Optimal probabilistic energy management in a typical micro-grid

Energy Conversion and Management 95 (2015) 314–325

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Optimal probabilistic energy management in a typical micro-grid based-on robust optimization and point estimate method Seyed Arash Alavi ⇑, Ali Ahmadian, Masoud Aliakbar-Golkar Faculty of Electrical Engineering, K. N. Toosi University of Technology, Seyyedkhandan, Dr. Shariati Ave., Tehran, Iran

a r t i c l e

i n f o

Article history: Received 6 December 2014 Accepted 11 February 2015 Available online 28 February 2015 Keywords: Uncertainty Robust optimization Point estimate method Energy management Micro-grid

a b s t r a c t Uncertainty can be defined as the probability of difference between the forecasted value and the real value. As this probability is small, the operation cost of the power system will be less. This purpose necessitates modeling of system random variables (such as the output power of renewable resources and the load demand) with appropriate and practicable methods. In this paper, an adequate procedure is proposed in order to do an optimal energy management on a typical micro-grid with regard to the relevant uncertainties. The point estimate method is applied for modeling the wind power and solar power uncertainties, and robust optimization technique is utilized to model load demand uncertainty. Finally, a comparison is done between deterministic and probabilistic management in different scenarios and their results are analyzed and evaluated. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The energy system studies include a wide range of issues from short term (e.g. real-time, hourly, daily and weekly operating decisions) to long term horizons (e.g. planning or policy making) [1]. Optimal operation of Micro-Grid (MG) is a managing optimization problem that should be managed by system operator with regard to the technical and economic constraints [2]. In today’s MG, due to many economic, environmental, and technical factors, the penetration of Distributed Energy Resources (DERs) such as Wind Turbines (WTs), Photo Voltaic cells (PVs), Micro-Turbines (MTs), biomass, and so on, have rapidly been increased in distribution networks [3]. On the other hand, renewable energy-based Distributed Generations (DGs) are always accompanying with uncertainty due to their dependency to the natural fluctuations. In addition, the prediction of system next day load demand always has errors due to the emergence of different consumers [4]. Therefore, the uncertainty modeling becomes the most consequential issue that system operators and administrators are faced with. Although the exact modeling of generation and consumption uncertainties has a high effect on optimizing of the operation strategy, many references solved this problem deterministically [5–9]. A comprehensive method for optimal power management of stand-alone hybrid energy system is proposed in [10], where the uncertainties are modeled using various possible scenarios for ⇑ Corresponding author. Tel.: +98 21 84062163; fax: +98 21 88462066. E-mail address: [email protected] (S.A. Alavi). http://dx.doi.org/10.1016/j.enconman.2015.02.042 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

wind speed and solar irradiation based on Weibull and Beta Probability Distribution Functions (PDFs), respectively. The differential evolution algorithm is used to handle the nonlinear optimization problem. A similar problem is presented in [11], where adaptive chaos clonal evolutionary programming is used to solve it and the uncertainty of renewables is modeled by fuzzy sets. However, the proposed method in [10,11] is appropriate for optimal scheduling and design of battery units only in stand-alone autonomous hybrid energy systems. The optimal management of batteries in MG is represented in [12,13]. Although both references employ Point Estimate Method (PEM) to investigate the uncertainty of wind and solar energy directly in cost objective function, no one considers power loss and reliability cost in the objective function. The uncertainty of wind power is modeled based on dependable capacity concepts in [14] but uncertainty of PVs is ignored. Moreover, none of the aforementioned papers consider load demand uncertainty. In addition, due to environmental perspective, the pollution of diesel generator should be considered in energy management procedure, because of its significant effects on the cost of operation. In this paper, an optimal strategy is proposed in order to perform an operation on a typical MG including renewable DGs (wind and solar power), conventional DGs (MT and diesel generator) and batteries. Furthermore, the PEM is applied for modeling the wind and solar power uncertainties according to Weibull and Beta PDFs, respectively and Robust Optimization (RO) is used to model the load demand uncertainty due to lack of the relevant PDFs in this case. The objective function of the problem is modeled as the

