An Adaptive Modified Firefly Optimisation Algorithm ...

Energy 51 (2013) 339e348

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An Adaptive Modified Firefly Optimisation Algorithm based on Hong’s Point Estimate Method to optimal operation management in a microgrid with consideration of uncertainties Sirus Mohammadi a, *, Babak Mozafari a, Soodabeh Solimani a, Taher Niknam b a b

Department of Electrical Engineering, Science and Research branch, Islamic Azad university, Tehran, Iran Department of Electronic and Electrical Engineering, Shiraz University of Technology, Modares, Shiraz, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 September 2012 Received in revised form 15 December 2012 Accepted 17 December 2012 Available online 16 January 2013

A probabilistic Energy Management system to optimize the operation of the Micro-Grid (MG) based on an efficient Point Estimate Method (PEM) is proposed in this paper. This method is used to model the uncertainty in the power generation of the wind farms and the Photovoltaic (PV) systems, the market prices and the load demands. PEMs constitute a remarkable tool to handle stochastic power system problems because good results can be achieved by using the same routines as those corresponding to deterministic problems, while keeping the computational burden low. For a system with m uncertain parameters, it uses 2m þ 1 calculations of cost function to calculate the statistical moments of cost function solution distributions by weighting the value of the solution evaluated at 2m þ 1 locations. The moments are then used in the probability distribution fitting. The normal, Beta and Weibull distributions are used to handle the uncertain input variables. Moreover, an Adaptive Modified Firefly Algorithm (AMFA) is employed to achieve an optimal operational planning with regard to cost minimization. Performance of the proposed method is verified using a typical MG. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Microgrid Point estimate Adaptive Modified Firefly Algorithm Operation

1. Introduction An MG (Micro-Grid) is an electric power distribution network with distributed Generations (DGs) capable of operating as a single controllable system. Consumers prefer DG units, which are due to their potential lower cost, higher service reliability, higher power quality, increased energy efficiency and energy independence [1]. Renewable energy sources like solar and wind can be interfaced through the DG modules with the MG system, which can operate in islanded mode and grid-connected mode [2]. The primary problems associated with the wind and solar energies are due to the nature of the sources, which are both time-varying and difficult to predict [3]. Moreover, there are uncertainties in the future load characteristics and market price. Several studies have been conducted to optimize the operation, load dispatch and management of Energy Storage Systems (ESS) of the MGs. These studies have neglected the uncertainties which exist in these parameters. Several researches have been performed to optimize the operation, load dispatch and management of ESS of the MGs. Linear * Corresponding author. Tel./fax: þ98 711 8427590. E-mail addresses: [email protected], (S. Mohammadi).

[email protected]

0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.12.013

programming algorithm is used in Ref. [4] to optimize MG operation cost and battery charge states. Chen et al. accordingly proposed a matrix real-coded genetic algorithm optimization module to achieve a practical method of smart management of ESS and load dispatch in MGs [5]. In the proposed operation management (OM) [6]a neutral network is incorporated to forecast power generation of a PV (Photovoltaic) energy source. Maximizing benefits owing to the energy pricing differences between on-peak and off-peak periods is gotten by electrical and thermal storage charge scheduling in Ref. [7]. The main shortcoming of the pervious studies on the energy management of MGs is neglecting the uncertainties which exist in the generation patterns, the future load characteristics and market price. Although employing Renewable Energy Sources obviates environmental concerns and fossil fuel consumption, they introduce uncertain power because of the stochastic wind and solar variation [8]. Moreover, in an open access power market, the degree of uncertainty of the load forecast error and market price can be even more perceptible [9]. By considering the unpredictable characteristics of wind turbine (WT) and PV power generation [10] along with the uncertainty of market price and load demand, the necessity of a deep investigation with a stochastic structure becomes more evident.

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Nomenclature The ith control vector/firefly The Lth element of the ith control vector/firefly the dimension of the control vector g absorption coefficient r distance between any two fireflies b0 initial attractiveness at r ¼ 0 ptGi , ptsj active power output of ith generator and jth storage device at time t active power bought/sold from/to the utility at time t ptGrid BtGt , Btsj bid of the ith DG source and jth storage device at hour t bid of utility at hour t BtGrid start up/shut-down costs for ith DG unit and jth SGi, Ssj storage device The amount of Dth load level pLD ptG;min , ptG;max minimum and maximum active power production of ith DG at hour t pts;min pts;max minimum and maximum active power production of jth storage at hour t ptgrid;min , ptgrid;max minimum and maximum active power production of the utility at hour t t ,W t1 Battery energy storage at time t and t1 Wess ess PCharge (Pdischarge) permitted rate of charge (discharge) through a definite period of time Dt Xi xi,L d

