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\title{Grey Wolf Optimizer Applied to Dynamic Economic Dispatch Incorporating Wind Power}
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             \author[1]{  Hardiansyah}

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\date{\small \em Received: 7 December 2019 Accepted: 5 January 2020 Published: 15 January 2020}

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\begin{abstract}
        


This article presents a new evolutionary optimization approach called gray wolf optimizer (GWO), which is based on gray wolf behavior for an optimal generating operation strategy. The GWO algorithm does not require any information about the gradient of the objective function, when searching for an optimal solution. The concept of the GWO algorithm, it seems a powerful and reliable optimization algorithm is applied to dynamic economic dispatch (DED) problem considering wind power. Many practical constraints of generators such as valve-point effects, ramp rate limits, and transmission losses are considered. The proposed algorithm is implemented and tested on two test systems that have 5-unit and 10-unit generators. The results confirm the potential and effectiveness of the proposed algorithm compared to various other methods are available in the literature. The results are very encouraging and prove that the GWO algorithm is a very effective optimization technique for solving various DED problems.

\end{abstract}


\keywords{gray wolf optimizer, dynamic economic dispatch, wind power, ramp rate limits, valvepoint effects.}

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\let\tabcellsep& 	 	 		 \par
Introduction he electric power system is one of the most vital needs in human life. The demand for electricity continues to increase causing electricity to be supplied by power plants to be very large. On the other hand renewable energy sources are the deciding factors in industrial development that can improve people's living standards. In addition, technological advances and developments have also contributed greatly to increasing electricity demand. Power system planning, power system management, and distribution of power system are required to meet consumer demand for an increase in the quantity and quality of electric power produced. Improving the quality of electric power is also very influential in increasing the efficiency and reliability of the system. Optimization of generator scheduling in the electric power system is very necessary, because the generation and distribution process in the electric power system requires a very large cost. Coordination between power plants is needed in an effort to optimize generator scheduling to get the minimum cost. Dynamic economic dispatch (DED) is the change in real-time load on an electric power system. The DED is a development of conventional ED involving ramp rate parameters. DED is used to determine the economic distribution of generating units within a certain timeframe of the generating units. The parameter to be considered is transmission losses. In fact, the distribution of electrical power to the load always causes power losses on the transmission line, therefore, transmission losses need to be calculated so that the generator can generate power that can meet the load requirements by considering the transmission loss. In general, the cost function for each generator is represented by a quadratic function, and the valve-point effect is ignored in solving the DED problem. If the DED problem includes the valve-point effect, then the problem becomes a non-convex optimization problem with nonconvex characteristics, which introduces difficulties in finding global optimal solutions \hyperref[b0]{[1]}\hyperref[b1]{[2]}\hyperref[b2]{[3]}.\par
Renewable energy is energy resource that comes from sustainable natural processes, such as energy from wind energy, solar energy, hydropower, biomass and geothermal energy. Renewable energy began to attract the attention of people and policy makers as an alternative energy resource after the world oil crisis in 1973. The use of renewable energy then rapidly developed when the United Nations Framework Convention on Climate Change (UNFCCC) was formed by the United Nations as a movement to reduce gas greenhouse. This institution continues to consistently voice the shift towards environmentally friendly energy through the Millennium Development Goals (MDGs) and Sustainable Development Goals (SDGs) issued by the United Nations. Climate change is currently a major concern of the world community due to its effect which causes an unnatural rise in world temperatures. The main cause of climate change is electricity production activities which are dominated by coal-fired power plants and natural gas power plants which account for around 30\% of total gas emissions that cause global warming. Wind energy is a clean and rapidly growing renewable energy resources. They have shown great prospects in decreasing fuel consumption as well as reducing pollutants emission. However, the expected wind power is difficult to predict accurately, primarily due to the intermittent nature of the wind speed, coupled with the highly non-linear wind energy conversion. In order to adjust unforeseeable nature of the wind power, planned productions and uses in electricity market must be improved during the real operation of the power system. Due to the intermittent characteristic of wind power, DED is very suited for formulate the problem of optimal scheduling of generating units by including wind power. Several related studies have been conducted to overcome the problem of ED and DED by including renewable energy sources to the power system \hyperref[b3]{[4]}\hyperref[b4]{[5]}\hyperref[b5]{[6]}\hyperref[b6]{[7]}\hyperref[b7]{[8]}\hyperref[b8]{[9]}\hyperref[b9]{[10]}\hyperref[b10]{[11]}.\par
