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This work proposes a novel nature-inspired algorithm called Ant Lion Optimizer (ALO). The ALO algorithm mimics the search mechanism of antlions in nature. A time domain based objective function is established to tune the parameters of the PI controller based LFC, which is solved by the proposed ALO algorithm to reach the most convenient solutions. A three-area interconnected power system is investigated as a test system under various loading conditions to confirm the effectiveness of the suggested algorithm. Simulation results are given to show the enhanced performance of the developed ALO algorithm based controllers in comparison with Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Bat Algorithm (BAT) and conventional PI controller. These results represent that the proposed BAT algorithm tuned PI controller offers better performance over other soft computing algorithms in conditions of settling times and several performance indices.

Load Frequency Control (LFC) is a crucial topic in power system operation and control of supplying sufficient and dependable electric power with better tone. An electric energy system must be maintained at a desired operating level characterized by nominal frequency and voltage profile, and this is achieved by close control of real and reactive powers generated through the controllable source of the system. The primary destination of the LFC is to maintain zero steady state errors for frequency deviation, and right tracking load demands in a multi-area power system [

In the event of an interconnected power system, any small sudden load change in any of the areas causes the variation of the frequencies of each and every area and likewise at that place is a fluctuation of power in tie line. The primary objective of Load Frequency Control (LFC) area is to maintain the right frequency, besides the desired power output (megawatt) in the interconnected power system and to monitor the change in tie line power between control areas. Thus, an LFC scheme incorporates an appropriate control system for an interconnected power system. It is heaving the capability to bring the frequencies of each area and the tie line powers back to original setpoint values or very nearer to set point values effectively after the load change because of the of conventional controllers. Nevertheless, the conventional controllers are having some demerits like; they are very sluggish in functioning. They do not care about the inherent nonlinearities of different power system components; it is very hard to decide the gain of the integrator setting according to changes in the operating point. The artificial intelligence control system delivers many advantages over the conventional integral controller. They are a lot faster than integrated controllers, and besides they give better stability response than integral controllers. Several control strategies for LFC of power systems have been proposed to maintain the frequency and tie-line power flow at their scheduled power values during normal and distributed conditions. Classical controllers are offered for LFC of power system [

In this proposed research work, a new form of multi-area power system with a combination of thermal, hydro and PV sources. A new algorithm proposed for load-frequency control, which can both reduce control time and diminish the value of frequency deviation during the active operation of power systems. By developing of industrial controllers, Proportional-Integral (PI) controllers is still one of the most popular controllers. A new approach addressed for load-frequency control of interconnected three area power systems by using an Ant Lion Optimizer (ALO) algorithm in this paper. The algorithm applied to optimize the PI parameters. Also, to adjust the PI controller, the ITAE is used as a cost function. The ITAE criterion was chosen due to it can determine a healthy weight for error signal in terms of time. This event can reduce settling time in the lowest value and damp fluctuations, quickly [

The system under study consists of three areas. Area one is a thermal non-reheat system, area two is a hydro system, and area three is photovoltaic (PV) system [

The dynamic model of Load Frequency Control (LFC) for a two-area interconnected power system is presented in this section. Each area of the power system consists of speed governing system, turbine, and generator as shown in _{tie}. The outputs are the generator frequency ∆f and Area Control Error (ACE) given by equation

where B is the frequency bias parameter.

To simplicity the frequency-domain analyzes, transfer functions are used to model each component of the area. Turbine is represented by the transfer function [

Transfer function of the governor is

Transfer function of the steam turbine is

Transfer function of the generator is

where K_{p} = 1/D and T_{p} = 2H/fD.