There are several techniques to deal with problems of uncertainty. The three main approaches are analytical, simulation and approximate methods [11]. The vast majority of techniques have been analytically based and simulation techniques have taken a minor role in specialized applications. The main reason for this is that simulation generally requires large amounts of computing time, whereas analytical models have been sufficient to provide planners and designers with the results needed to make objective decisions [12]. Moreover, analytical models require some mathematical assumptions in order to simplify the problem. Simulation methods estimate uncertainty by simulating the actual process and random behavior of the system. The main drawback of the simulation is the great number of simulations required to attain convergence. Approximate methods give an approximate description of the statistical properties of output random variables. These methods provide a satisfactory balance between speed and precision. The PEM (Point Estimate Method) can be used to calculate the statistical moments of a random quantity that is a function of one or several random variables and had been used in the transfer capability uncertainty computation [13]. The main advantage of point estimate is the use of deterministic routines for solving probabilistic problems and overcoming the difficulties associated with the lack of perfect knowledge of the probability functions of stochastic variables, since these functions are approximated using only their first few statistical moments (mean, variance, skewness, and kurtosis). The first PEM was developed by Rosenblueth in 1975 [13] for symmetric variables and was later revisited [14] to consider asymmetric variables. Since then, several methods that improve the original Rosenblueth’s method have been presented [15]. In practical power system problems, the number of input random variables (IRVs) involved is high [16]. Therefore, the Rosenblueth’s original methods, as well as recent and more accurate (PEMs) based on the Rosenblueth’s approach, is not appropriate because the number of simulations could be even greater than those in the Monte Carlo simulation [17]. Moreover, the number of

hcharge (hdischarge) charge (discharge) efficiency of the battery Wess,min (Wess,max) lower and upper bounds on battery energy storage Pcharge,max (Pdischarge,max) maximum rate of charge (discharge) during definite period of time Dt The scheduled spinning reserve at time t Rt m number of the input random variable of 2 m þ 1 method m,s the mean and the standard deviation v wind speed c the scale parameter of Weiball function j the shape parameter of Weiball function location parameter of Weiball function v0 the cut-in speed of wind vci the cut-out speed of wind vco the wind speed vr the rated output power of WT pr t, k time interval and iteration index n total number of optimization variables NT total number of hours total number of generation and storage units Ng, Ns total number of load levels ND status of unit i at hour t uti

simulations to be performed by using the PEMs developed by Hong [18] or Harr [19] grows linearly with the number of IRVs. Although Harr’s method is appropriate for correlated variables, it is limited to symmetric variables. In this study, one of Hong’s PEMs is devised to optimize the OM of MGs under uncertainty parameters. Mostly, the 2m PEM has been used to deal with probabilistic load-flow computation [20]. The 2m scheme generally fails to give satisfactory results when the number of uncertain input variables is high [21].

2. Hong’s PEM Hong’s PEMs focus on the statistical information provided by the first few central moments of a problem IRV on k points for each variable, named concentrations. By using these points and the function F, which relates input and output variables, information about the uncertainty associated with problem output random variables can be obtained. The kth concentration (pl,k ,wl,k)of a random variable pl can be defined as a pair composed of a location pl,k and a weight wl,k. The location pl,k is the kth value of variable pl at which the function F is evaluated. The weight wl,k is a weighting factor which accounts for the relative importance of this evaluation in the output random variables. By using Hong’s methods, the function F has to be evaluated only k times for each IRV pl at the k points made up of the kth location pl,k of the IRV pl and the mean (m) of the m1 remaining input variables, i.e., at the k points. In other words, the deterministic problem has to be solved k times for each IRV pl, and the difference among these problems is the deterministic value pl,k assigned to pl, while the remaining IRVs are fixed to their corresponding mean. The number k of evaluations to carry out depends on the scheme used. Therefore, the total number of evaluations of F is k  m. Specific schemes of Hong’s PEM take into account one more evaluation of function F at the point made up of the mIRVs means (mp1,mp2,.,mpl,.,mpm). Therefore, for these schemes, the total number of evaluations ofF is k  m þ 1.

S. Mohammadi et al. / Energy 51 (2013) 339e348

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Let pl be a random variable with Probability Distribution Function (PDF) fpl, then the k concentrations (pl,k,wl,k) of the mIRVs are obtained from the statistical input data. The location pl,k to be determined is [21]:

pl;k ¼ mpl þ xl;k spl

(1)

The weight wl,k and xl,k are obtained by solving the nonlinear system of equations [21]

8 K  j P > > > w x ¼ ll;j ; j ¼ 1; .; 2k  1 > < k ¼ 1 l;k l;k

(2)

> K   > P 1 > > wl;k xl;k ¼ : m k¼1

Fig. 1. 2m þ 1 scheme.

