In this research, a whale-optimized fuzzy PID controller was developed to manage automatic generation control in multiple-area electrical energy systems with an availability-based tariff (ABT) pricing scheme. The objective of this work is to minimize the power production costs, area control errors (ACEs), and marginal costs of the multiple-area electrical energy system with real-time load and frequency variation conditions. The generation of power, deviation of power in the tie line, and deviation of frequency of the interconnected three-area electrical energy system, including the hydrothermal steam power plant and gas power plant, will be measured and analyzed rigorously. Based on the output from the whale optimization, the fuzzy PID controller regulates the deviation of power in the tie line and the deviation of frequency of the interconnected three-area electrical energy system. The reliability and suitability of the proposed optimization, i.e., whale-optimized fuzzy PID controller, are investigated against already presented methods such as particle swarm optimization and genetic algorithms.

The importance of the responsibility of power and energy automatic generation control is in supplying the stipulated energy and power to the electrical loads with the least momentary fluctuation. However, the challenge is to regulate the power flow in the tie line and oscillation in frequency during various load demands. To damp out the oscillation, an optimal controller has been developed for automatic generation control (AGC) [

Various currently synchronized markets are expected to develop into a multiple-source structure, the implications of power sector deregulation on LFCs [

To solve the problems of automatic production operation, an integral controller based on fuzzy logic is designed, and a hybrid genetic algorithm- (GA-) fuzzy controller is required for the multiple-area electrical system model of thermal power plants [

In [

In [

Adaptive weight is utilized in PSO to optimize the parameter of the PID-controller parameter for multiple-area electrical systems running in the open-market scenario [

The possibility has been raised of determining the results of physical limits, such as the governor dead band (GDB) and production rate limitation (GRT) nonlinearity of thermal device reheat turbines, by using a GSA to optimize the AGC PI/PID controller parameters with ITAE yield-improved performance [

From the above literature review, there is ample scope in AGC for the availability-based tariff pricing scheme. In this work, whale optimization is utilized to find the optimal parameter of a multiple-area electrical system with a fuzzy PID controller to minimize the generation cost, marginal cost, and ACE of the multiple-area network with real-time load and frequency variation conditions. The overall system is developed in the MATLAB/Simulink toolbox. Various test analyses are investigated to test the suitability of the proposed whale-optimized fuzzy-controlled three-area power system.

A multiple-area electrical system can be separated into numeral power pools interrelated by a common link kenned as tie lines. The area frequency drops when the rise in perturbation, or, is kept at zero. Likewise, the frequency may rise if the load decreases, and vice versa. On the other hand, the desirable frequency to keep the constant is such that

Tie-line interconnected electrical system.

In an interconnected electrical system consisting of quite a lot of pools, the responsibility of the AGC is to split the loads between the system, station, and generator to subsequently attain the most economy and in addition to controlling the scheduled interchange of tie-line powers. It does this while keeping the frequency sensible. Throughout major transient turbulence and emergencies, the AGC’s role is to bypass, and protective relays are activated to bring back the electrical network in normal mode. The bulky dimensions of the electrical system are usually separated into different power area stands on the standard of coherency. The control areas are interconnected through the common link and are called power tie lines. Under irregular conditions of the power grid, they are used for contractual energy sharing between the control areas.

Effective area management is needed for frequency variation in some regions of the united scheme in order to regulate its production and to reestablish the deviation of frequency and power in the tie line. Both control groups sustain the other control groups owing to significant generation or load changes. An interconnected power grid seeks to overcome ACEs for effective operation in terms of net power exchange and frequency [

Equipped with simplicity in the primary control loop, the LFC’s power shift in group 1 is met up by the generation rise in both groups combined with a shift in the reduction in frequency and power in the tie line. The power system is attained at the standard frequency in the usual working conditions in the demands of regions. For the usual mode, a simple control technique is to set the frequency at a standard level (50 Hz) and retain the power flow in the tie line while each area can take up changes in its individual load. The classical LFC relies on the influence of the tie line bias, where each region tries to nullify the ACE. The ACE is a combination of power in the tie line and frequency error in the linear scale and is expressed in Equation (