S.A. Alavi et al. / Energy Conversion and Management 95 (2015) 314–325

315

Nomenclature X OF(t) T NDG NST NLD NEV NF Nbus NPVs NPVp CF OPR t CF EMI t CF RLB t C grid;t PGrid;t xout ði; kÞ

vector of the optimization variables objective function ($) total number of hours total number of distributed generations total number of storage systems total number of load demands total number of failure events total number of feeders total number of busses number of parallel cells in PV module number of series cells in PV module operation cost of MG at hour t emission cost of MG at hour t reliability cost of MG at hour t price of utility at hour t active power bought/sold from/to the utility at hour t binary variable associated to the ith load demand outage due to kth failure event kk average failure rate of kth failure event CO cost function of costumer outage ($) hi type of the ith load demand rk outage time of kth failure event (h) Pf ðtÞ probability of failure event at tth hour Rð:Þ standard ramp function PL;i ðtÞ power demand of the ith load demand at hour t (kW) PRES restored power of the ith load demand by all the units m;i (kW) PGi ðtÞ active power output of ith DG at time t (kW) Psj ðtÞ active power output of jth storage at time t (kW) PLoss;f ðtÞ lost power of each feeder f at hour t(kW) PGi;min ðtÞ minimum active power production of ith DG at hour t PGi;max ðtÞ maximum active power production of ith DG at hour t Psj;min ðtÞ minimum active power production of jth storage at hour t Psj;max ðtÞ maximum active power production of jth storage at hour t PGrid;min ðtÞ minimum active power production of the utility at hour t PGrid;max ðtÞ maximum active power production of the utility at hour t Pcharge;t permitted rate of charge through t Pdischarge;t permitted rate of discharge through t Pcharge;max maximum rate of charge of battery Pdischarge;max maximum rate of discharge of battery SOCðtÞ battery state of charge at time t SOC min lower bounds on battery state of charge SOC max upper bounds on battery state of charge V bus;min minimum allowed operation voltage (p.u.) V bus;max maximum allowed operation voltage (p.u.)

operation, pollution and reliability cost of the MG that is optimized for 24-day ahead operation. In addition, the proposed MG can be operated in islanding mode if it is technically and economically practicable. The optimal operation problem is modeled as an optimization problem and it is solved using Particle Swarm Optimization (PSO) algorithm under technical constraints. The remainder of the paper is organized as follows: in section 2 the optimal energy management of the MG and uncertainty modeling are formulated. The proposed methodology and numerical studies are presented in sections 3 and 4, respectively. Finally, the conclusion remarks are given in section 5.

V bus;y ðtÞ If ;z ðtÞ If ;max C Fuel C o&m C st C Inv C CO2 C SO2 C NOx CO2;i ðtÞ

voltage of yth bus at time t current of zth feeder at time t maximum current of each feeder fuel cost ($) operation and maintenance cost ($) start up cost ($) investment cost ($) externality cost of generating CO2 ($/lb) externality cost of generating SO2 externality cost of generating NOx carbon dioxide pollutants of ith DG unit at hour t (lb/kW h) SO2;i ðtÞ sulfur dioxide pollutants of ith DG unit at hour t (lb/kW h) NOx ðtÞ nitrogen oxide pollutants of ith DG unit at hour t (lb/kW h) Pmt ðtÞ active power generated by micro-turbine at time t Pwt ðtÞ active power generated by wind turbine at time t PDg ðtÞ active power generated by diesel generator at time t aDg ; bDg ; cDg cost coefficients of diesel generator Prated Dg PR;w V ci Vr V co d n CF wt C wc;battery C Pow:Elec:

gbat Pbat ðtÞ V base Dt PPV ðtÞ PPV ;STC GT ðtÞ GT STC

c

T jSTC Tj T amb NOCT C SollarPanel PFCAP DG;i;L PFCAP ST;i;L

maximum active power generated by diesel generator nominal active power of wind turbine cut in speed of the wind turbine rated speed of the wind turbine cut-off speed of the wind turbine interest rate life time of unit capacity factor of wind turbine battery wear cost ($) cost of power electronic ($) charge and discharge efficiency of the battery active charge and discharge of battery bus voltage of battery time step active power generated by PV at time t maximum test power for the STC (standard test conditions) solar radiation on tilted module plane (W/m2) of PV at time t solar radiation for STC (standard test conditions) power-temperature coefficient reference cell temperature (°C) of PV cell temperature of PV (°C) environmental temperature normal operating cell temperature (°C) solar panel cost ($/W) useful free capacity of the Lth DG source for ith load demand restoration (kW) useful free capacity of the Lth storage unit for ith load point restoration (kW)