Over the past few years, a number of approaches have been developed to solve this DED problem using mathematical programming, namely, the lambda iteration method, linear programming, quadratic programming and the gradient projection method \hyperref[b11]{[12]}\hyperref[b12]{[13]}\hyperref[b13]{[14]}. Most of the methods that have been applied do not apply to non-convex or non-smooth cost functions. Many heuristic optimization techniques known such as genetic algorithms (GA), simulated annealing (SA), differential evolution (DE), particle swarm optimization (PSO), artificial bee colony (ABC) algorithm, hybrid evolutionary programming (EP) and sequential quadratic programming (SQP), deterministically guided PSO, hybrid PSO and SQP, hybrid seeker optimization algorithm and sequential quadratic programming (SOA-SQP), imperialist competitive algorithm (ICA), hybrid harmony search (HHS) algorithm, artificial immune system (AIS), and glowworm swarm optimization (GSO) have been successfully used to solve the DED problems \hyperref[b14]{[15]}\hyperref[b15]{[16]}\hyperref[b16]{[17]}\hyperref[b17]{[18]}\hyperref[b18]{[19]}\hyperref[b19]{[20]}\hyperref[b20]{[21]}\hyperref[b21]{[22]}\hyperref[b22]{[23]}\hyperref[b23]{[24]}\hyperref[b24]{[25]}\hyperref[b25]{[26]}\hyperref[b26]{[27]}\hyperref[b27]{[28]}.\par
More recently, a new meta-heuristic search algorithm, called Gray Wolf Optimizer (GWO) \hyperref[b28]{[29]}, has no affinity for sticking to local optimal points in complex multimodal optimization problems and which provides a more diverse search of space the solution. The GWO is based on gray wolf behavior. Better optimal solutions with lower computational loads can be found at GWO compared to the stochastic search techniques mentioned above. In this paper, the GWO algorithm has been applied to solve the DED problem considering wind power. The performance of the proposed approach has been demonstrated in the 5-unit and 10unit generating systems. The results obtained from the proposed algorithm are compared with other optimization results reported in the literature. The comparison shows that the proposed GWO-based approach provides the best solution in terms of minimum production cost and power loss. 
\section[{II.}]{II.} 
\section[{Problem Formulation}]{Problem Formulation}\par
The objective of DED problem is to find the optimal schedule of output powers of online generating units with predicted power demands over a certain period of time to meet the power demand at minimum operating cost. The objective function of the DED problem is, ( )T t N i c P b P a P F F T t N i i t i i t i i T t N i t i t i T , ,\textbf{2 , 1 ; , , 2 , 1 for ) ( 1 1 , 2 , 1 1}\par
, ,? ? = = + + = = ?? ?? = = = = (1)\par
where F i,t (in \$/h) is the operating cost of ith unit at time interval t, a i , b i , and c i are the cost coefficients of generating ith unit, P i,t (in MW) is the real power output of generating ith unit at time period t, and N is the number of generators. T is the total number of hours in the operating horizon. The fuel cost function of ith unit with valve-point effects is represented as follows \hyperref[b8]{[9,}\hyperref[b20]{21,}\hyperref[b21]{22]}:( ) ( ) ?? = = ? ? ? ? ? ? ? ? ? × × + + + = T t N i t i i i i i t i i t i i T P P f e c P b P a F 1 1 , min , , 2 , sin ) (\textbf{(2)}\par
where F T (in \$/h) is total operating cost of generation including valve point loading, e i and f i are fuel cost coefficients of ith unit reflecting valve-point effects. 
\section[{a) Power Balance}]{a) Power Balance}\par
For power balance, an equality constraint should be satisfied. The total generated power should be the same as total load demand plus the total line loss.\par
( )t L t D N i t w t i P P P P , ,\textbf{1}\par
, ,+ = + ? =\textbf{(3)}\par
where P w,t is power output of wind farm at time interval t; P D,t is the load demand at time interval t; P L,t is the transmission loss at time interval t that can be represented using the B-coefficients:?? = = = N i N j t j ij t i t L P B P P 1 1 , , ,\textbf{(4)}\par
where B ij , is the loss-coefficient matrix. 
\section[{b) Generation Limits}]{b) Generation Limits}\par
Generation output of each generator should lie between minimum and maximum limits. The corresponding inequality constraint for each generator is max , , min , i t i i P P P ? ? \hyperref[b4]{(5)} where P i, min and P i, max are the minimum and maximum capacity of unit i, respectively. 
\section[{c) Ramp Rate Limits}]{c) Ramp Rate Limits}\par
The actual operating ranges of all on-line units are restricted by their corresponding ramp rate limits. The ramp-up and ramp-down constraints can be written as (  {\ref 6}) and \hyperref[b6]{(7)}, respectively.up i t i t i R P P , 1 , , ? ? ? (6) down i t i t i R P P , , 1 , ? ? ? (7)\par
where P i,t and P i,t-1 are the present and previous power outputs, respectively. R i,up and R i,down are the ramp-up and ramp-down limits of unit i. The fuel cost is minimized subjected to the following constraints:\par
To consider the ramp rate limits and power output limits constraints at the same time, therefore, equations (  {\ref 5}), (  {\ref 6}) and (  {\ref 7}) can be rewritten as follows:\} , min\{ \} , max\{ , 1 , max , , , 1 , min , up i t i i t i down i t i i R P P P R P P + ? ? ? ? ? (8) III. 