This system can be solved by the procedure developed by Ref. [22]. In this system ll,j is the ratio of the jth moments about the mean of pl toðsk Þj , that is

M ðp Þ

ll;j ¼  j l j spl

(3)

ZN 

j pl  mpl fpl dpl

(4)

N

Note that ll,1 ¼ 0,ll,2 ¼ 1 and ll,3,ll4 are, respectively, the skewness and kurtosis of pl. Once all the concentrations (pl,k,wl,k)are obtained, the function F is evaluated at the points(mp1,mp2,.,ml,k,.,mpm) yieldingZ(l,k), where Z is the vector of output random variables. The jth moment of the output random variables. m X K m X K   X X E Zj y wl;k  ½Zi ðl; kÞj ¼ wl;k l¼1 k¼1

l¼1 k¼1

ij h   Fi mp1 ; .; pl;k ; .; mpm (5)

The Standard Deviation (SD) of the Z is

sZ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi   varðzÞ ¼ E Z 2  ðEðZÞÞ2

(6)

2.1. 2m þ 1 scheme(K ¼ 3 and xl,3 ¼ 0) 2m þ 1 scheme is described with the aid of Fig. 1.If three concentrations (K ¼ 3) are used for each random variable, and one of the locations of the concentrations is fixed at its mean value, we can match only the first four moments of the marginal PDF of the random variables. The solution of the system is [20]:

xl;k ¼

ll;3 2

þ ð1Þ

3k

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ll;4  l2l;3 ; k ¼ 1; 2 and xl;3 ¼ 0 4 (7)

It can be seen that the standard location values of this scheme do not depend on the number of IRVs. Weights are:

wl;k ¼

w0 ¼

m X l¼1

where

Mj ðpl Þ ¼

By noting that m of 3m concentrations are located in the same point (mp1,mp2,.,ml,k,.,mpm) with the sum of the weights equal to w0:

ð1Þ3k   xl;k xl;1  xl;2

k ¼ 1; 2 and

wl;3 ¼

1  m l

l;4

1  ll;3

wl;3 ¼ 1 

m X

1

l ¼ 1 ll;4

 ll;3

2

(9)

Load demand, WT& PV power generation and market price are some of the most uncertain variables in the new deregulated power systems especially in the MGs. For instance, the uncertainty of load can root from many different variables like weather conditions, temperature variations, humidity, and programs pursued by the governments. A deterministic analysis must be run for each point (mp1, mp2, .,mpl, ., mpm). Note that a deterministic routine may be used to carry out the computations because only deterministic values are involved. The cost function solution can be expressed as follows:

  Zðl; kÞ ¼ Fi mp1 ; mp2 ; .; pl;k ; .mpm

(10)

The function F transfers the uncertainty from the IRVs to the output random variables and Z(l,k) is the vector of output random variables associated with the Kth concentration of random variable pl. The vector of output random variables is used to estimate the raw moments of the output random variables as follows:

EðZÞyEðZÞ þ wl;k Zðl; kÞ     E Z j yE Z j þ wl;k ðZðl; kÞÞj

(11)

The process ends once all the concentrations of all IRVs are taken into account. In order to obtain the PDF and the Cumulative Density Function (CDF) of the output random variables, the Grame Charlier expansion is used [23].To improve the optimization process, AMFA (an Adaptive Modified Firefly Algorithm) is utilized. 3. Original firefly algorithm (OFA) The OFA is inspired from the behavior of firefly insects in the summer of tropical areas. FA is a meta-heuristic optimization algorithm, first introduced at Cambridge University based on three key ideas [24]: 1) Given that any two fireflies may be attracted by each other, all fireflies are supposed to be unisex. 2) The firefly with less brightness is attracted toward the firefly with more brightness. 3) In the case that no firefly with more brightness is recognized, the firefly can move randomly in the search space.

2

(8)

FA is a population-based optimization algorithm which has many similarities with the other population-based algorithms such

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as Artificial Bee Colony, Particle Swarm Optimization and Bacteria Foraging Optimization. However, the existence of some characteristics like low dependability of the algorithm on the adjusting parameters, the appropriate ability of local search and the simplicity of both idea and implementation distinguish FA from the rest. 3.1. Distance between fireflies The distance among the fireflies in the air is just similar to the distance among the fireflies’ vectors in the optimization search space. In this paper, the distance between ith and jth firefly is calculated in the Cartesian framework as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d uX   2 rij ¼ Xi  Xj  ¼ t xi;L  xj;L

(12)

L¼1

3.2. Attractiveness of firefly Indeed, as the distance between two fireflies increases, less light can be seen by the fireflies. In order to simulate this firefly’s behavior, any monotonically decreasing function as Eq. (13) can be used [25]:

bðrÞ ¼ b0  expðgrm Þ; m  1

(13)