Transfer function of the hydraulic turbine is

The system under investigation consists of three area interconnected power system as shown in _{1} and B_{2} are the frequency bias parameters; ACE_{1}, ACE_{2}, and ACE_{3} are area control errors; u_{1}, u_{2}, and u_{3} are the control outputs from the controller; R_{1} and R_{2} are the governor speed regulation parameters in pu Hz; T_{g}_{1 }and T_{g}_{2} are the speed governor time constants in sec; DP_{V}_{1} and DP_{V}_{2} are the change in governor valve positions (pu); DP_{g}_{1} and DP_{g}_{2} are the governor output command (p.u). T_{t}_{1} and T_{t}_{2} are the turbine time constant in sec; DP_{t}_{1} and DP_{t}_{2} are the change in turbine output powers; d_{PD}_{1}, d_{PD}_{2}, and d_{PD}_{3} are the load demand changes; DP_{Tie} is the incremental change in tie line power (p.u); KP_{S}_{1}, KP_{S}_{2,} and KP_{S}_{3} are the power system gains; TP_{S}_{1}, TP_{S}_{2}, and TP_{S}_{3} are the power system time constant in sec; T_{12}, T_{23}, and T_{31} are the synchronizing coefficient and Df_{1}, Df_{2} and Df_{3} are the system frequency deviations in Hz.

Despite significant strides in the development of advanced control schemes over the past two decades, the conventional Proportional-Integral (PI) controller and its variants remain an engineer’s preferred choice because of its structural simplicity, reliability, and the favorable ratio between performance and cost. Beyond these benefits, it controllers also offers simplified dynamic modeling, lower user-skill requirements, and minimal development effort, which are issues of substantial importance to engineering practice. As the name suggests, the PI algorithm consists of three basic modes, the proportional mode, and integral mode. A proportional controller has the effect of reducing the rise time, but never eliminates the steady-state error. An integral control has the effect of eliminating the steady-state error, but it may make the transient response worse. The design of PI controller requires determination of the two parameters, Proportional gain (KP) and Integral gain (KI) [

The error inputs to the controllers are the respective area control errors (ACE) are:

The control inputs of the power system u_{1} and u_{2} are the outputs of the controllers. The control inputs are obtained as:

In the design of a PI controller, the objective function is first defined based on the desired specifications and constraints. The design of objective function to tune PI controller is based on a performance index that considers the entire closed loop response. Typical output specifications in the time domain are peak overshooting, rise time, settling time, and steady-state error. Four kinds of performance criteria usually considered in the control design are the Integral of Time multiplied Absolute Error (ITAE), Integral of Squared Error (ISE) and Integral of Absolute Error (IAE).

The sum of time multiple absolute errors in ACE is considered as a performance index. Hence, J can be:

where J is the objective function and

The ALO algorithm also finds superior optimal designs for the majority of classical engineering problems employed, showing that this algorithm has merits in solving constrained problems with separate search spaces. The main inspiration of the ALO algorithm comes from the foraging behavior of antlion’s larvae [

The ALO algorithm mimics interaction between antlions and ants in the trap. To model such interactions, ants are required to move over the search space, and antlions are allowed to hunt them and become fitter using traps. Since ants move stochastically in nature when searching for food, a random walk is chosen for modeling ants’ movement as follows:

where cumsum calculates the cumulative sum, n is the maximum number of iteration; t shows the step of random walk (iteration in this study), and r(t) is a stochastic function defined as follows:

where t shows the step of random walk (iteration in this study) and rand is a random number generated with uniform distribution in the interval of [0, 1].

The position of ants is saved and utilized during optimization in the following matrix:

where M_{Ant} is the matrix for saving the post of each ant, A_{i,j}_{ }shows the value of the j^{th} variable (dimension) of i^{th} ant; n is the number of ants, and d is the number of variables. It should be noted that ants are similar to particles in PSO or individuals in GA. The position of an ant refers the parameters for a particular solution. Matrix M_{Ant} has been considered to save the post of all ants (variables of all solutions) during optimization. For evaluating each ant, fitness (objective) function is utilized during optimization and the following matrix stores the fitness value of all ants:

where M_{OA} is the matrix for saving the fitness of each ant, A_{i,j}_{ }shows the value of the j^{th} dimension of the i^{th} ant; n is the number of ants, and f is the objective function. In addition to ants, we assume the antlions are also hiding somewhere in the search space. In order save their positions and fitness values, the following matrices are utilized:

where M_{Antlion} is the matrix for saving the post of each antlion, AL_{i,j}_{ }shows the j^{th} dimension’s value of i^{th} antlion; n is the number of antlions, and d is the number of variables (dimension).