\section[{Grey Wolf Optimizer}]{Grey Wolf Optimizer}\par
Grey Wolf Optimizer (GWO) is a new population based meta-heuristic algorithm proposed by  {\ref Mirjalili et al. in 2014 [29]}. The grey wolves mostly like to live in a pack and one of the most important features is their very strict social hierarchy. The main leader of the pack is called alpha. The alpha wolf is the most predominant wolf in the pack as his/her orders were followed by rest of the pack. The alpha wolf is one of the most important members in terms of managing the pack.\par
The second important one is called beta. They are also known as sub-ordinate wolves as they help alpha in their respective work. They act as advisor to alpha and commander to the rest of the wolves in the pack. The third one are called delta. They submitted themselves to the alphas and betas but dominate the omegas. The fourth one which are lower ranking wolves are called omega. They have to submit themselves to all other members in the pack.\par
In another important thing among the grey wolves is their hunting mechanism which includes tracking, chasing, encircling and harassing the prey until they stop moving. Then they attack the prey. The mathematical model of this model is discussed as following. 
\section[{a) Social Hierarchy}]{a) Social Hierarchy}\par
In order to mathematically model the social hierarchy of wolves when designing GWO that would consider the first fitness solution as alpha (?), the second best solution as beta (?), and the third best solution as delta (?). The rest of the solutions are assumed as omega (?). The hunting mechanism is decided by ?, ?, and ?, and the ? wolves have to follow them. 
\section[{b) Encircling Prey}]{b) Encircling Prey}\par
As the grey wolves encircle prey during the hunt, so their mathematical model which represents their encircling behavior is discussed as below: is linearly decreased from 2 to 0. The grey wolf can update their position according to equation (  {\ref 9}) and \hyperref[b9]{(10)}.) ( ) ( t X t X C D w p ? ? ? ? ? ? = (9) D A X t X p w ? ? ? ? ? ? = + ) 1 (\textbf{(10} 
\section[{c) Hunting}]{c) Hunting}\par
As we know that the grey wolf firstly recognizes the prey and then encircles them to hunt. The hunt is usually decided by alpha and beta, delta also participate in hunting occasion. So mathematically in the hunting procedure we take alpha, beta and delta as the best candidate solution and omega have to update its position according to the best search agent. The mathematical model for hunting is shown below:) ( 1 t X X C D ? ? ? ? ? ? = ? ? (13) ) ( 2 t X X C D ? ? ? ? ? ? = ? ? (14) ) ( 3 t X X C D ? ? ? ? ? ? = ? ? (15) ? ? D A X X ? ? ? ? ? ? = 1 1 (16) ? ? D A X X ? ? ? ? ? ? = 2 2 (17) ? ? D A X X ? ? ? ? ? ? = 3 3 (18) ( ) 3 1 3 2 1 X X X t X ? ? ? ? + + = + (19) where ? X ? is the position of the alpha, ? X ? is the position of the beta, ? X ? is the position of the delta, 1 C ? , 2 C ? , 3 C ? , 1 A ? , 2 A ? , and 3 A ? are all random vectors, X\par
? is the position of the current solution, and t is the iteration number. 
\section[{d) Search for Prey}]{d) Search for Prey}\par
As we know that the grey wolves finishes their hunt by attacking the prey. In mathematical model we have A ? is a random variable having values in the interval [-2a, 2a] where a is decreased from 2 to 0 over the course of iterations. When the random value of A ? are in [-1, 1] then the next position of search agent is between its current position and position of prey. The pseudo code of the GWO algorithm is presented in Figure 1.  IV. 
\section[{Simulation Results}]{Simulation Results}\par
In order to demonstrate the performance of the GWO algorithm, two testing systems consisting of a 5unit and 10-unit generating system with non-smooth cost functions are taken into account. The GWO algorithm is implemented in MATLAB 2016a on a Pentium IV personal computer with a processor speed of 3.6 GHz and 4 GB RAM. The time horizon for scheduling is one day divided into 24 periods every one hour. The iteration performed for each test case is 1000 for the 5-unit system and 500 for the 10-unit system; and the number of search agents (population) taken in both test cases is 30. 