2. Problem statement This paper presents a short-term energy management scheme in a typical MG including renewable and conventional DGs. The objective function is minimizing the MG total operational cost (generation cost of DGs, the cost of purchasing energy from the upstream grid and the reliability cost) and the pollution cost. The cost of expected Energy Not Supplied (ENS) is considered as the reliability cost that it is depended on the load demand type and interruption duration. After the failure event, MG operates in islanding mode and supplies the load demand optimally if it is

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justified economically and technically, otherwise the load demand will be interrupted. Therefore, the ENS computation depends on islanding mode operations in failure event period [15]. Moreover, in ENS computing, it is assumed that just feeders have failure rates and busses failure events are ignored. The decision variables are the optimal amounts of output power in each DG, the battery scheduling as well as the amounts of exchanging energy with upstream network in each interval of operating period, which is formulated in Eq. (1), and it should be solved under technical constraints:

Min:f ðXÞ ¼

T X OF t

ð1Þ

t¼0

OF t ¼ CF OPR þ CF EMI þ CF RLB t t t CF OPR ¼ t

NDG NST X X CF OPR CF OPR i;t þ j;t þ C grid;t P Grid;t i¼1

CF EMI ¼ t

ð2Þ

ð3Þ

j¼1

jPdischarge;t j 6 Pdischarge;max

ð10Þ

– The battery State of Charge (SOC) should not be violated a pre-determined maximum and minimum value [16].

SOC min 6 SOCðtÞ 6 SOC max

8t 2 T

ð11Þ

– The battery is authorized to change its charging and discharging state just one time per operation period [17]. 2.1.4. MG technical constraints According to Eqs. (12) and (13), the busses voltage and feeders current should not be violated an authorized value during the MG optimal operation.

V bus;min 6 V bus;y ðtÞ 6 V bus;max 0 6 If ;z ðtÞ 6 If ;max

y ¼ 1; 2; . . . ; Nbus

z ¼ 1; 2; . . . ; NF

8t 2 T

8t 2 T

ð12Þ ð13Þ

ð4Þ 2.2.1. MT cost modeling

XX

EMI The MT operating cost ðCF OPR Mt;t þ CF Mt;t Þ includes fuel, maintenance, start up, investment and pollution costs can be obtained as follows [18]:

(

rk T X X xout ði; kÞ  kk  COðhi ; r k Þ  Pf ðtÞ

i2NLD k2N EV

R½PL;i ðt þ lÞ 

X

)

t¼1 l¼1

PRES m;i ðt; l; r k Þ

ð5Þ

m2NDS

In Eq. (1), X T ¼ ½X 1 ; X 2 ; . . . ; X T  is the variables state vector that includes active power producing by each DG, charge and discharge power of battery and the active power exchanging with the upstream distributed network. Moreover, in Eq. (5), N DS ¼ fN DG ; N ST g is the set of total number of DGs and storage systems. 2.1. Constraints

2.1.1. Power balance constraint The total generated power in each interval should be equal to the total load demands, power stored in the battery bank and the total losses of feeders as Eq. (6): NDG NST NLD NF X X X X PGi ðtÞ þ Psj ðtÞ þ P Grid ðtÞ ¼ PLk ðtÞ þ PLoss;f ðtÞ : j¼1

CF OPR Mt;t ¼ C Fuel;mt þ C o&m;mt þ C st;mt þ C inv ;mt

k¼1

8t

f ¼1

2T

ð6Þ

ð15Þ 2.2.2. Diesel generator cost modeling EMI The diesel generator operating cost ðCF OPR Dg;t þ CF Dg;t Þ includes fuel and pollution costs and can be formulated as follows [11]:

( C Fuel;mt ¼

ð16Þ

aDg P2Dg þ bDg PDg þ cDg

if 0 < PDg < Prated Dg

0

if P Dg ¼ 0

 PDg ðtÞ

ð18Þ

2.2.3. Wind generation cost modeling According to the ‘‘speed-power’’ characteristic of WT (Fig. 1), the output power of WT depends on wind speed that can be formulated as Eq. (19) [19]:

2.1.2. Power generation constraints The maximum and minimum generated power of each unit should be limited according to Eqs. (7)–(9).