In each iteration, after the steps mentioned in 3.2e3.4 are implemented, the total firefly population should be improved as follows: Suppose the best and the worst individual of the firefly popuIter & X Iter , respectively. For each firefly lation in each iteration is XBest worst i in the population, three fireflies Xq1, Xq2and Xq3 are selected from the fireflies’ population randomly such that q1 s q2 s q3 s i. Now, the following two new individuals are generated:

  XMute1 ¼ Xq1 þ D  Xq2 Xq3  Iter  X Iter XMute2 ¼ XMute1 þ D  XBest Worst

 XBest;1 ¼ xBest;1 ; xBest;2 ; .; xBest;d xImprove1;j ¼

xImprove2;j ¼

One of the main ideas of FA is that the firefly with less brightness is attracted to the firefly with more brightness. The movement of the jth firefly (less brightness) toward the ith firefly (with more brightness) is mathematically formulated as follows:

xImprove3;j ¼

(14)

This equation consists of three segments:  The first segment is the current position of the jth firefly.  The second segment simulates the brightness of the ith firefly seen by the jth firefly.  The third segment allows the jth firefly to move randomly in the entire search space when no brighter firefly is visible around it. The constant value a as the randomization parameter is in the range of (0.1). According to recent works, it is shown that the tuning of the two variables b0 and g depends on the characteristics of the investigated problem. Consequently, there is no accurate formulation to adjust these parameters to all types of optimization problems. Meanwhile, some keys are regarded in all cases. As the distance between two fireflies increases, each firefly’s attractiveness reduces in the other’s view. Similarly, as the absorption coefficient g reaches zero, the attractiveness coefficient (b) moves to b0. Consequently, the distance between two fireflies does not have any effect on the light intensity which simulates a local or global solution. This limiting feature corresponds to the OFA [26]. 3.4. Modified Firefly Algorithm FA has many benefits that make it a powerful tool in optimization applications. However, a new modification process for improving the total ability of FA in both local and global exploration as well as reducing the possibility of trapping in local optimal is proposed. The key idea in the proposed modification process is based on

(15)

Where D is a random number in the range [0, 1]. Now using, XMute1 & XMute2, the following five fireflies are produced:



3.3. Movement of the fireflies

  Xj ðtÞ ¼ Xj þ b0  expðgr m Þ  Xi  Xj þ uj  1 uj ¼ a rand  2

 Improving the diversity of the population through 2 mutations and 3 cross over operations.  Encouraging the total firefly population to move toward the best promising local or global individual.

xImprove4;j ¼

xMute1;j ; xBest;j ;

if k1  k2 if k1 > k2

(16)

xMute1;j ; xj ;

if k3  k2 if k3 > k2

(17)

xBest;j ; xj ;

if k4  k3 if k4 > k3

xMute1;j ; xMute2;j ;

if k5  k4 if k5 > k4

XImprove;5 ¼ j  XWorst þ z  ðXBest  XWorst Þ

(18)

(19) (20)

Where k1,k2,k3,k4,k5,j and z are random values in the range [0, 1]. The best firefly among XImprove,1, XImprove,2, XImprove,3, XImprove,4, XImprove,5 is compared with the ith firefly (Xi). If it is better than Xi, it replaces X; otherwise, it will remain in its position. 3.5. Adaptive tuning of a or adaptive MFA Eq. (21) showed that the randomization parameter (a) is utilized to control the algorithm for a random search while the neighboring fireflies are not seen by the given firefly. In fact, a manages the random movement of each firefly which is randomly chosen in the range of [0, 1]. A large a encourages the algorithm to search for the optimum solution through the faraway search space while a small a facilitates the local search. Therefore, an appropriate value for the randomization parameter (a) can give a satisfactory balance between global and local search. Therefore, in this paper an adaptive control procedure is introduced to improve the total ability of the algorithm for both local and global search. In this regard, by the use of the experience from several running of the algorithm, a heuristic function which changes over each iteration is obtained as follows:

aIterþ1 ¼



1 2kmax

1=kmax

aIter

(21)

Given the above function, the value of a changes as time passes so to provide a good balance between the local and global search during the optimization process.

S. Mohammadi et al. / Energy 51 (2013) 339e348

343

4. Problem formulation

4.2.2. Real power generation capacity

The total operating cost of the MG includes the fuel costs of units as well as their start-up/shut-down costs. The mathematical model of such problem can be expressed as follows.

ptGi;min  ptGi  ptGi;max ptsj;min  ptsj  ptsj;max ptgrid;min  ptGrid  ptgrid;max 4.2.3. Spinning reserve

4.1. Objective function

Ng X

The objective function can be formulated as follows:

Min f ðXÞ ¼

NT X

þ

8 Ng h NT < X X

Costt ¼

t¼1

(25)

:

t¼1

i¼1

i

uti ptGi BtGi þ SGi uti  ut1

i

Ns h

i X

utj ptsj Btsj þ Ssj utj  ut1

þ ptGrid BtGrid j

j¼1

9 = ;

