Differential Evolution

Global optimization problem through continuous areas is often present in the whole of the scientific area. Generally, to optimize a specific characteristic of the system with choosing parameters in the system. To simply and ease, a parameters system generally described as a vector.

The basic approach of the optimization problem is initiated by designing the objective function that could exhibit the objectives problem include with constraint limitations. Mostly the objective function defines the optimization problem as a minimization task. In those example problems, the objective function is precisely considered as a cost function. If the function is nonlinear and non-differentiable, direct search methods are the choice.

After the random variable generated, the decision must be made whether appropriate with derived parameter or not. Most of the standard direct search approach

criterion, all of the latest parameter vectors is feasible if it reduces the value of the cost function. Even though the greedy search works converges quite fast, it has the risk fell in a local minimum. Executing some vectors simultaneously, better parameter settings could help other vectors avoid local minima. Generally, people need the practical minimization method to fulfill four requirements:

(1) Able to handle linear, nonlinear, differentiable, non-differentiable, and multimodal cost functions.

(2) Parallel ability to solve computation with intensive cost functions.

(3) Simple and Ease to use, for example, less organize variables to adjust the optimization. These Variables would also be robust and easy to select.

(4) Well convergence characteristic, for example, consistent convergence to the global minimum in continues stand-alone test.

Having some of the requirements mentioned above, the Differential Evolution was designed to be a stochastic direct search method to meet the first requirement.

Utilize with direct search technique have the benefit of being simply implemented to optimization experiment where the cost value is derived from a physical test rather than a computer simulation.

The second requirement is necessary for demanding computational optimizations, as an instance, the optimal function evaluation may obtain from time variants (hours to days), and it is often implemented in finite element simulation or integrated circuit design. in order to achieve feasible results in an appropriate time consumes, the only reasonable way is to apply a parallel computer. Differential

Evolution meet those demand by implement stochastic variable in a vector population which able to complete independently.

To complete the third requirement, if the optimization method is automatically arranged it will be advantages because only little input is required from the user. The new variable vectors are generated by considering highest variables and shorten around the low variable. The new variable will substitute their old variables if they fit with the function value with reduced cost compared to their previous. These methods allow the search space, i.e. the polyhedron, to expand and contract without special control variable settings by the user.

The last requirement number 4 needs good convergence characteristic, there are mandatory for a well construct optimization algorithm. Although plenty methods found is theoretically express the convergence characteristic of a global minimization method, just comprehensive attempt within several conditions may exhibit whether a minimization approach can meet the requirement.

As a parallel direct search approach, the Differential Evolution starts with a number of population (NP) candidate solutions, which may be represented as

x_i, G ,i=1,2,. . . , NP (14)

where i index denotes the population and G denotes the generation to which the population belongs. The Differential Evolution process depends on the manipulation

In the population for every generation G. NP does not change during the optimization work. The initial variable population is randomly chosen and should cover the entire variable space. Normally, an identical probability distribution for all random variable except the stated variable. If the preliminary solution is available, the first population could be produced by increase the normal distributed random variation to the nominal solution xnom,0. Differential Evolution results in latest variable vectors by including the weighted difference between two population vectors to a third vector. This operation is called by mutation.

Next is the variable mixing is referred by crossover. The mutated variables vector mixed with the variables in the another decisive vector, the target vector, to yield the so-called trial vector.

The last operation is selection, when the test vector results in a lower cost function value than the objective vector, the test vector replaces the objective vector in the next generation. Each population vector has to provide once as the objective vector so that NP tournament occurred in one generation. Next following paragraph explains Differential Evolution’s basic strategy specifically:

1. Mutation

As the main operator of Differential Evolution, the execution of this process causes Differential Evolution dissimilar from other Evolutionary algorithms. The Differential Evolution mutation process uses the vector differentials between the current population variable for specifying both the value and way of perturbation used to the individual subject of the mutation process. The mutation process at each generation

begins by randomly selecting three individuals in the population. For each target vector x_{i ,G ,i}=1,2,. . . , NP , a mutant vector is generated according to

x_{i ,G +1}=x_{r 1 ,G}+F .(x_{r 2 ,G}−x_{r 3 ,G}) (15)

with random indexes r 1, r 2 , r3 ∈ {1, 2, . . . , NP}, integer, mutually different and F > 0. The randomly chosen integers r 1, r 2 , r3 are also chosen to be different from the running index i, so that NP must be greater or equal to four to allow for this condition. F is a real and constant factor ∈ [0, 2] which controls the amplification of the differential variation (xr 2,G−xr 3, G) .