8t 2 T;

8i 2 NDG

Psj;min ðtÞ 6 PSj ðtÞ 6 Psj;max ðtÞ 8t 2 T; 8j 2 NST PGrid;min ðtÞ 6 P Grid ðtÞ 6 PGrid;max ðtÞ 8t 2 T

ð7Þ ð8Þ ð9Þ

2.1.3. Battery constraints In order to perform an optimal operation and increase the battery lifetime, the following constraints should be considered:

ð17Þ

CF EMI Dg;t ¼ ðC CO2  CO2Dg ðtÞ þ C SO2  SO2Dg ðtÞ þ C NOx  NOxDg ðtÞÞ

where Psj ðtÞ and P Grid ðtÞ can be negative or positive in observing and injecting power periods respectively.

PGi;min ðtÞ 6 PGi ðtÞ 6 PGi;max ðtÞ

ð14Þ

CF EMI Mt;t ¼ ðC CO2  CO2mt ðtÞ þ C SO2  SO2mt ðtÞ þ C NOx  NOxmt ðtÞÞ  P mt ðtÞ

CF OPR Dg;t ¼ C Fuel;Dg

The typical MG should operate under technical constraints as follows:

i¼1

jPcharge;t j 6 Pcharge;max ;

2.2. MG operation cost modeling

NDG X CF EMI i;t i¼1

CF RLB ¼ t

– According to Eq. (10), the charge and discharge rate of battery in each one-hour interval of whole operation period should be in a limited boundary.

Fig. 1. Speed-power characteristic of WT.

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S.A. Alavi et al. / Energy Conversion and Management 95 (2015) 314–325

8 0 > >   > > 2

> P R;w > > : 0

if

v < V ci

if V ci < v < V r if V r < v < V co if

ð19Þ

v > V co

A, B and C are coefficients which are extracted from [19]: The cost of generating wind power includes investment cost, operation and maintenance cost. It can be formulated as MT cost and computed as Eq. (20):

during the faults. This operation scenario of battery is similar to DGs except that the available power of battery during failure events depends on the amounts of its stored power in past. The battery should be discharged during the faults and supply the load buses according to their priority. The power which will be produced by battery in Eq. (5), computed in Eq. (27).

PRES ST;i ðt; l; r k Þ ¼

X

¼

dð1þdÞ I  ð1þdÞ þ C o&m;wt n 1

P R;w  CF wt  8760

 Pwt  t

¼

0;

ð21Þ

The battery SOC is modeled according to Eq. (22) [21].

  Pbat ðtÞ  Dt SOC ðt þ Dt Þ ¼ SOCðtÞ þ gbat V base

ð22Þ

2.2.5. PV cost modeling The outage power of PV depends on cells temperature and solar irradiance in Maximum Power Point (MPP) linearly that can be formulated as follows [22]:

    GT ðtÞ  PPV ðtÞ ¼ PPV ;STC   1  c  T j  T jSTC  NPVs  N PVp GT STC ð23Þ GT ðtÞ  ðNOCT  20Þ GT STC

ð24Þ

According to Eq. (25), solar power generating cost includes investment, power electronic interfaces, operation and maintenance, and installation costs.