(22)

where X ¼ [X1 X2 . Xt . XNT] and Xt are state variables vector including active powers of units and their related states and can be described as follows:

h

X t ¼ ptG1 ;ptG2 ;.;ptGNg ;pts1 ;pts2 ;.;ptsNs ;ut1 ;ut2 ;.;utNs þNg

i

(23)

i ¼ 1

uti ptGi;max þ

Ns X j ¼ 1

utj ptsj;max þ ptgrid;max 

t t1 Wess ¼ Wess þ hcharge Pcharge Dt 



1

hdischarge

4.2.1. Power balance

5.1. Wind speed modeling

i¼1

Ns X j¼1

ptsj þ ptGrid ¼

ND X D¼1

PLt D

(24)

Pdischarge Dt

t W Wess;min  Wess ess;max Pcharge;t  Pcharge;max ; Pdischarge;t  Pdischarge;max

5. Modeling the variability of sources

ptGi þ

D ¼ 1

PLt D þ Rt

(26)

4.2.4. Energy storage limits As there are some limitations on charge and discharge rate of storage devices during each time interval, the following equation and constraint can be considered:

4.2. Constraints

Ng X

ND X

(27)

(28)

The Weibull distribution function can be used to simulate the wind speed behavior. The Weibull PDF is formulated as follows [27]:

Fig. 2. A typical LV microgrid.

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S. Mohammadi et al. / Energy 51 (2013) 339e348



Table 1 The limits and bids of the installed DG.

m ¼ lG 1 þ

ID

Type

Min power (kW)

Max power (kW)

Bid (Vct/kWh)

Start-up/shut-down cost (Vct)

1 2 3 4 5 6

MT PAFC PV WT Bat Utility

6 3 0 0 30 30

30 30 25 15 30 30

0.457 0.294 2.584 1.073 0.38 e

0.96 1.65 0 0 0 e

Hour

Price(Ect/kWh)

1 2 3 4 5 6 7 8 9 10 11 12

0.23 0.19 0.14 0.12 0.12 0.20 0.23 0.38 1.50 4.00 4.00 4.00

13 14 15 16 17 18 19 20 21 22 23 24

1.50 4.00 2.00 1.95 0.60 0.41 0.35 0.43 1.17 0.54 0.30 0.26

8  j1  v j > < pr k1 v þ k2 ; vE < vr ¼ ; k1 ¼ &k ¼ k1 vci > Pr ; vr < v  vco vr  vci 2 > : 0; v > vco

(32)

The PDF for the wind output is

(29)

8 Zvci ZN >   > > > > ¼ 0 ¼ pðv  v  v Þ þ pðv  vÞ ¼ f ðvÞdv þ f ðvÞdv p p co > 0 w;i ci > > > > vco v0 > > > > pw  k2 > > > < Zvr Zk1 > pð0 < pwi  pr Þ ¼ pðvci  v < vr Þ ¼ f ðvÞdv ¼ f ðvÞdv > > > > > v v > ci ci > > > Zvco >   > > > p p ¼ pðv ¼ p < v  v Þ ¼ f ðvÞdv > r r co w;i > > : vr

(33)

With m,s as

Fig. 3. The forecasted values of the load demand, market price and PV and WT power production.

S. Mohammadi et al. / Energy 51 (2013) 339e348 Table 3 Comparison of objective function value evaluated in SOP1 for 20 trails (Deterministic Framework).

345

Table 5 Comparison of objective function value evaluated in SOP3 for 20 trails (Deterministic Framework).

Method

Best solution (Vct)

Worst solution (Vct)

Average (Vct)

SD (Vct)

Mean simulation time (Sec)

Method

Best solution (Vct)

Worst solution (Vct)

Average (Vct)

SD (Vct)

Mean simulation time (Sec)

GA [1] PSO [1] FSAPSO [1] CPSO-T [1] CPSO-L [1] AMPSO-T [1] AMPSO-L [1] FA AMFA

277.7444 277.3237 276.7867 275.0455 274.7438 274.5507 274.4317 275.3491 264.7600

304.5889 303.3791 291.7562 286.5409 281.1187 275.0905 274.7318 281.7841 264.7600

290.4321 288.8761 280.6844 277.4045 276.3327 274.9821 274.5643 277.6191 264.7600

13.4421 10.1821 8.3301 6.2341 5.9697 0.3210 0.0921 2.5383 0

e e e e e e e 12.938 8.572

GA PSO FSAPSO FA AMFA

334.8694 327.7211 326.4291 318.6401 299.4124

345.0211 340.3123 335.4931 332.9012 299.4124

336.2912 331.2102 331.4301 322.9102 299.4124

17.6310 13.1244 10.6621 5.1471 0

14.291 14.283 13.281 13.428 8.839

5.3. The PV variability The variability of the PV source is a function of the solar insolation that has been modeled, using a beta distribution function. The corresponding PDF of the variable si is formulated as:

a ¼ m

 ð1  mÞm

s

b ¼ ð1  mÞ

f ðsi Þ ¼



1

ð1  mÞm

s

(34) 1

(35)

sia1 ð1  si Þb1 Gða þ bÞ GðaÞGðbÞ

(36)