\section[{a) Test System 1}]{a) Test System 1}\par
In this section a 5-unit system is tested considering the valve-point effects, the ramp rate limits, and transmission losses. All technical data generating units are given in Appendix, which is taken from \hyperref[b15]{[16]}. The optimal dispatch of real power for the given scheduling horizon using the proposed GWO algorithm is given in Table \hyperref[tab_1]{1}. Figure 2 shows the convergence characteristic of GWO technique for DED problem. The comparison results between the proposed GWO algorithm and other methods are shown in Table \hyperref[tab_2]{2}. It is clear that the proposed GWO algorithm has achieved lower minimum production cost.  \hyperref[b15]{[16]} 47356 APSO \hyperref[b24]{[25]} 44678 DE \hyperref[b16]{[17]} 43213 ICA \hyperref[b24]{[25]} 43117.05 PSO \hyperref[b18]{[19]} 50124 HHS \hyperref[b25]{[26]} 43154.8554 ABC \hyperref[b19]{[20]} 44045.83 GSO \hyperref[b27]{[28]} 43414.12 AIS \hyperref[b24]{[25]} 44385.43 GWO 42709.4563 In this section a 10-unit system is tested considering the valve-point effects, the ramp rate limits, and transmission losses. All technical data generating units are adopted from \hyperref[b29]{[30]}, as given in Appendix. The optimal dispatch of real power for the given scheduling horizon using proposed GWO algorithm is given in Table \hyperref[tab_3]{3}. Table \hyperref[tab_4]{4} shows hourly production cost and power loss obtained from GWO algorithm. Figure  {\ref 3} shows the cost convergence characteristic of GWO technique for 10-unit system. The comparison of different methods with the proposed GWO algorithm in terms of the best cost is given in Table \hyperref[tab_5]{5}. Clearly from the results, the proposed GWO algorithm produces a higher quality solution in terms of minimum production costs.  \hyperref[b26]{[27]} 2596847.38 PSO \hyperref[b26]{[27]} 2580148.25 MBFA \hyperref[b26]{[27]} 2544523.21 AIS \hyperref[b26]{[27]} 2500684.32 GWO 2463046.3595 c) DED with wind power In testing the following system, wind power connected to the network is considered. The total installed capacity of wind power connected to the network is 100 MW, with a total of 50 wind turbines \hyperref[b10]{[11]}. The best results obtained from the proposed GWO technique for the DED model without and with wind power are summarized in Table \hyperref[tab_6]{6}. The cost convergence characteristics of the DED model with wind power for the two systems are shown in Figures  {\ref 4 and 5}, respectively.\par
To realize the rationality of the integration of wind power into the power system, the comparison results of the two DED models are presented in Table \hyperref[tab_6]{6}.\par
From Table \hyperref[tab_6]{6}, it can be seen that when compared to the DED model without wind power for the 5-unit system, the savings in operating costs per day are obtained 2780.5154 \$ and transmission losses reduced by 25.7935 MW (down 13.2982\%). For the 10-unit system, the operating cost savings per day were 128069.3605 \$ and transmission losses were reduced by 121.0233 MW (9.2037\% decrease).   
\section[{Conclusion}]{Conclusion}\par
This paper has successfully applied the GWO algorithm to solve the DED problem. Different constraints such as the valve-point effects, ramp rate limits, and transmission loss are taken into consideration to solve the DED problem without and with wind power. The feasibility of the proposed method was demonstrated with 5-unit and 10-unit generating system and compared with other optimization methods reported in the literature. The results obtained show that the GWO algorithm has a much better performance in terms of minimum production cost. The main advantage of the proposed GWO algorithm is the good ability to find the best solution.      \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.85\textwidth}}
Initialize the grey wolf population X i (i=1, 2, ..., n)\\
Initialize a, A, and C\\
Calculate the fitness of each search agent\\
X a = the best search agent\\
X ß = the second best search agent\\
X ? = the third best search agent\\
while (t < Max number of iterations)\\
for each search agent\\