CF OPR PV;t ¼ C Inv ;PV þ C o&m þ C Ins;PV þ C SollarPanel;PV

ð25Þ

2.2.6. Reliability cost modeling In current work, the ENS value is considered as reliability index, and the MG operator is penalized according to this index. The penalty cost value depends on the not supplied energy levels. The proposed algorithm tries to decrees penalty cost like the other MG costs by optimal operation. Therefore, the islanding mode operation capability will be important. DGs and battery are used to restore lost loads in failure events periods. After each failure event the faulty parts will disconnect from the rest of MG by the circuit breakers located at two sides of feeders. The DGs located at the faulty parts should not only supply lost loads according to their priority, but also have the ability of supplying them (in this paper all load demands have the same priority). The restoration power of DGs, as mentioned in Eq. (5), can be computed as Eq. (26) after restoration of faulty part. [15]

X

yDG ði; LÞ  P FCAP DG;i;L ;

yDG ði; LÞ

L2NDG

¼

Otherwise

2.3. Uncertainty modeling

CF OPR Battery;t ¼ C o&m;battery þ C Inv ;battery þ C Pow:Elec:;battery þ C wc;battery

PRES DG;i ðt; l; r k Þ ¼

ð27Þ

1; if ith load is connected to Lth Storage

ð20Þ

2.2.4. Battery cost modeling The operational cost of batteries includes wear cost, operation and maintenance cost, investment cost and the power electronic interfaces cost as Eq. (21). The wear cost will be computed based on [20].

T j ¼ T amb þ

yST ði; LÞ

L2NST

n

CF OPR Wt;t

yST ði; LÞ  P FCAP ST;i;L ;

1; if ith load is connected to Lth DG 0; Otherwise

The uncertainty is defined as the probability of difference between the forecasted value and the real value. The exact modeling of renewable resources and load uncertainties has a significant effect on MG operational cost. There are different methods to deal with uncertainties such as analytical methods, and approximate methods and so on, which can be used in different problems optimally with regard to their application. 2.3.1. Wind and solar power uncertainties modeling based on PEM method PEM method fundamental PEM is one of the approximate methods, which deals with uncertainties. This method is favorite for power system operators because of its high accuracy in computing and its needless to have information of random variables PDF. Km + 1 Hong’s PEM Concentrating statistical information of a random variable on K points, which a function ðFÞ establishes a connection between input variables (probabilistic (XÞ and deterministic (cÞÞ and output variables, is the main idea of PEM. Function F (Z ¼ F ðp1 ; p2 ; . . . ; pl ; . . . ; pm ; cÞÞ depends on all input variables (mÞ. According to Fig. 2, in Hong’s PEM, for each random variable ðpl Þ, function F should be computed Km + 1 times where K is the number of points and l = 1, 2, 3, . . ., m. The K points related to the m random variables ðpl;k ; xl;k Þ will be computed based on statistical information and the PDF of that variable. This computed pair data for each random variable includes pl;k and its weighting factor xl;k , which states the weight of the location in output results. The location pl;k is computed as follow:

pl;k ¼ lpl þ nl;k rpl

ð28Þ

z Z=f(P1,P2,…,Pl,…,Pm)

(z(l,2), l2)

(z(l,3), 0)

(z(l,1), l1)

fpl ð26Þ

In this paper, the storage system is used to supply not only load demands at peak time, but also required energies of load demands

pl

Fig. 2. 2m + 1 Scheme.

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S.A. Alavi et al. / Energy Conversion and Management 95 (2015) 314–325

where lpl , rpl , nl;k denote the mean, the standard deviation of input

Start

random variable pl and the standard location of kth point respectively. The standard location nl;k and weighting factors xl;k will be computed by solving the nonlinear equations set which are stated in Eq. (29) [23].

8 K X > > > xl;k ¼ m1 > < k¼1

Input micro-grid data, selling and purchasing energy costs, DGs operation and pollution costs, battery operation cost, the mean value and variance of wind speed and solar irradiance at the next day

Uncertainty modeling of wind power and solar power using 2 m +1 PEM and extracting [(w 1,w 2,w 3),(Pw 1,Pw 2,Pw 3)], [(w 1,w2,w3),(Pv 1,Pv 2,Pv 3)] using Eqs. (28-31) and (33, 34)

ð29Þ

K > X   > > > xl;k nl;k j ¼ kl;j :

j ¼ 1; . . . ; 2K  1

k¼1

Define

and determine the uncertain hours of load demand randomly

where kl;j is the jth standard central moment of random variable pl with density function f pl . The standard central moments can be computed as follow:

h =1

Extract the h th sample of load demand profile using Eqs. (42-46)

Mj ðp Þ kl;j ¼  l j

ð30Þ

rpl

Initialize particles of PSO

where M j ðpl Þ, is calculated by Eq. (31).