6. Simulation results The case study is considered a typical Low Voltage MG as portrayed in Fig. 2.The system data is extracted from [1], where a complete data set can be found. A variety of DG sources, such as Micro Turbine (MT), proton-exchange membrane FC (PEM-FC), WT, PV and NickeleMetaleHydride (NiMH) battery are installed in the network. It is assumed that all DG sources produce active power at unity power factor. Also, the thermal load is not considered in the proposed MG system. Moreover, there is a power exchange link between the utility and the MG during the time step in the study period based on the decisions made by the MG Central Controller (MGCC). Table 1 offers the minimum and maximum production limits and the bid coefficients of the DG units in the MG. Table 2 offers the real-time market energy prices for the examined period of time. The hourly forecasted load demand, the normalized forecasted output power of WT and PV and the hourly forecasted market price for a typical day can be found in Fig. 3. The total load demand within the MG for a typical day is 1695 kW. In this study, the power output of PV and WT are considered equal to their maximum Table 4 Comparison of objective function value evaluated in SOP2 for 20 trails (Deterministic Framework). Method

Best solution (Vct)

Worst solution (Vct)

Average (Vct)

SD (Vct)

Mean simulation time (Sec)

GA [1] PSO [1] FSAPSO [1] FA AMFA

277.7444 277.3237 276.7867 274.3740 261.2340

304.5889 303.3791 291.7562 281.0733 261.2340

290.4321 288.8761 280.6844 276.6021 261.2340

13.4421 10.1821 8.3301 2.7294 0

e e e 13.239 8.738

available power at each hour of the day. The AMFA is implemented to find the optimal solutions of the deterministic and probabilistic OM problem for the mentioned MG. Also to better understand our study, three different Statuses of Operation (SOP) are considered and discussed as described in the following. In the first SOP (SOP1), it is supposed that all the power units are in service. The main purpose of this scenario is to compare the performance of the AMFA with that of other methods mentioned in Ref. [1]. In the second SOP (SOP2), the Battery is similar to SOP1, but the power units are allowed to shut down for flexible OM. In the third SOP (SOP3), similar to SOP2, all the units can be in the ON/OFF mode, but the initial charge of the Battery is zero. Consequently, Battery should be charged in the first hours to be able to discharge in the other hours of the day. All the simulations are carried out by MATLAB 7.12 on a Pentium-IV, core i3, 2.13 GHz PC with 3 GB RAM. 6.1. Deterministic analysis In the deterministic OM of the MGs, it is assumed that the output power of the WT and the PV units are equal to their forecasted values and the reaming value of the load demand is satisfied by the other DG units. At first, to show the superiority of the AMFA over the other methods, the simulation results are implemented in a deterministic environment. Table 3 gives the comparison between the AMFM and the methods presented in Ref. [1]. It can be seen that the proposed AMFA has much better performance and

Table 6 Best solutions obtained for deterministic framework using AMFA (SOP3). Time (Hour)

DG sources (kWh) PV

WT

FC

MT

Battery

Utility

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 0 0 0 0 0 0 0.2000 3.7500 7.5250 10.4500 11.9500 23.9000 21.0500 7.8750 4.2250 0.5500 0 0 0 0 0 0 0

1.7850 1.7850 1.7850 1.7850 1.7850 0.9150 1.7850 1.3050 1.7850 3.0900 8.7750 10.4100 3.9150 2.3700 1.7850 1.3050 1.7850 1.7850 1.3020 1.7850 1.3005 1.3005 0.9150 0.6150

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

20.2150 18.2150 18.2150 19.2150 24.2150 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 6 0

30 30 30 30 30 27.9150 21.7850 16.5050 30 30 28.7750 21.6400 14.1850 18.5800 30 30 7.1030 3.7850 1.3020 4.7850 30 0 1.9150 4.6150

30 30 30 30 30 30 30 30 19.5350 20.6150 30 30 30 30 23.6600 15.5300 29.7680 30 30 30 13.3005 9.6995 30 30

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S. Mohammadi et al. / Energy 51 (2013) 339e348 Table 8 Expected power generation for DGs evaluated by 2m þ 1 method for 20 trails (SOP1). Time (Hour)

DG sources (kWh) PV

WT

FC

MT

Battery

Utility

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 0 0 0 0 0 0 0.2008 3.7496 7.4971 10.4400 11.8757 23.8624 21.1427 7.8647 4.2293 0.5539 0 0 0 0 0 0 0