where M_{OAL} is the matrix for saving the fitness of each antlion, AL_{i,j} shows the j^{th} dimension’s value of i^{th} antlion; n is the number of antlions, and f is the objective function. During optimization, the following conditions are applied:

Ants move around the search space using different random walks.

Random walks are applied to all the dimension of ants.

Random walks are affected by the traps of antlions.

Antlions can build pits proportional to their fitness (the higher fitness, the larger pit).

Antlions with larger pits have the higher probability of catching ants.

An antlion can catch each ant in each iteration and the elite (fittest antlion).

The range of random walk is decreased adaptively to simulate sliding ants towards antlions.

If an ant becomes fitter than an antlion, this means that it is caught and pulled under the sand by the antlion.

An antlion repositions itself to the latest caught prey and builds a pit to improve its chance of catching another prey after each hunt.

The ALO algorithm is defined as a three-tuple function that approximates the global optimum for optimization problems as follows [

where A is a function that generates the random initial solutions, B manipulates the initial population provided by the function A, and C returns true when the end criterion is satisfied. The functions A, B, and C are defined as follows:

where M_{Ant} is the matrix of the position of ants, M_{Antlion} includes the position of antlions; M_{OA} contains the corresponding fitness of ants, and M_{OAL} has the fitness of antlions.

The pseudo codes the ALO algorithm is defined as follows:

where a^{i} is the minimum of the random walk of the i^{th} variable, ^{t}^{h} variable, is the minimum of the i^{th} variable at i^{th} iteration, and indicates the maximum of the i^{th} variable at t^{th} iteration. ^{th} iteration, ^{th} iteration, and ^{th} ant at t^{th} iteration. ^{th} antlion at t^{th} iteration, t shows the current iteration and ^{th} ant at t^{th} iteration.

In the ALO algorithm, the antlion and ant matrices are initialized randomly using the function A. In every iteration; the function B updates the position of each ant on an antlion selected by the roulette wheel operator and the elite. The boundary of position updating is first defined proportionally to the current number of iteration. Two random walks then accomplish the updating position around the selected antlion and elite. When all the ants randomly walk, they are evaluated by the fitness function. If any of the ants become fitter than any other antlions, their positions are considered as the new posts for the antlions in the next iteration. The best antlion is compared to the best antlion found during optimization (elite) and substituted if it is necessary. These steps iterative until the function C returns false.

Theoretically speaking, the proposed ALO algorithm can approximate the global optimum of optimization problems due to the following reasons:

Exploration of the search space is guaranteed by the random selection of antlions and random walks of ants around them.

Exploitation of search space is ensured by the adaptive shrinking boundaries of antlions’ traps.

There is a high probability of resolving local optima stagnation due to the use of random walks and the roulette wheel.

ALO is a population-based algorithm, so local optima avoidance is intrinsically high.

The intensity of ants’ movement is adaptively decreased over the course of iterations, which guarantees convergence of the ALO algorithm.

Calculating random walks for every ant, and every dimension promotes diversity in the population.

Antlions relocate to the position of best ants during optimization, so promising areas of search spaces are saved.

Antlions guide ants towards promising regions of the search space.

The best antlion in each iteration is stored and compared to the best antlion obtained so far (elite).

The ALO algorithm has very few parameters to adjust.

The ALO algorithm is a gradient-free algorithm and considers problem as a black box.