Update the position of the current search agent by equation (19)\\
end for\\
Update a, A, and C\\
Calculate the fitness of all search agents\\
Update X ? , X ? , and X ?\\
t=t+1\\
end while\\
Return X ?\\
( ) Volume XX Issue IV Version I\\
of Researches in Engineering\\
Global Journal\end{longtable} \par
 
\caption{\label{tab_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.024096385542168672\textwidth}P{0.10542168674698794\textwidth}P{0.1180722891566265\textwidth}P{0.11927710843373494\textwidth}P{0.11987951807228917\textwidth}P{0.11927710843373494\textwidth}P{0.14698795180722893\textwidth}P{0.09698795180722891\textwidth}}
H\tabcellsep P1 (MW)\tabcellsep P2 (MW)\tabcellsep P3 (MW)\tabcellsep P4 (MW)\tabcellsep P5 (MW)\tabcellsep Cost (\$)\tabcellsep Ploss (MW)\\
1\tabcellsep 27.4519\tabcellsep 98.5642\tabcellsep 112.6621\tabcellsep 124.9061\tabcellsep 50.0400\tabcellsep 1290.9632\tabcellsep 3.6243\\
2\tabcellsep 40.8780\tabcellsep 20.6864\tabcellsep 112.6565\tabcellsep 124.8953\tabcellsep 139.7611\tabcellsep 1377.0230\tabcellsep 3.8773\\
3\tabcellsep 10.0011\tabcellsep 93.0222\tabcellsep 112.4978\tabcellsep 124.6033\tabcellsep 139.6305\tabcellsep 1390.6017\tabcellsep 4.7549\\
4\tabcellsep 60.1566\tabcellsep 98.3944\tabcellsep 112.6397\tabcellsep 124.8896\tabcellsep 139.7547\tabcellsep 1585.5829\tabcellsep 5.8351\\
5\tabcellsep 10.0244\tabcellsep 88.7822\tabcellsep 112.0767\tabcellsep 124.8338\tabcellsep 228.9681\tabcellsep 1617.1250\tabcellsep 6.6853\\
6\tabcellsep 50.1727\tabcellsep 98.5283\tabcellsep 112.7020\tabcellsep 124.9175\tabcellsep 229.5269\tabcellsep 1781.1620\tabcellsep 7.8474\\
7\tabcellsep 73.6823\tabcellsep 98.4360\tabcellsep 112.6268\tabcellsep 209.7858\tabcellsep 139.7856\tabcellsep 1784.5556\tabcellsep 8.3165\\
8\tabcellsep 12.3970\tabcellsep 98.7988\tabcellsep 112.6697\tabcellsep 209.8054\tabcellsep 229.5890\tabcellsep 1798.0200\tabcellsep 9.2598\\
9\tabcellsep 49.5491\tabcellsep 98.5680\tabcellsep 112.6757\tabcellsep 209.7783\tabcellsep 229.5974\tabcellsep 1978.6326\tabcellsep 10.1685\\
10\tabcellsep 72.2391\tabcellsep 20.0936\tabcellsep 112.6555\tabcellsep 209.8019\tabcellsep 300.0000\tabcellsep 2135.0457\tabcellsep 10.7901\\
11\tabcellsep 74.9901\tabcellsep 22.4924\tabcellsep 123.6426\tabcellsep 210.0779\tabcellsep 300.0000\tabcellsep 2244.7025\tabcellsep 11.2030\\
12\tabcellsep 74.9978\tabcellsep 124.6737\tabcellsep 112.6965\tabcellsep 209.7741\tabcellsep 229.5776\tabcellsep 2180.7454\tabcellsep 11.7197\\
13\tabcellsep 64.1287\tabcellsep 98.5337\tabcellsep 112.5886\tabcellsep 209.8145\tabcellsep 229.4943\tabcellsep 1997.0867\tabcellsep 10.5597\\
14\tabcellsep 49.6763\tabcellsep 98.5417\tabcellsep 112.6029\tabcellsep 209.7535\tabcellsep 229.5338\tabcellsep 1978.2501\tabcellsep 10.1681\\
15\tabcellsep 12.4498\tabcellsep 98.6583\tabcellsep 112.8169\tabcellsep 209.8146\tabcellsep 229.5189\tabcellsep 1797.7365\tabcellsep 9.2584\\
16\tabcellsep 21.4368\tabcellsep 98.5737\tabcellsep 112.7391\tabcellsep 124.9316\tabcellsep 229.5195\tabcellsep 1654.7180\tabcellsep 7.2007\\
17\tabcellsep 11.9769\tabcellsep 83.8383\tabcellsep 30.9181\tabcellsep 208.9142\tabcellsep 229.6487\tabcellsep 1660.5675\tabcellsep 7.2962\\
18\tabcellsep 42.6229\tabcellsep 21.2725\tabcellsep 112.7108\tabcellsep 209.8011\tabcellsep 229.5037\tabcellsep 1797.6510\tabcellsep 7.9110\\
19\tabcellsep 12.5602\tabcellsep 98.5976\tabcellsep 112.7763\tabcellsep 209.8092\tabcellsep 229.5146\tabcellsep 1797.6550\tabcellsep 9.2580\\
20\tabcellsep 64.1452\tabcellsep 98.4801\tabcellsep 112.6121\tabcellsep 209.8090\tabcellsep 229.5131\tabcellsep 1997.1149\tabcellsep 10.5595\\
21\tabcellsep 54.9786\tabcellsep 20.3704\tabcellsep 174.9802\tabcellsep 209.8063\tabcellsep 229.4998\tabcellsep 2086.0725\tabcellsep 9.6354\\
22\tabcellsep 47.2316\tabcellsep 98.4822\tabcellsep 112.6528\tabcellsep 124.8810\tabcellsep 229.5265\tabcellsep 1773.6759\tabcellsep 7.7741\\
23\tabcellsep 56.9070\tabcellsep 98.5339\tabcellsep 112.6500\tabcellsep 124.9057\tabcellsep 139.7739\tabcellsep 1581.7362\tabcellsep 5.7705\\
24\tabcellsep 10.0019\tabcellsep 80.8739\tabcellsep 112.2489\tabcellsep 124.8239\tabcellsep 139.5408\tabcellsep 1423.0320\tabcellsep 4.4894\\