2. Crossover

After the mutation process done, next proceed to the crossover operation. The perturbed individual, vi , G+1=(v_{1 i , G+1}+v_{2i , G+1}, .. . , v_{¿,G +1}) , and the previous population member xi ,G +1=(x_{1 i ,G +1}+x_{2 i , G+1}, .. . , x_{¿,G +1}) ,are subject to the crossover operation, that finally generates the population of candidates, or “trial” vectors

ui ,G+1=(u1 i ,G +1+u2 i ,G +1, . . ., u¿,G +1)

uj , i .G +1=

{

^{vj ,i .G +1if randx}j ,i . G+1^{j≤Cr⩗ j=k}

(16)

Where, j = 1. . . n, k ∈ {1, . . ., n} is a random parameter’s index, chosen once for each i, and the crossover rate, Cr ∈ [0, 1], the other control parameter of Differential Evolution, is set by the user. Figure 11 below showed example of crossover process in 6 dimension vectors.

Fig 11. Crossover Process Illustration Using 6 Variable Dimension.

3. Selection

In Differential Evolution the selection operation also discrete with others variant of Evolutionary Algorithm. The following population generation is picked from the individual in current population and its corresponding trial vector as to the rule below:

x_{i. G+ 1}=

{

^{ui .G+1if f (uxi .G+1}i .G ^{)≤ f ( xi . G)}

(17)

Thus, every individual in the test population appealed with their counterpart in the current population. Lowest individual objective function value will survive from the competition selection to the population in the next generation. As a result, all the individuals of the next generation are as good or better than their counterparts in the current generation. In Differential Evolution the test vector is not compared against all

the individuals in the current generation, but only against one individual, its counterpart, in the current generation.

CHAPTER 3 PROPOSED APPROACH

3.1 Model Description

The objective in this proposed approach is to implement the renewable energy and hybrid system into smart grid management. With the randomness of the wind and solar, the grid system should able to cover all of the load demand with the help of storage system. The wind and solar generated output matrices will match with load variance matrices to find the efficient component sizing should be installed [6]. The difference value from power generation load demand will consider the storage system should supply the load demand or charging. When the generation power exceeds more power, the surplus energy will manage to store in the storage system. However, when the generating power needs extra energy the storage system will support to supply load demand.

Figure 12. Hybrid Renewable System in Smart Grid Interconnection.

The considering the load demand should be helpful to avoid the peak average rate and maintain the power system in balance, the distribution of load demand also takes into account for this simulation as the load demand management in power scheduling method. In this approach, that load distribution will represent as the load shifting. The flexibility and the effectivity of the storage system will also have determined by the rate of load shifting and the charging/discharging storage.

Using Evolutionary Algorithm such as Genetic Algorithm or Differential Evolution would be suitable to attain the optimal capacity of installed wind energy source, solar energy source, and storage system capacity. Initially, the huge population will be constrained, selected from the feasible area and evaluated by the fitness function, then the crossover and mutation function will proceed to achieve the optimal solution.

The randomness of renewable energy such as wind and solar would be difficult for the evolutionary algorithm to find the optimal solution. The Evolutionary Algorithm will face to the huge computational burden for the uncertain result and will be difficult to converge to global optima. To minimize that computational burden, we need to perform some statistical methods or simulation to achieve the point estimation from the uncertain model then the optimal solution of renewable energy able to solve with evolutionary algorithm [9]. The details of proposed approach illustrated in the flowchart below and will explain in the next paragraph.

Figure 13. Flowchart Proposed Optimization Method in GA.

First of all, the input data historical solar irradiance, wind speed, and load demand are used to find the probability model which is represent the nature of wind

No

Evaluating Fitness Function, Constraints and calc max(Sc) Penalty Factor

speed, solar irradiance, and load demand [13]. Using curve fitting we can obtain the probability model of the nature source that will we use to determine the component sizing for each scale of the wind and solar generating power. For the load demand, the normal distribution or Gaussian distribution is applied to model lower bound and upper bound of load variance. Having the parameters from the curve fitting we can generate the pseudo random number with the probability function. Those random number generated will reassemble the nature data that fitted in curve fitting.

Wind Speed Data (mph)

Figure 14. Fitting Wind Distribution.

With Weibull distribution the wind distribution fitting will obtain parameters as follow:

Log likelihood: -59.8279

Mean: 5.02301 Variance: 11.0614

Parameter Estimate Std. Err.