1.7831 1.7856 1.7808 1.7952 1.7817 0.9178 1.7833 1.3017 1.7883 3.0786 8.8145 10.4399 3.8957 2.3723 1.7800 1.3096 1.7831 1.7835 1.3062 1.7871 1.3002 1.2960 0.9136 0.6148

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 30.0000 30.0000 28.8243 21.4942 14.6001 18.1171 30.0000 30.0000 30.0000 6.0000 6.0000 6.0000 30.0000 30.0000 6.0000 6.0000

15.6094 17.9483 17.7870 16.9868 12.0226 3.6267 2.1581 7.6355 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 29.9350 22.6922 30.0000 30.0000 30.0000 1.8183 10.7289

30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 19.7523 20.6916 29.9971 30.0000 30.0000 30.0000 23.5160 15.7916 6.7684 19.8299 30.0000 19.1881 13.1249 20.3385 30.0000 30.0000

in the day can be seen on the behavior of MT as an expensive power unit. In fact, at low load level, MT is forced to reduce its power production to the minimum value. In contrast, at high load levels where the market bid surpasses the bid offered by the MT, the optimization problem has decided to increase the power production of MT which is completely right. In the SOP2, the MGCC let the MG start-up or shut-down each DG at each of the hours. It is evident that as a result of more flexibility in SOP2, it is expected that total cost would be reduced. The simulation results are shown in Table 4. Similar to SOP1, the satisfying performance of the AMFA over the other algorithms is evident. Also, by comparing the CPU time, it can be seen that the utilization of the proposed Fig. 4. Pseudocode of the 2m þ 1 scheme.

stability in comparison to the other algorithms. By comparing the SD of the results found by AMFA with that of the others, the high stability and reliability of the proposed algorithm can be deduced. From Fig. 3, it is seen that from 1 AM to 6 AM, the MG experiences a low electrical load demand in which the utility sells the electrical energy with lower cost. Therefore, it is beneficial that the NiMHBattery starts to charge at these hours. On the other hand, from about 7 AM, the load demand and the market bid increase. This behavior of the market and load consumption forces the NiMHBattery to start to supply as much as possible so that the power production of the market would be reduced and therefore the total cost is improved. The influence of load variation in different hours

Table 7 Expected value of total cost evaluated by 2m þ 1 method.

SOP1 SOP2 SOP3

The algorithm

Best solution (Vct)

Worst solution (Vct)

Average (Vct)

SD (Vct)

Simulation time (Sec)

FA AMFA FA AMFA FA AMFA

274.4532 260.3290 273.1763 261.925 320.7684 301.039

282.3119 264.1983 280.1639 262.9134 331.1117 301.8623

277.4552 261.8223 276.1352 262.1941 322.1231 300.1213

2.41 0. 012 2.11 0.04 4.22 0.025

0.98 0.78 1.13 1.18 1.44 1.56

Table 9 Expected power generation for DGs evaluated by 2m þ 1 method for 20 trails (SOP2). Time(Hour)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

DG sources (kWh) PV

WT

FC

MT

Battery

Utility

0 0 0 0 0 0 0 0.2008 3.7496 7.4971 10.4400 11.8757 23.8624 21.1427 7.8647 4.2293 0.5539 0 0 0 0 0 0 0

1.7831 1.7856 1.7808 1.7952 1.7817 0.9178 1.7833 1.3017 1.7883 3.0786 8.8145 10.4399 3.8957 2.3723 1.7800 1.3096 1.7831 1.7835 1.3062 1.7871 1.3002 1.2960 0.9136 0.6148

30.0000 26.2081 27.7448 29.7506 30.0000 25.3631 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 23.9104 30.0000 30.0000 30.0000 30.0000 30.0000

0 0 0 0 0 0 0 0 30.0000 30.0000 28.8243 21.4942 14.6001 18.1171 30.0000 30.0000 30.0000 6.9346 6.8226 0 30.0000 30.0000 0 0

9.6094 8.1564 9.5318 10.7375 6.0226 6.5953 8.1581 13.6355 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 28.9634 29.8675 30.0000 30.0000 4.4026 4.7289

30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 19.7523 20.1929 29.9958 30.0000 30.0000 30.0000 23.5160 15.7916 6.7684 18.8303 29.9933 25.3206 13.1249 20.3385 29.5297 30.0000

S. Mohammadi et al. / Energy 51 (2013) 339e348 Table 10 Expected power generation for DGs evaluated by 2m þ 1 method (SOP3). Time (Hour)

DG sources (kWh) PV

WT

FC

MT

Battery

Utility

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 0 0 0 0 0 0 0.2008 3.7496 7.4971 10.4400 11.8757 23.8624 21.1427 7.8647 4.2293 0.5539 0 0 0 0 0 0 0