M j ðpl Þ ¼

Z

1

1

 j pl  lpl f pl dpl

k=1

ð31Þ

Start the k th iteration of PSO

It should be noticed that kl;1 ¼ 0; kl;2 ¼ 1, and kl;3 ; kl;4 are the skewness and kurtosis of pl , respectively. How to solve Eq. (29) is completely described at [24]. After computing all pairs ðpl;k ; xl;k Þ, the output function Z will be computed for each variable and for each concentrated point Zðl; kÞ based on Fðlp1 ; lp2 ; . . . ; pl;k ; . . . ; lpm Þ. The output random variable in

Calculate the objective function for the operation period using Eq. (1)

Calculate the expected values of power flow results (DGs, battery and Grid optimal operating points) using Eq. (32)

jth moment will be computed according to Eq. (32).

EðZ j Þ ffi

m X K X

h 

xl;k  F lp1 ; lp2 ; . . . ; pl;k ; . . . ; lpm

i j

Update the global and previous particle and velocity

ð32Þ

l¼1 k¼1

Move the particles

In current work, (2m + 1) Hong’s PEM scheme (K ¼ 3; nl;k ¼ 0Þ is applied for wind and PV power uncertainties. K ¼ 3 is selected for each input random variable, and the third point is placed in mean value of the variable. Then, the first forth moments of the random variable PDF should be computed. The essential equations to compute nl;k and xl;k are taken from [25] as follow:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kl;3 3 3k kl;4  k2l;3 k ¼ 1; 2 nl;3 ¼ 0 nl;k ¼ þ ð1Þ 4 2 ð1Þ3k 1 1 xl;k ¼ ; xl;3 ¼  m kl;4  k2l;3 nl;k ðnl;1  nl;2 Þ

Is PSO algorithm converged ?

ð33Þ

Is PSO algorithm converged ?

ð34Þ

x0 ¼

xl;3 ¼ 1 

l¼1

1

l¼1

kl;4  k2l;3

Yes

Modify the robust optimal solution

No

h=h+1

No

Is PSO algorithm converged ? Yes

function F in this point and its new weighting factor should be computed (x0 Þ. m X

k=k+1

Yes

According to Eq. (28), setting nl;3 ¼ 0, yields pl;k ¼ lpl , therefore the third point of all random variables will be their mean values   lp1 ; lp2 ; . . . ; lpl ; . . . ; lpm . So there is one more computation of

m X

No

Save the most robust results

ð35Þ

WT and PV uncertainties modeling In this paper, wind speed and solar irradiance are considered as two random variables and function F is power flow equations which their results are the voltages of busses and the exchanging power with upstream distributed network in slack bus (the slack bus is the 20/0.4 kV substation bus). Weibull PDF and Beta PDF are applied to deal with uncertainties related to the wind speed and solar irradiance, respectively. Solar irradiance modeling According to [26], Beta PDF is the best distribution function to model solar irradiance that is formulated as follow:

Finish

Fig. 3. The proposed algorithm flow chart.

( f b ðxÞ ¼

CðaþbÞ CðaÞCðbÞ

0

 xa1 ð1  xÞb1

for 0 6 x 6 1; a P 0; b P 0 Otherwise ð36Þ

In Eq. (36), x is the amount of solar irradiance in kW/m2, a and b are Beta PDF parameters and f b ðxÞ is the Beta PDF of x. a and b will be

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S.A. Alavi et al. / Energy Conversion and Management 95 (2015) 314–325

Power Utility Grid

20 Kv 400 V SD

400 V

PCC

Bus. 1

Bus. 11

Feeder 3. Commercial

Bus. 8

Bus. 3

Feeder 2. Residential

Feeder 1. Residential

Bus. 7

Bus. 2

Bus. 6 Bus. 12 Disel Generator

Bus. 9

Bus. 4

Bus. 13 Bus. 5

Bus. 10 Bus. 14

Micro Turbine

PV

Battery Bank Wind Turbine Fig. 4. The proposed MG topology.