The model of the three area interconnected power system under study has been developed using MATLAB/Sim- ulink software platform as shown in

A step increase in demand of 10% and 20% applied at t = 0s in area-1, and the system dynamics responses are shown in Figures 4-9 and for 20% step load in area 2 illustrated in Figures 10-15. It is evident from both

conditions that, when ITAE is used as an objective function, the system performance with proposed controller is better than that of GA, PSO and BAT based PI controller on ITAE criteria. So, it can be concluded that the performance of proposed ALO is superior to that of GA, PSO, and BAT from fitness function minimization point of view. The response with ALO based PI controller is the best among the other alternatives as best system performance is obtained from desired control specification point of view.

Sensitivity analysis is carried out to study the robustness the system to wide changes in the operating conditions and system parameters [_{PS} and T_{PS}. The power system parameters are calculated for different loading conditions as given in Appendix A. The change in tie-line power is simulated by changing the synchronizing coefficient T_{12}. Figures 16-45 shows the exchanged power between three area under this load exchange. The controller parameters, error performance indices like ITAE, ISE, IAE, frequency and tie line power deviations with various controllers under nominal load and sensitivity load test conditions with time constants of speed governor (T_{g}), turbine (T_{t}), tie-line power (T_{12}) and hydraulic governor coefficients (T_{h1} and T_{h3}) are varied from their nominal values in the range of +25% to −25%. The power system parameters and time constants are calculated for the varied condition and used in the simulation model. In all the cases the controller parameters obtained using the objective function J is considered due to its superior performance. The results obtained are depicted in

ALO algorithm is suggested in this paper to tune the parameters of PI controllers for LFC problem. An integral time absolute error of the ACE for all areas is chosen as the objective function to enhance the system response in terms of the settling time and overshoots. The major contributions of this paper are:

Establishment of the dynamic model for three area power system considering with LFC based Ant Lion Optimizer algorithm to assure the superiority of the PI controller over GA, PSO, BAT and conventional integral controller throughout different disturbances for various signals.

Parameter Variation | % Change | Controller | Performance Index | Settling Time (Ts) in sec | ||||
---|---|---|---|---|---|---|---|---|

ISE | IAE | ITAE | ΔF1 | ΔF2 | ΔF3 | |||

Case A | 10% load change in area 1 | Conv PI | 687.2543 | 2.4023e+04 | 4.8761e+05 | 113.30 | 94.29 | 110.9 |

GA PI | 149.1328 | 5.0897e+03 | 5.5266e+04 | 40.95 | 70.59 | 69.76 | ||

PSO PI | 146.5642 | 4.9364e+03 | 6.9875e+04 | 23.18 | 49.43 | 53.44 | ||

BAT PI | 156.8874 | 5.0013e+03 | 3.8638e+04 | 20.63 | 20.87 | 21.72 | ||

ALO PI | 149.0129 | 4.2245e+03 | 2.9450e+04 | 16.45 | 17.03 | 17.81 | ||

20% load change in area 1 | Conv PI | 2.5796e + 003 | 4.5427e+004 | 8.7791e+005 | 116.4 | 86.23 | 111.4 | |

GA PI | 598.5967 | 1.0471e+004 | 1.0981e+005 | 61.12 | 49.66 | 52.87 | ||

PSO PI | 589.3889 | 1.0487e+004 | 1.5803e+005 | 54.43 | 94.19 | 79.20 | ||

BAT PI | 653.4579 | 1.0627e+004 | 7.4717e+004 | 42.39 | 36.34 | 41.10 | ||

ALO PI | 598.5334 | 8.7707e+003 | 6.1928e+004 | 21.87 | 18.83 | 18.35 |

Parameter Variation | % Change | Controller | Performance Index | Settling Time (Ts) in sec | ||||
---|---|---|---|---|---|---|---|---|

ISE | IAE | ITAE | ΔF1 | ΔF2 | ΔF3 | |||

Case B | +25% in TG | Conv PI | 2.4170e+003 | 6.5980e+004 | 3.2888e+06 | 197.9 | 198 | 197.2 |