The overall number of interactions between the adjacent control areas is determined by the frequency bias

As a result,

Here,

The ABT typically contains the following balanced pricing scheme: the cost of capacity, electricity costs, and unplanned interchange (UI). All three payments are included in the AGC of the electrical system, and the lowest-cost activities are valued. The ABT with the integrated power structure is discussed in the manuscript. Hydrothermal, gas power plant, and steam power plant systems are integrated control systems. Centered on the ABT process, the functionality of AGC is resolutely investigated in a unified electrical system. The objective fitness is definitive for examining the ABT progression, and the frequency regulation is also specified for the assessment [

The cost of unscheduled interchange (UI) is often available in real time and is regulated by the inverse curve of cost-frequency [

UI cost vs. frequency graph.

The marginal cost of the generation unit in the multiple-area electrical system is represented by the following equation,

The small variation in marginal cost

The small change in profit,

An AGC three-area device block diagram is shown in Figure

AGC for three-area system with ABT.

The fixed structures of fuzzy PID controllers with steady gains are optimized to a particular operating point and provide optimal performance for that operating point. The design of a controller will require proper selection of fuzzy PID output gain constants accordingly, so that deviations in the ACE, UI cost, and marginal cost will be minimized, while profit cost is maximized. The deviation in frequency replication is due to the increase in demand of consumers. If the output replication is more prominent than the authoritative ordinance, the machine will incline to raise in speed, making the frequency elevate, and vice-versa. Hence, the control engineers take appropriate action in tuning the gain constants by monitoring the rise or fall in frequency. The gain constants of the controller depend on the number of loads predicted for that machine. The load prognostication is predicated on the injective authorization of the particular machine obtained during the particular period. In short, fuzzy PID controllers are included in the design of the LFC to enable the turbine-governor system to take corrective action immediately after the load. Figure

Fuzzy PID controller tuning by whale optimization algorithm.

The fitness function for the three-area AGC with the availability-based tariff pricing scheme is expressed in the following Equation (

The minimize objective function is focused to:

There is no need for gradient information due to its ease of implementation, and practically, it can be broadly applied to many disciplines using metaheuristic optimization algorithms to gain interest. The efficiency of humpback whales is part of the latest whale optimization metaheuristic optimization method. Two steps of the search process are discovery and exploitation. Maintaining a proper balance between these two search processes is a very difficult challenge during the development of every metaheuristic algorithm.

Typically, whales are the largest mammals and one of the world’s most majestic creatures. Researchers have discovered that within the brains of whales, there are certain cells that are identical to human spindle cells. Because of this testimony, whales with instincts are regarded as the most intelligent types. Humpback whales are some of the largest whales, and their exceptional hunting practice has made them popular above all other whales. By shaping individual bubbles along a “9”-shaped path called the bubble-net feeding process, searching is accomplished. First, in this process, whales dive 12 m down, continue to form a bubble in a spiral shape around the prey, and swim up to the surface again [

Whales are ready to surround their victims and inform them in order to haunt its place to arrive at the desired solution. The goal—prey, in this context—is believed to be the most excellent candidate result at present. In this way, various numerical equations are defined as follows [

The distance is determined between the whales situated at (

There is a 50 percent chance of deciding between the bubble-net hunt methods to inform the location of whales in optimization as humpback whales swarm through the prey. The following equations explain the method described in the mathematical model, where the probability for each surrounding mode is denoted by

Flowchart of whale optimization algorithm.

In this section, the simulation result verification of a whale optimization algorithm-optimized fuzzy PID controlled three-area electrical system with an availability-based tariff pricing scheme is investigated and compared with GA and PSO. The parameters used for the three-area electrical system are shown in Table

Specification of three-area electrical system.

Area | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 0.06 | 2.4 | 0.425 | 120 | 18 | 0.28 | 0.08 | 0.02 | 1.243 |

2 | 0.08 | 2.7 | 0.37 | 115 | 25 | 0.33 | 0.07 | 0.02 | 1.658 |

3 | 0.06 | 2.5 | 0.4 | 112.5 | 20 | 0.3 | 0.072 | 0.02 | 1.356 |

The three-area electrical system was designed and modeled in MATLAB Simulink, and it is depicted in Figure

MATLAB Simulink model of the three-area electrical system.