computed by use of mean value ðlÞ and standard deviation (rÞ of random variable x according to Eqs. (37) and (38):



lð1 þ lÞ 1 r2

lð1 þ lÞ  1 b ¼ ð1  lÞ r2

a¼l

ð37Þ ð38Þ

Wind speed modeling According to the [26], Weibull PDF is the best distribution function to model wind speed behavior, which is formulated in Eq. (39).



k1 h V wind k i  k V wind c f w ðV wind Þ ¼ e c c

ð39Þ

where k is called the shape index, and c is called the scale index.

2.3.2. Load uncertainties modeling based on RO RO fundamental RO, first was introduced by Soyster in 1973, uses uncertain boundaries to describe the input variables uncertainties. Applying this method leads to the proper decisions which will be optimized for the worst case of probabilistic input variable into the determined boundary. The main application of this method is when there is not enough data for the input variable PDF. If function Z ¼ f ðxÞ depends on uncertain input x, because of the lack of a certain PDF of x, its uncertainty will be modeled as an uncertain boundary x 2 UðxÞ which according to Eq. (40), x can choose every value in boundary U.

x 2 UðxÞ ¼ fxjjx  xj 6 ^xg

ð40Þ

In Eq. (40), x and ^x are the forecasted value and the possible maximum difference between x and x. RO seeks a way not only to

S.A. Alavi et al. / Energy Conversion and Management 95 (2015) 314–325

Residential Load (%),Commercial Load (%), Price of Purchasing Energy (%) Price of Selling Energy (%)

320

100 Price of Purchasing Energy Price of Selling Energy Commercial Load Residental Load

90 80 70 60 50 40 30 20 2

4

6

8

10

12

14

16

18

20

22

24

Time (h) Fig. 5. Load and price profiles.

45

Wind power Solar power

40 35

(kW)

30 25 20 15 10 5 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (h) Fig. 6. Mean value of wind power and solar power.

Table 1 Result of running program in Sc.2, Sc.3, Sc.4 with C ¼ 5. Time (h)

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

Scenario 2

Scenario 3

Scenario 4

Diesel Generator

Micro Turbine

Battery

Grid

Diesel Generator

Micro Turbine

Battery

Grid

Diesel Generator

Micro Turbine

Battery

Grid

0 0 0 0 0 0 95.53 92.1 0 0 92.26 0 0 81.41 87.09 75.09 0 321.38 400 400 352.12 327.49 98 0

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

39.27 0 39.27 39.27 33.14 0 39.27 0 0 0 0 0 0 0 0 0 0 39.27 39.27 39.27 38.99 0 0 0

133.14 72.77 112.94 115.11 107.26 106.44 83.45 106.76 263.61 261.07 176.06 266.89 174.08 179.93 172.22 176.13 263 0 30.95 9.9 0 0 188.62 164.04

0 0 0 0 0 0 94.86 85.46 0 84.94 0 86.26 87.05 0 0 0 0 310.87 400 383.1 382.72 400 0 0

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

0 0 0 39.27 20.31 0 39.27 39.27 0 0 0 0 0 0 0 0 0 39.27 39.27 39.18 0 0 0 0

89.1 64.45 62.63 110.58 86.55 103.1 78.97 150.66 258.21 163.1 263 172.55 179 246.87 243.27 237.56 250.15 0 16.28 0 0 89.14 278.59 155.95

0 0 0 0 0 0 99.6 0 0 0 91.38 81.17 81.94 0 0 76.44 0 351.3 400 400 335.69 306.72 0 0

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

0 39.27 0 0 0 39.27 39.27 0 0 0 0 39.27 0 0 0 0 0 0 39.27 0 39.27 0 39.27 14.14

89.05 122.95 48.02 70.34 65.82 143.39 74.21 177.12 258.21 248.56 171 217.56 184.1 246.87 243.27 160.68 250.15 0 16.28 22.93 0 0 214.6 141.51

Voltage (p.u)

Voltage (p.u)

Fig. 7. Buses voltage profiles in Sc.1 (a), Sc.2 (b), Sc.3 (c), and Sc.4 (d).