GA PI | 199.5821 | 6.3954e+03 | 6.4610e+04 | 53.24 | 59.72 | 54 | ||

PSO PI | 191.7430 | 6.2585e+03 | 8.6006e+04 | 43.71 | 43.01 | 42.93 | ||

BAT PI | 258.5478 | 7.8110e+03 | 5.7540e+04 | 36.65 | 37.23 | 36.79 | ||

ALO PI | 186.7742 | 5.2499e+03 | 3.6980e+04 | 21.85 | 22.32 | 22.03 | ||

+25% in TT | Conv PI | 1.8112e+003 | 5.1996e+004 | 2.0302e+006 | 198.4 | 195.4 | 195.5 | |

GA PI | 622.5414 | 1.7977e+04 | 2.9229e+05 | 124.6 | 112.7 | 120.4 | ||

PSO PI | 469.5161 | 1.3650e+04 | 2.0004e+05 | 107.8 | 91.41 | 87.41 | ||

BAT PI | 749.5638 | 2.1911e+04 | 3.7435e+05 | 97.23 | 93.34 | 77.53 | ||

ALO PI | 401.8087 | 1.1103e+04 | 1.1486e+05 | 68.12 | 62.06 | 63.37 | ||

+25% in T12 | Conv PI | 1.2332e+003 | 4.0432e+004 | 1.3645e+006 | 196.4 | 157.1 | 190.3 | |

GA PI | 231.3837 | 7.4623e+03 | 7.6473e+04 | 54.35 | 57.87 | 53.6 | ||

PSO PI | 217.9085 | 7.0637e+03 | 9.3569e+04 | 38.36 | 44.41 | 41.76 | ||

BAT PI | 291.4573 | 8.8849e+03 | 6.9190e+04 | 28.31 | 38.75 | 38.52 | ||

ALO PI | 210.3520 | 6.0441e+03 | 4.3933e+04 | 23.15 | 27.63 | 27.03 | ||

−25% in TG | Conv PI | 199.7606 | 8.7988e+003 | 1.1584e+005 | 77.34 | 73.84 | 70.61 | |

GA PI | 125.8109 | 4.7644e+03 | 5.2134e+04 | 28.31 | 27.84 | 46.64 | ||

PSO PI | 123.2376 | 4.7568e+03 | 7.6474e+04 | 22.53 | 17.83 | 39.92 | ||

BAT PI | 153.5863 | 5.4350e+03 | 3.9674e+04 | 18.93 | 15.35 | 20.99 | ||

ALO PI | 126.7282 | 3.9297e+03 | 2.8650e+04 | 15.19 | 12.74 | 15.91 | ||

−25% in TT | Conv PI | 134.1685 | 7.4446e+003 | 1.0388e+005 | 60.22 | 70.9 | 72.75 | |

GA PI | 99.6813 | 4.6223e+03 | 5.4438e+04 | 38.39 | 39.46 | 50.73 | ||

PSO PI | 94.9256 | 4.4384e+03 | 7.6097e+04 | 27.72 | 28.62 | 38.69 | ||

BAT PI | 98.9754 | 4.2241e+03 | 3.2559e+04 | 21.42 | 21.71 | 31.41 | ||

ALO PI | 96.4345 | 3.5071e+03 | 2.7416e+04 | 13.85 | 16.87 | 19.61 | ||

−25% in T12 | Conv PI | 184.3827 | 8.2345e+003 | 1.0851e+005 | 54.14 | 72.27 | 69.4 | |

GA PI | 121.6113 | 4.6104e+03 | 5.1095e+04 | 22.36 | 35.03 | 46.77 | ||

PSO PI | 118.7871 | 4.6084e+03 | 7.4763e+04 | 18.55 | 23.73 | 22.84 | ||

BAT PI | 154.6238 | 5.5114e+03 | 4.0791e+04 | 16.7 | 21.16 | 20.19 | ||

ALO PI | 122.1739 | 3.7845e+03 | 2.7824e+04 | 12.12 | 18.1 | 17.61 |

Parameter Variation | % Change | Controller | Performance Index | Settling Time (Ts) in sec | ||||
---|---|---|---|---|---|---|---|---|