MATLAB Simulink model of the fuzzy PID controller.

MATLAB Simulink model of the availability-based tariff pricing scheme.

The output-scaling factors of the fuzzy PID controller, i.e.,

Convergence graph for WOA.

Optimization results of GA, PSO, and WOA.

Algorithm | Mean | Standard deviation | Worst | Best | Computation time (sec) | |||
---|---|---|---|---|---|---|---|---|

GA | 0.268 | 0.283 | 0.229 | 1.352 | 0.1799 | 1.94964 | 1.233 | 964 |

PSO | 0.258 | 0.121 | 0.221 | 1.154 | 0.2490 | 1.81754 | 1.187 | 850 |

WOA | 0.259 | 0.98 | 0.233 | 1.023 | 0.13572 | 1.40512 | 1.092 | 752 |

A developed three-area power system with an availability-based tariff system was tested in three different operating conditions: a constant load condition, with a sudden change in load conditions in each area and a sudden load change in conditions in all areas. The constant load condition

Deviation frequency response of area 1, area 2, and area 3 for constant load condition.

The sudden load disturbance is created in area 1, i.e., the load change in area 1 is from 0.2 p.u to 0.4 pu at 50 seconds, and the load profile in area 2 and area 3 is maintained at 0.2 pu. The corresponding response of this tie line power deviation and the frequency deviation of the three areas is shown in Figure

Response of the system under sudden load change from 0.2 to 0.4 p.u in area 1 at 50 seconds.

Frequency deviation

Tie line power deviation

The sudden load disturbance is created in area 2, i.e., the load change in area 1 from 0.4 p.u to 0.2 pu at 40 seconds, and load profile area 1 and area 3 is maintained at 0.2 pu. The corresponding response of such a tie line power deviation and frequency deviation of the three areas is shown in Figure

Response of the system under sudden load change from 0.4 to 0.2 p.u in area 2 at 40 seconds.

Frequency deviation

Tie line power deviation

The sudden load disturbance is created in area 3, i.e., the load change in area 1 is from 0.5 p.u to 0.9 pu at 30 seconds, and the load profiles in area 1 and area 2 are maintained at 0.2 pu. The corresponding response to such a tie line power deviation and frequency deviation of the three areas is shown in Figure

Response of the system under sudden load change from 0.5 to 0.9 p.u in area 3 at 30 seconds.

Frequency deviation

Tie line power deviation

The sudden load disturbance is created in all the areas, i.e., the load change is from 0.2 p.u to 0.4 p.u in area 1 at 30 seconds, the load change is from 0.4 p.u to 0.2 p.u in area 2 at 60 seconds, and the load change is from 0.5 p.u to 0.9 p.u in area 3 at 90 seconds. The corresponding response to such a tie line power deviation and frequency deviation of the three areas is shown in Figure

Response of the system under sudden load change from 0.2 to 0.4 p.u in area 1 at 30 seconds, load change from 0.4 to 0.2 p.u in area 2 at 60 seconds, and load change from 0.5 to 0.9 p.u in area 3 at 90 seconds.

Frequency deviation

Tie line power deviation

In this paper, a whale optimization algorithm was utilized to tune the parameter of a fuzzy PID controller to minimize the marginal cost, unscheduled interchange cost, and ACE while maximizing the profit of a three-area electrical system. The overall system model was designed and developed using the MATLAB Simulink software. The whale optimization algorithm was compared with GA and PSO to test the suitability of the whale optimization algorithm. The three-area power system with ABT was tested with WOA fuzzy PID, PSO fuzzy PID, and GA fuzzy PID controllers in different operating conditions. Based on the test result of the simulation, the three-area electrical system controlled by the whale optimization algorithm-optimized fuzzy PID controller has less overshoot, less undershoot, a faster settling time, and a faster recovery time than the GA fuzzy PID and PSO fuzzy PID control.

All data are available for readers.

The authors declare that they have no conflicts of interest.