α 5.5816 0.7782 β 1.54152 0.250919

Estimated covariance of parameter estimates:

A B

A 0.605595 0.0611139

B 0.0611139 0.0629604

Solar Radiation (KW)

Figure 15. Fitting Solar Distribution.

Log likelihood: -165.429 Domain: 0 < y < Inf Mean: 362.564 Variance: 127224

Parameter Estimate Std. Err.

α 1.03324 0.263384 β 350.9 113.87

Estimated covariance of parameter estimates:

a b

a 0.0693711 -23.5592

b -23.5592 12966.4

Load Demand (MW)

Figure 16. Fitting Load Distribution.

Log likelihood: -236.221 Domain: -Inf < y < Inf

Mean: 21289.6

mu 21289.6 949.002 sigma 4649.14 693.053

Estimated covariance of parameter estimates:

mu sigma mu 900606 8.88776e-11 sigma 8.88776e-11 480323

Proceed to the next step, the number generated from those random value can be calculated as the energy input for the wind turbine and PV arrays and load variance as the function described in chapter 2. The calculation of the generated probability density function will have little differences. Therefore, to avoid the Evolutionary Algorithm in high computational burden which will make difficult to find global optima we need to sample the generation output and take the estimation point or the most frequent appears.

The simple way we could use Mode or Multimodal to take the sample value from the generating power which depends on from the probability function. The other approach we could use the Monte Carlo Simulation to take a sample in detailed analysis.

The Mode is the number of most often appears in the data sampling. Mode with discrete probability distribution is a number of x which have probability mass function in the maximum value or sometimes called by maximum likelihood sample.

Similar to the statistical method, average and median, the numerical value of the mode is similar with mean and median in the normal distribution, and might be different in highly inclined distributions. Having data sample [5,5,5,6,7,8,4,6,7,8,6,3,6], the Mode

value will result with 6, if we set the most value more than 2 modes it called multimodal and for those set data will result 6 and 5.

Mode example:

Sample N = [ Pw1, Pw 2, Pw 2, Pw 2, Pw 3, Pw 4 ] (18)

Mode = Pw 2

Multimodal example:

Sample N = [ Pw1, Pw 2, Pw 2, Pw 2, Pw 3, Pw 3, Pw 4, Pw 5 ] (19) Mutimodal = Pw 2 & Pw3

Monte-Carlo simulation is one of tool to perform the analytical simulation method, this method is simple and quite accurate to find the estimation point of the probability. Monte-Carlo methods are well known and computationally intensive family of algorithms that has been heavily used in the last few decades for solving a broad class of physical, mathematical and statistical problems. In Monte-Carlo simulation, random numbers are sampled from the probability distributions output. Each set of samples Pw 1 is iterating until Pw n and the resulting yields from the sample is registered. The historical data of the wind and solar resources as input is fitted by curve fitting and model the random number that prior to distribution function and normalized as power output. Next the Monte-Carlo simulation is proceeding the simulation of power generation output and take the sample to define the estimation point f (x1, ..., xn;

generation, the estimation point is use for variable to decide the scaling parameters of the generation and calculate storage system needed with the load shifting allowable on the system.

Sample = [ Pw 1, Pw 2, Pw 3, … Pw n]

thus (20)

Ew = Number of success Pw 1, Pw 2, Pw3, … Pwn ntotal sampling

After we obtain the estimation point of the output generation we use that estimation point into the searching variable. Where the basic objective function actually is to minimize all of the cost which is also include the capital expense whether installation cost IEw , IEpv , IEs and operation cost OEw , OEpv ,OEs . The αw , αpv value is determine how many wind and solar generating power scale should be installed, S is declare what size of storage capacity should be installed. The EP∧P is indicate the balance cost system of energy and power rating, then the base optimal capacity and cost function is:

F(x )=Min {(IEw +OEw)∙αw +(IEpv+OEpv)∙ αpv+(IEs +OEs)∙ S+ EP+P } (21)

Where:

Ew = expense for each installed Wind Generating Scale ($)

Epv = expense for each installed Solar Generating Scale ($) Es = expense for each installed storage capacity ($)

IEw = Installation Cost of Wind Generation ($)

IEpv = Installation Cost of PV Generation ($) IEs = Installation Cost of Storage System ($)

OEw = Operation Cost of Wind Generation ($) OEpv = Operation Cost of PV Generation ($) OEs = Operation Cost of Storage ($)

αw = Installed Wind Generation Scale

αpv = Installed Solar Generation Scale S = Installed Storage Capacity (MW)

EP = Power Rating Cost ($) P = Power Rating (MW)

Because most of the utility company include the installation and operating cost become one as expense of the component, thus we can simplify by:

F(x )=Min {Ew ∙ αw+Epv ∙ αpv+ Es∙ S+EP+P } (22)

The result of variable αw , αpv , Pwt , Ppvt is determine from the random estimation of wind and solar radiation input.

