as the iterations increase, ϕ 1 and ϕ 2 are random vectors in range of [0, 1].

VI. RESULTS AND DISCUSSION
A two-area interconnected power system from Ali and Abd-Elazim [50] is examined in this research. In the MAT-LAB environment, all simulations are carried out. To convert a multi-objective illustration to a single-objective illustration, rank-sum-weight approach is employed. The Overall objective function is given in (16), whereas (9), (10), (11), and (12)  depict the controller's boundary conditions. Appendix A and Appendix B contain all of the parameters for both the areas and controller, respectively. For time-domain simulations, six different test scenarios with a variety of load perturbations are taken into account. The following is a list of the six test scenarios considered: • Test Scenario 1: At t = 0 s, a step load variation of +0.05 p.u. is examined in area-1, with no load change in area-2.
• Test Scenario 2: At t = 0 s, a step load variation of +0.05 p.u. is examined in area-2, with no load change in area-1.
• Test Scenario 3: At t = 0 s, a step load variation of +0.05 p.u. is examined in both the areas.   • Test Scenario 4: At t = 0 s, a step load variation of +0.050 p.u. is considered in area-1, whereas at t = 0 s, a −0.05 p.u. step load variation is considered in area-2.
• Test Scenario 5: At t = 0 s, a step load variation of +0.05 p.u. is considered in area-1, whereas at t = 0 s, a +0.10 p.u. step load variation is considered in area-2. • Test Scenario 6: At t = 0 s, a step load variation of +0.10 p.u. is considered in area-1, whereas at t = 0 s, a +0.05 p.u. step load variation is considered in area-2. For various loading conditions as discussed in Test Scenario 1 to Test Scenario 6, the results are provided in Tables 2-7. In these tables, the performance of GWO-based PID controller is also compared with other controllers tuned using differential evolution (DE), elephant herding optimization (EHO), Nelder-Mead simplex (NMS), membrane computing (MC), and Luus-Jaakola (LJ) algorithms. The values of overall objective function, , and three sub-objetives, 1 , 2 , and 3 , as provided in (16), are included in these findings. The controller parameters κ pg , κ ig , κ dg and η for which the minimal value of the objective function is attained are also provided in these tables. These tables also indicate the settling times and peak overshoots of the variations in the frequencies and tie-line power.  For fair comparison of controllers designed using GWO, DE, EHO, NMS, MC, and LJ algorithms, a comparative study is also performed. For all six test circumstances stated above, every optimization algorithm is run consecutively 50 times. During simulations, a total 100 solutions are considered for each algorithm. The results for comparative study are provided in Table 8.
The simulation results for Test Scenario 1 are presented in Table 2. Fig.(s) 3, 4, and 5 demonstrate the deviations in frequencies of area-1, area-2, and change in tie-line power achieved respectively under this loading scenario. The GWO-based PID controller provides the smallest value of the objective function, , when compared to other algorithmbased controllers. In addition, all three sub-objectives, 1 , 2 , and 3 are determined to be minimal for the GWO-based PID controller. Further study of the simulation results reveals that  the GWO-based PID controller achieves the shortest settling times for frequency fluctuations in area-1 and area-2. Table 3 tabulates the simulation results for Test Scenario 2. Under the loading scenario described in Test Scenario 2, the suggested GWO-based PID controller achieves the lowest value of . Along with the overall objective function being the least value, the sub-objective 2 is determined to be the least. Fig.(s) 6, 7, and 8 illustrate the changes in frequencies of area-1, area-2, and tie-line power, respectively. Fig.(s) 6, 7, and 8 also indicate that the GWO-based PID controller outperforms the other five controllers.
Under the circumstances described in Test Scenario 3, variations in frequencies of area-1, area-2, and tie-line power obtained using the proposed GWO-based PID controller are shown in Fig.(s) 9, 10, and 11, respectively. Table 4 tabulates the simulation results for Test Scenario 3. The suggested GWO-based PID controller achieves the lowest value of  as well as all sub-objectives, 1 , 2 , and 3 . As a result, the suggested GWO-based PID controller easily outperforms the other five controllers.
Similarly, the simulation results for Test Scenario 4 are presented in Table 5. The proposed GWO-based PID controller achieves the minimal values of the objective function, , as well as all sub-objectives 1 , 2 , and 3 . Fig.(s) 12, 13, and 14 illustrate the deviations in frequencies of area-1, area-2, and tie-line power, respectively. The settling time of frequencies in area-1 and area-2, as well as the settling time of tie-line power are minimal in these plots. From the Test Scenario 4 findings, it is clear that the suggested GWO-based PID controller outperforms the other controllers investigated.
The deviations in frequencies of area-1, area-2, and tie-line power obtained using the proposed GWO-based  PID controller for the loading condition described in Test Scenario 5 are shown in Fig.(s) 15, 16 and 17, respectively. Table 6 tabulates the simulation results for this Test Scenario 5. The suggested GWO-based PID controller produces the lowest values of the objective function, , as well as all three sub-objective functions, 1 , 2 , and 3 . As a result, the proposed GWO-based PID controller is easily outperforming the existing controllers in Test Scenario 5.
The overall efficacy of the proposed GWO-based PID controller is evaluated using comparative statistical analysis. For each test scenario, the best, mean, worst, and standard deviation are calculated. The acquired values are listed in Table 8. It is evident from this table that the proposed GWO-based PID controller delivers the best value in all the scenarios. The DE-based controller is the next best option, followed by the MC-based controller. In the test scenarios, namely Test VOLUME 10, 2022   Scenario 1, Test Scenario 2, Test Scenario 4, Test Scenario 5, and Test Scenario 6, the proposed GWO-based PID controller obtains the minimal mean value of objective function. In Test Scenario 1, Test Scenario 2, and Test Scenario 4, the proposed GWO-based PID controller has the lowest standard deviation values, whereas the DE-based controller have the least standard deviation values in Test Scenario 3 and Test Scenario 5 along with NMS-based controller having the least standard deviation value in Test Scenario 6. According to this statistical analysis, the suggested GWO-based PID controller is more effective than the other five controllers developed using the DE, EHO, NMS, MC, and LJ algorithms.    Table 7 tabulates the simulation results obtained using the proposed GWO-based PID controller for the loading  condition described in Test Scenario 6. The overall objective function, , and all three sub-objectives, 1 , 2 , VOLUME 10, 2022   Fig.(s) 18, 19, and 20, respectively. As can be seen in the plots, the suggested GWO-based PID controller clearly achieves the shortest settling time for frequency variations in area-1, area-2, and power deviations in tie-line. As a result, it is evident that the suggested GWO-based PID controller outperforms the other five controllers in this test instance also.