1.7831 1.7856 1.7808 1.7952 1.7817 0.9178 1.7833 1.3017 1.7883 3.0786 8.8145 10.4399 3.8957 2.3723 1.7800 1.3096 1.7831 1.7835 1.3062 1.7871 1.3002 1.2960 0.9136 0.6148

29.7506 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000

20.3000 17.9029 18.2130 19.0132 23.9774 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 14.3651 0

29.6600 29.8512 30.0000 30.0000 30.0000 27.6267 21.8419 16.3645 29.9466 30.0000 28.8243 21.4942 14.5318 18.1171 30.0000 30.0000 6.1809 4.2351 1.3078 4.8119 28.7301 0.8924 0.1834 4.7289

30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 19.6989 20.1929 29.9958 30.0000 29.9317 30.0000 23.5160 15.7916 29.4124 30.0000 30.0000 30.0000 11.8549 10.5539 30.0000 30.0000

modification as well as the adaptive procedure has enhanced the optimization ability of the proposed algorithm sufficiently. In SOP3, the situation is somewhat more complicated. Since the NiMH-Battery does not have any initial electrical charge, then at first it has to be charged to be able to discharge at the next required hours. The simulation results for SOP3 are shown in Table 5. The optimal operating points of the DGs for SOP3 are shown in Table 6. According to the deterministic results, the satisfying performance of AMFA was deduced.

347

6.2. Probabilistic analysis The 2m þ 1 scheme is implemented to model the uncertainties in the generation patterns, the market prices and the load demand in the mentioned MG. In this scheme, the first and the second moments, E(S1) and E(S2), of the optimal production cost of the OM problem are computed; thereafter, the obtained moments are employed to estimate the solution PDF and the CDF fitting. The algorithm of proposed method is shown in Fig. 4. As explained in Fig. 1, the 2m þ 1 scheme generates two estimated points for each IRV. One of these points value is under the forecasted value of the corresponding IRVs and the other one is over it; then using these points along with the mean location, the first and the second moments of the output random variable are obtained. The coefficient of variation, defined as the SD and mean value ratio, is used to indicate the dispersion of the random variables [20]. Load demand and Market prices are modeled as normal distribution with SD of 5% [12,21]. The PDF of the production cost is considered normal. Table 7 shows the effectiveness of the proposed modifications on the Firefly. From the results summarized in this table, the improvement of the Firefly algorithm, choosing the updating strategy and the dependent random coefficient can be deduced. It can be seen that the consideration effect of uncertainty has increased to values of the objective function in all the cases such that the total expected cost is increased. The small values of the SD of the AMAF in three SOPs confirm their robustness and reliability. In Table 8, for SOP3 the expected power generation for DGs evaluated by 2m þ 1 method is shown for 20 trails in a stochastic environment. It can be seen that the consideration effect of uncertainty has increased to values of the objective function in all the cases such that the total expected cost is increased. From this table, the appropriate performance of the proposed method is evident. Also, in Tables 8e10 the aggregated power production for each of the DGs is shown. Figs. 5 and 6 portray the PDFs and the CDFs for

Fig. 5. PDF and CDF of expected cost for SOP1&2.

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S. Mohammadi et al. / Energy 51 (2013) 339e348

Fig. 6. PDF and CDF of expected cost for SOP3.

Fig. 7. PDF of expected cost for SOP1 using seven-step approximation method.

SOP1, SOP2 and SOP3. Moreover, Fig. 7 shows the PDF for SOP1 using a seven-step approximation method [12]. These figures demonstrate that the presented method could find optimal production cost distribution under uncertain environments. At last, to verify the impact of the considering uncertainties in the decision making, the Expected Value of Perfect Information (EVPI) [28] for the three SOPs are calculated. The EVPI represents how much the MGCC would be willing to spend cost to cover the uncertainty in the stochastic processes under consideration. The EVPI is 5.343, 7.211 and 9.453 for SOP1, SOP2 and SOP3, respectively. 7. Conclusion This paper proposes a probabilistic approach to taking uncertainty in the market prices, the load demand and the electric power generation of the WT and PV units to optimize the OM of the MGs. The 2m þ 1 PEM along with a novel optimization algorithm based on the AMFA has been proposed to optimize the problem. The proposed AMFA makes uses a powerful modification process to enhance the diversity of the firefly population as well as a selfadaptive technique to increase the ability of the algorithm to move toward the promising global optimal solution quickly. The proposed techniques were applied to a typical MG which operates in grid-connected mode and equipped with storage devices. The OM problem of the MG is solved and analyzed in both the deterministic and probabilistic manner. The information obtained would provide system planning engineers with more confidence in making judgments concerning system expansion plans. References [1] Moghaddam A, Seifi A, Niknam T, Alizadeh P. Multi-objective operation management of a renewable MG (micro-grid) with back-up micro-turbine/ fuel cell/battery hybrid power source. Energy 2011;36:6490e507.

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