Pwt=Pw ∙αw (23)

Ppvt=Ppv ∙ αpv (24)

Where:

Pw = Estimation Number of Wind Power (MW)

Ppv = Estimation Number of Solar Power (MW) αw = Scale of Wind Generate Power Rate Needed

Pwt = Total Wind Output Generated (MW) Ppvt = Total Solar Output Generated (MW)

For the balance of power system, the both of renewable energy should capable of supplying the variance of load demand, as illustrated on the figure 12. The generated power from the wind and solar power is obtained from the wind and the solar probability before, then the generated output will be matched with load variance. Both of renewable energy also need to generate extra energy in order to deliver the energy surplus ES into storage system as the system grid already fulfilled, then we need to apply the following constraint:

The variance of load demand Lv value is obtained from the random number with Gaussian distribution which is have specific upper and lower total load value of a day. To planning the good power generation, the total power output should also capable to supply in every predicted situation, as for illustration showed on figure below:

Object 282

Figure 17. Ideal Energy Generation Planning.

In term of minimization, the result for solar generation is almost always not preferred due with the same cost amount the wind will have better output. We can observe those of evidence in stochastic modeling of [5]. Having huge both of energy resources the wind become preferable than solar. Because of those condition in this research we try to make the solar generation take into account, consider the output of solar generation should be useful when there no windy days or in some area which does not have strong wind but has a lot of solar radiance. For those following criteria we add constraint below which is Lmin is act as minimum rate (in percentage) of load demand should be able covered by solar generation:

∑

t =1 T

Ppvt ≥(

∑

t =1 T

Lv )∙ Lmin (27)

The difference from output power and load will determine whether storage will charge or discharge. That decision is calculated with Pwt + Ppvt – Lv > 0,

when the energy surplus is below than zero Pwt + Ppvt – Lv < 0, and the energy storage at the time is larger than the storage minimum capacity (Sc > Smin) it will allow to discharge.

Smin < Sc < Smax (28)

The maximum of storage capacity will determine by charging or discharging value, the illustration showed in figure 18. the number energy stored is not all amount of the energy surplus, the loses energy transfer from the grid will affect the stored energy.

Then the efficiency of the storage system will be implemented in the calculation system.

With d as allowed rate value of charging/discharging (in percentage) and ηs as the efficiency of storage, the equation to determine storage capacity in charging and discharging is:

Charging Pwt + Ppvt – Lv > 0

Sc=(1−d ) Smin+ηs(Pwt +Ppvt−Lv +Lst) (29)

Discharging Pwt + Ppvt – Lv < 0

Sc=(1−d ) Smin+(Pwt+Ppvt −Lv +Lst) (30)

Where:

Sc = Storage Capacity (MW)

d = Charging/Discharging Rate Value (%) ηs = Efficiency of Storing (%)

Lst = Load Shifting (MW)

Figure 18. Storage Sizing Illustration.

The other case, when the storage is below of their minimal capacity or even empty, it is mean the load demand is higher and need to use advantages of load shifting as the implementation of demand scheduling method. The rate of allowable load shifting is also determining how much storage capacity will decide, the larger load shifting rate allowed the storage capacity needed will be lower. The load shifting will simply add as constraints in the simulation, with Lm as the modified load after some load move on the next time (t), Lnext as the next total load after added by moved load.

The shifted load should be positive and also should have the limit rate shifting allowed which is expressed all below:

Lm=Lv−Lst (31)

Lnext=Lv+Lst (32)

0 ≤ Ls ≤ δ ∙ Lv (33)

Min Capacity Max Capacity 2 Max Capacity 1

Object 336

Figure 19. Illustration of Modified Load in the Load Variance.

The total amount of load shifting should not violate the maximum allowable rate. In the other hand the shifted load will move into next time load variance, thus the total load in the next time is sum of the load on that day itself and the moved load from last time. In this condition, the moving load should not make the load in the next time larger than the variance of the daily load.

Finally, the result is taken from decision variable that shows value for the wind, solar, and storage capacity at the time, how much load shifting that executed due to

Finally, the result is taken from decision variable that shows value for the wind, solar, and storage capacity at the time, how much load shifting that executed due to

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