VII. CONCLUSION
For tackling the problem of AGC, GWO assisted rank-sumweight method based controller is proposed in this article. The controller design takes into account three objectives. These separate objectives result in a multi-objective formulation. This multi-objective formulation is converted into single objective formulation using a decision making tool, rank-sum-weight method. The suggested GWO assisted ranksum-weight method based controller is compared to controllers tuned using differential evolution (DE), elephant herding optimization (EHO), Nelder-Mead simplex (NMS), membrane computing (MC), and Luus-Jaakola (LJ) algorithms for ensuring the effectiveness. The effectiveness of suggested GWO assisted rank-sum-weight method based controller is tested for six distinct scenarios involving a variety of perturbations in the range of mild load variations to large load variations in interconnected areas. The numerical results demonstrate the superiority and efficacy of the suggested GWO-based PID controller in comparison with other optimization based controllers. The improved nature of the suggested controller for an AGC issue is demonstrated by time-domain simulations for variations in the frequencies of the regions under various scenarios. A comparative statistical analysis is performed for all the scenarios to assess the overall effectiveness of the suggested GWO-based PID controller.
The examination of AGC issue with non-linear models and implementation of renewable energies is the future topic of this research. The efficacy of rank-sum-weight method should also be investigated for other multi-objective formulations in the area of AGC. Moreover, other optimization algorithms [52], [53] should also be investigated for single-objective and multi-objective formulations [54], [55] of AGC problem.

DATA AVAILABILITY
My manuscript has no associated data.