\tabcellsep \tabcellsep \multicolumn{2}{l}{Total cost \& losses}\tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{42709.4563 193.9628}\end{longtable} \par
 
\caption{\label{tab_1}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.16190476190476188\textwidth}P{0.2833333333333333\textwidth}P{0.12142857142857141\textwidth}P{0.2833333333333333\textwidth}}
Method\tabcellsep Fuel cost (\$)\tabcellsep Method\tabcellsep Fuel cost (\$)\\
SA\tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_2}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.85\textwidth}}
Year 2020\\
35\\
of Researches in Engineering ( ) Volume XX Issue IV Version I F\\
H 1 150.0153 135.1646 81.6951 P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) 78.1106 171.7151 157.6784 130.0000 120.0000 21.1887 10.0431 P9 (MW) P10 (MW) 2 150.0339 135.0000 88.1448 99.2764 210.5885 159.5589 130.0000 120.0000 21.5715 18.2262 3 150.0220 135.4325 145.4896 143.4040 242.7314 160.0000 130.0000 120.0000 48.7274 10.6402 4 150.0218 136.1829 226.6413 212.9782 243.0000 160.0000 130.0000 120.0000 39.1587 23.4546 5 150.0237 138.2234 262.7324 217.9014 242.8597 160.0000 129.9846 119.7323 75.2897 22.6374 6 150.1772 137Global Journal\\
© 2020 Global Journals\end{longtable} \par
 
\caption{\label{tab_3}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.02725450901803607\textwidth}P{0.2282565130260521\textwidth}P{0.16182364729458917\textwidth}P{0.042585170340681364\textwidth}P{0.22995991983967934\textwidth}P{0.1601202404809619\textwidth}}
H\tabcellsep Cost (\$)\tabcellsep Ploss (MW)\tabcellsep H\tabcellsep Cost (\$)\tabcellsep Ploss (MW)\\
1\tabcellsep 60618.6976\tabcellsep 19.6109\tabcellsep 13\tabcellsep 141137.7122\tabcellsep 84.4640\\
2\tabcellsep 64038.9120\tabcellsep 22.4001\tabcellsep 14\tabcellsep 121076.9722\tabcellsep 70.5684\\
3\tabcellsep 71273.7775\tabcellsep 28.4469\tabcellsep 15\tabcellsep 104451.0947\tabcellsep 58.3985\\
4\tabcellsep 79124.4204\tabcellsep 35.4374\tabcellsep 16\tabcellsep 87490.3301\tabcellsep 43.5525\\
5\tabcellsep 83318.3071\tabcellsep 39.3846\tabcellsep 17\tabcellsep 83283.8259\tabcellsep 39.3461\\
6\tabcellsep 91979.1098\tabcellsep 48.0402\tabcellsep 18\tabcellsep 91920.5131\tabcellsep 48.0091\\
7\tabcellsep 97395.8194\tabcellsep 52.9469\tabcellsep 19\tabcellsep 104451.0565\tabcellsep 58.3995\\
8\tabcellsep 104451.4185\tabcellsep 58.4004\tabcellsep 20\tabcellsep 141139.6650\tabcellsep 84.4762\\
9\tabcellsep 121076.9983\tabcellsep 70.5655\tabcellsep 21\tabcellsep 121076.9045\tabcellsep 70.5676\\
10\tabcellsep 141138.1957\tabcellsep 84.4707\tabcellsep 22\tabcellsep 91902.9362\tabcellsep 48.0466\\
11\tabcellsep 152498.7538\tabcellsep 92.0713\tabcellsep 23\tabcellsep 75066.7055\tabcellsep 31.7975\\
12\tabcellsep 165433.1451\tabcellsep 100.0750\tabcellsep 24\tabcellsep 67701.0884\tabcellsep 25.4654\end{longtable} \par
 
\caption{\label{tab_4}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.3090909090909091\textwidth}P{0.5409090909090909\textwidth}}
Method\tabcellsep Fuel cost (\$)\\
GA\tabcellsep \end{longtable} \par
 
\caption{\label{tab_5}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{P{0.20218446601941747\textwidth}P{0.24344660194174755\textwidth}P{0.06601941747572815\textwidth}P{0.2640776699029126\textwidth}P{0.07427184466019418\textwidth}}
Models\tabcellsep \multicolumn{2}{l}{5-unit system Fuel cost (\$) Ploss (MW)}\tabcellsep \multicolumn{2}{l}{10-unit system Fuel cost (\$) Ploss (MW)}\\
DED without wind power\tabcellsep 42709.4563\tabcellsep 193.9628\tabcellsep 2463046.3595\tabcellsep 1314.9416\\
DED with wind power\tabcellsep 39928.9419\tabcellsep 168.1693\tabcellsep 2334976.9990\tabcellsep 1193.9183\\
V.\tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_6}Table 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{A4} \par 
\begin{longtable}{P{0.03299086757990867\textwidth}P{0.03299086757990867\textwidth}P{0.13778538812785388\textwidth}P{0.056278538812785384\textwidth}P{0.08247716894977168\textwidth}P{0.103824200913242\textwidth}P{0.12905251141552512\textwidth}P{0.1261415525114155\textwidth}P{0.03978310502283105\textwidth}P{0.10867579908675798\textwidth}}