ISE | IAE | ITAE | ΔF1 | ΔF2 | ΔF3 | |||

Case B | +25% in TH1 | Conv PI | 640.3085 | 2.2469e+04 | 4.3259e+05 | 98.35 | 94.86 | 111.5 |

GA PI | 150.4418 | 5.2925e+03 | 5.9267e+04 | 20.4 | 47.49 | 42.11 | ||

PSO PI | 147.4219 | 5.2543e+003 | 8.3200e+004 | 17.86 | 40.39 | 33.02 | ||

BAT PI | 174.1908 | 5.7051e+003 | 4.3449e+004 | 15.01 | 21.88 | 22.36 | ||

ALO PI | 149.2697 | 4.3745e+003 | 3.2572e+004 | 13.27 | 16.17 | 19.63 | ||

−25% in TH1 | Conv PI | 646.2135 | 2.2851e+04 | 4.4357e+05 | 108.3 | 99.33 | 112.8 | |

GA PI | 153.6146 | 5.3245e+03 | 5.0693e+04 | 42.94 | 48.67 | 57.82 | ||

PSO PI | 147.4235 | 5.1995e+003 | 7.1503e+004 | 28.81 | 36.01 | 31.45 | ||

BAT PI | 171.7490 | 5.5536e+003 | 3.6383e+004 | 22.43 | 29.57 | 21.43 | ||

ALO PI | 186.3932 | 5.4897e+003 | 3.6976e +004 | 18.56 | 19.92 | 18.56 | ||

+25% in TH3 | Conv PI | 635.4319 | 2.2416e+04 | 4.3628e+05 | 123.8 | 110.8 | 117 | |

GA PI | 150.0007 | 5.3351e+03 | 6.1740e+04 | 50.28 | 64.02 | 63.65 | ||

PSO PI | 147.6909 | 5.4360e+003 | 9.6678e+004 | 34.61 | 39.47 | 35.77 | ||

BAT PI | 169.7062 | 5.5613e+003 | 4.1018e+004 | 21.65 | 28.48 | 21.95 | ||

ALO PI | 161.6022 | 4.7642e+003 | 3.5182e+004 | 16.54 | 19.56 | 16.78 | ||

−25% in TH3 | Conv PI | 663.1553 | 2.3353e+04 | 4.5034e+05 | 105.7 | 100.9 | 116.9 | |

GA PI | 152.9414 | 5.2504e+03 | 4.9655e+04 | 42.27 | 51.07 | 52.19 | ||

PSO PI | 149.0362 | 5.0918e+003 | 6.2321e+004 | 33.5 | 37.65 | 34.96 | ||

BAT PI | 208.4893 | 6.7761e+003 | 4.7161e+004 | 24.92 | 22.36 | 21.23 | ||

ALO PI | 149.2290 | 4.3562e+003 | 2.9897e+004 | 18.23 | 19.15 | 19.73 |

The robustness of the controller is confirmed through parameter variations. ALO outperforms GA, PSO, BAT in solving LFC problem due to only one parameter required to fine-tune.

On the other hand, GA deals with a population of solutions, thus leading to the disadvantage of requiring a significant number of function evaluations, extensive computational time and gets trapped in local minimum solution. PSO suffers from weak local search ability, and the algorithm may lead to possible entrapment in local minimum solutions. Also, Bat algorithm exploitation stage too quickly by varying loudness and pulse rates quickly, it can result in stagnation after some initial stage.

The capability of the developed controllers to compensate the communication time delay and preserve its satisfactory performance is demonstrated.

The effectiveness of the controller regarding various indices and settling time is proved.

Nominal parameters of the three area system investigated are [

R. Satheeshkumar,R. Shivakumar, (2016) Ant Lion Optimization Approach for Load Frequency Control of Multi-Area Interconnected Power Systems. Circuits and Systems,07,2357-2383. doi: 10.4236/cs.2016.79206