1\tabcellsep \tabcellsep 410\tabcellsep 7\tabcellsep 626\tabcellsep 13\tabcellsep 704\tabcellsep 19\tabcellsep \tabcellsep 654\\
2\tabcellsep \tabcellsep 435\tabcellsep 8\tabcellsep 654\tabcellsep 14\tabcellsep 690\tabcellsep 20\tabcellsep \tabcellsep 704\\
3\tabcellsep \tabcellsep 475\tabcellsep 9\tabcellsep 690\tabcellsep 15\tabcellsep 654\tabcellsep 21\tabcellsep \tabcellsep 680\\
4\tabcellsep \tabcellsep 530\tabcellsep 10\tabcellsep 704\tabcellsep 16\tabcellsep 580\tabcellsep 22\tabcellsep \tabcellsep 605\\
5\tabcellsep \tabcellsep 558\tabcellsep 11\tabcellsep 720\tabcellsep 17\tabcellsep 558\tabcellsep 23\tabcellsep \tabcellsep 527\\
6\tabcellsep \tabcellsep 608\tabcellsep 12\tabcellsep 740\tabcellsep 18\tabcellsep 608\tabcellsep 24\tabcellsep \tabcellsep 463\\
Unit\tabcellsep P i,min (MW)\tabcellsep P i,max (MW)\tabcellsep R i,up (MW/h)\tabcellsep R i,down (MW/h)\tabcellsep a i (\$/MW 2 hr)\tabcellsep b i (\$/MWhr)\tabcellsep c i (\$/hr)\tabcellsep e i (\$/hr)\tabcellsep f i (rad/MW)\\
1\tabcellsep 150\tabcellsep 470\tabcellsep 80\tabcellsep 80\tabcellsep 0.1524\tabcellsep 38.5397\tabcellsep 786.7988\tabcellsep 450\tabcellsep 0.041\\
2\tabcellsep 135\tabcellsep 470\tabcellsep 80\tabcellsep 80\tabcellsep 0.1058\tabcellsep 46.1591\tabcellsep 451.3251\tabcellsep 600\tabcellsep 0.036\\
3\tabcellsep 73\tabcellsep 340\tabcellsep 80\tabcellsep 80\tabcellsep 0.0280\tabcellsep 40.3965\tabcellsep 1049.9977\tabcellsep 320\tabcellsep 0.028\\
4\tabcellsep 60\tabcellsep 300\tabcellsep 50\tabcellsep 50\tabcellsep 0.0354\tabcellsep 38.3055\tabcellsep 1243.5311\tabcellsep 260\tabcellsep 0.052\\
5\tabcellsep 73\tabcellsep 243\tabcellsep 50\tabcellsep 50\tabcellsep 0.0211\tabcellsep 36.3278\tabcellsep 1658.5692\tabcellsep 280\tabcellsep 0.063\\
6\tabcellsep 57\tabcellsep 160\tabcellsep 50\tabcellsep 50\tabcellsep 0.0179\tabcellsep 38.2704\tabcellsep 1356.6592\tabcellsep 310\tabcellsep 0.048\\
7\tabcellsep 20\tabcellsep 130\tabcellsep 30\tabcellsep 30\tabcellsep 0.0121\tabcellsep 36.5104\tabcellsep 1450.7045\tabcellsep 300\tabcellsep 0.086\\
8\tabcellsep 47\tabcellsep 120\tabcellsep 30\tabcellsep 30\tabcellsep 0.0121\tabcellsep 36.5104\tabcellsep 1450.7045\tabcellsep 340\tabcellsep 0.082\\
9\tabcellsep 20\tabcellsep 80\tabcellsep 30\tabcellsep 30\tabcellsep 0.1090\tabcellsep 39.5804\tabcellsep 1455.6056\tabcellsep 270\tabcellsep 0.098\\
10\tabcellsep 10\tabcellsep 55\tabcellsep 30\tabcellsep 30\tabcellsep 0.1295\tabcellsep 40.5407\tabcellsep 1469.4026\tabcellsep 380\tabcellsep 0.094\\
\tabcellsep \tabcellsep \multicolumn{6}{l}{Table A-5: Load demand for 24 hours (10-unit system)}\tabcellsep \tabcellsep \\
\multicolumn{2}{l}{Time}\tabcellsep Load\tabcellsep Time\tabcellsep Load\tabcellsep Time\tabcellsep Load\tabcellsep Time\tabcellsep \tabcellsep Load\\
\multicolumn{2}{l}{(h)}\tabcellsep (MW)\tabcellsep (h)\tabcellsep (MW)\tabcellsep (h)\tabcellsep (MW)\tabcellsep (h)\tabcellsep \tabcellsep (MW)\\
1\tabcellsep \tabcellsep 1036\tabcellsep 7\tabcellsep 1702\tabcellsep 13\tabcellsep 2072\tabcellsep 19\tabcellsep \tabcellsep 1776\\
2\tabcellsep \tabcellsep 1110\tabcellsep 8\tabcellsep 1776\tabcellsep 14\tabcellsep 1924\tabcellsep 20\tabcellsep \tabcellsep 1972\\
3\tabcellsep \tabcellsep 1258\tabcellsep 9\tabcellsep 1924\tabcellsep 15\tabcellsep 1776\tabcellsep 21\tabcellsep \tabcellsep 1924\\
4\tabcellsep \tabcellsep 1406\tabcellsep 10\tabcellsep 2022\tabcellsep 16\tabcellsep 1554\tabcellsep 22\tabcellsep \tabcellsep 1628\\
5\tabcellsep \tabcellsep 1480\tabcellsep 11\tabcellsep 2106\tabcellsep 17\tabcellsep 1480\tabcellsep 23\tabcellsep \tabcellsep 1332\\
6\tabcellsep \tabcellsep 1628\tabcellsep 12\tabcellsep 2150\tabcellsep 18\tabcellsep 1628\tabcellsep 24\tabcellsep \tabcellsep 1184\end{longtable} \par
 
\caption{\label{tab_7}Table A - 4 :}\end{figure}
 			\footnote{© 2020 Global Journals} 		 		\backmatter  			 
\subsection[{Appendix}]{Appendix}			 			  				\begin{bibitemlist}{1}
\bibitem[Cheng and Zhang ()]{b8}\label{b8} 	 		‘A dynamic economic dispatch model incorporating wind power based on chance constrained programming’.  		 			W Cheng 		,  		 			H Zhang 		.  	 	 		\textit{Energies}  		2015. 8 p. .  	 
\bibitem[Attaviriyanupap et al. ()]{b20}\label{b20} 	 		‘A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function’.  		 			P Attaviriyanupap 		,  		 			H Kita 		,  		 			E Tanaka 		,  		 			J Hasegawa 		.  	 	 		\textit{IEEE Transactions on Power Systems}  		2002. 17  (2)  p. .  	 
\bibitem[Berahmandpour et al. ()]{b6}\label{b6} 	 		‘A new method for real time economic dispatch solution including wind farm’.  		 			H Berahmandpour 		,  		 			. M Sh 		,  		 			H Kuhsari 		,  		 			Rastegar 		.  	 	 		\textit{Renewable Energy Research and Applications (RERA)},  				2020. 1 p. .  	 
\bibitem[Jiang et al. ()]{b9}\label{b9} 	 		‘A novel improved particle swarm optimization approach for dynamic economic dispatch incorporating wind power’.  		 			W Jiang 		,  		 			Z Yan 		,  		 			Z Hu 		.  	 	 		\textit{Electric Power Components and Systems},  				2011. 39 p. .  	 
\bibitem[Balamurugan and Subramanian ()]{b17}\label{b17} 	 		‘An improved differential evolution based dynamic economic dispatch with nonsmooth fuel cost function’.  		 			R Balamurugan 		,  		 			S Subramanian 		.  	 	 		\textit{Journal of Electrical Systems}  		2007. 3  (3)  p. .  	 
\bibitem[Somuah and Khunaizi ()]{b13}\label{b13} 	 		‘Application of linear programming re-dispatch technique to dynamic generation allocation’.  		 			C B Somuah 		,  		 			N Khunaizi 		.  	 	 		\textit{IEEE Transactions on Power Systems}  		1990. 5  (1)  p. .  	 
\bibitem[Victoire and Jeyakumar ()]{b21}\label{b21} 	 		‘Deterministically guided PSO for dynamic dispatch considering valve-point-effect’.  		 			T. Aruldoss Albert Victoire 		,  		 			A Ebenezer Jeyakumar 		.  	 	 		\textit{Electric Power Systems Research}  		2005. 73  (3)  p. .  	 
\bibitem[Balamurugan and Subramanian ()]{b16}\label{b16} 	 		‘Differential evolution-based dynamic economic dispatch of generating units with valve-point effects’.  		 			R Balamurugan 		,  		 			S Subramanian 		.  	 	 		\textit{Electric Power Components and Systems},  				2008. 36 p. .  	 
\bibitem[Hindi and Ghani ()]{b12}\label{b12} 	 		‘Dynamic economic dispatch for large scale power systems: a Lagrangian relaxation approach’.  		 			K S Hindi 		,  		 			M R Ghani 		.  	 	 		\textit{Electrical Power \& Energy Systems}  		1991. 13  (1)  p. .  	 
\bibitem[Hemamalini and Simon ()]{b19}\label{b19} 	 		‘Dynamic economic dispatch using artificial bee colony algorithm for units with valve-point effects’.  		 			S Hemamalini 		,  		 			S Simon 		.  	 	 		\textit{European Transactions on Electrical Power}  		2011. 21 p. .  	 
\bibitem[Han et al. ()]{b0}\label{b0} 	 		‘Dynamic economic dispatch: feasible and optimal solutions’.  		 			X S Han 		,  		 			H B Gooi 		,  		 			D S Kirschen 		.  	 	 		\textit{IEEE Transactions on Power Systems}  		2001. 16  (1)  p. .  	 
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\end{bibitemlist}
 			 		 	 
\end{document}