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optimize the cost function but also to reach optimized solution with high probability, when there is error in forecasted value of variable x [27–31]. Load uncertainty modeling In this paper, the load demands of busses ðPload Þ that can fluctuate into a boundary are considered as random variables, also we consider load demand with not adequate information of its PDF but bounded. Therefore, with regard to [1,30] RO will be a proper method to model load demand uncertainty. The RO modeling procedure can be explained as follow: For each load demand in each interval a boundary like as Eq. n (41) and a forecasted value plbt , is associated.

^lcbt ; p ^ltbt  plbt 2 ½p

ð41Þ

^ltbt and p ^lcbt are upper and lower bound of load demand In Eq. (41), p for every load bus in any b and t. In RO, the cost function will be optimized for the worst case of possible event while reaching this worst case is a complicated problem. In this method, a Conservation Degree (CD) (C) is defined (C is the number of time periods that plbt n is allowed to be different from its forecasted value plbt ) that in this paper, C 2 ½0; T. If C = 0; therefore, the real value of load demand is close to the forecasted value in every period of operating and if C = t, so in t arbitrary hours in the operating time period the value of real load demand is different from the forecasted value [27,28]. With increasing C, the optimization process will be more robust while the optimization cost increases. Thus, selecting the suitable C for each optimization problem in each work condition is different. In order to determine the uncertain set, first, axillary variables should be defined like Eqs. (42)–(44) and then the uncertain set will be defined. 2500

2344.8

Cost ($)

2000 1500

1445.9

1448.9

1457.8

Sc.2

Sc.3

Sc.4 , =5

1000 500 0

Sc.1

Fig. 8. MG operational cost in Sc.1, Sc.2, Sc.3, Sc.4 with C = 5.

8 þ if plbt ¼ p^ltbt > < zbt ¼ 1  ^lcbt z ¼1 if plbt ¼ p > : bt n þ  zbt ¼ zbt ¼ 0 if plbt ¼ plbt c n c ^lbt plbt ¼ plbt  p

ð42Þ ð43Þ

t

^ltbt  plnbt plbt ¼ p

ð44Þ

Using Eq. (42)–(44), the uncertain set will be defined as (45). ( ) T X  þ  n t c U ¼ pl 2 RjBjT : zbt þ zbt 6 Cb ;pl ¼ plbt þ zþbt plbt  zbt plbt 8b 2 B; 8t 2 T t¼1

ð45Þ

Then, using Monte Carlo sampling method, different scenarios of load demand in each hour will be generated by a random variable (rv) subject to uniform distribution over [0 1] like Eq. (46):

8 n c þ  if r v 6 Cb =2T > < pl ¼ plbt  plbt ; zbt ¼ 0; zbt ¼ 1; n t zþbt ¼ 1; zbt ¼ 0; if r v P 1:2  Cb =2T pl ¼ plbt þ plbt ; > : n zþbt ¼ zbt ¼ 0; Otherwise pl ¼ plbt ; ð46Þ Therefore, the problem will be solved with different scenarios for deterministic pls and the possible worst event case will be optimized and will be recommended as the most robust solution. 3. Proposed methodology based on PSO algorithm Optimal energy management in a MG is a combinatorial problem which should be solved using powerful meta-heuristic methods. Due to the suitable capabilities in dealing with such problems, the PSO algorithm has been employed to solve the problem [32,33]. PSO is a swarm intelligent algorithm and it is based on the movement of some groups of particles, which share their explorations among themselves. Each individual in PSO called particle flies in the searching space with a velocity, which is dynamically adjusted according to its own flying experience and its components’ flying experience. The details of PSO can be found in [34,35]. The optimal energy management procedure is shown in Fig. 3. Using PDFs for modeling the uncertainties related to the wind power and solar power, the PEM determines three estimated points for wind power using Weibull distribution and three estimated points for solar power using Beta distribution with their weighting factors. This data will be saved into a matrix, then using RO method, different scenarios according to the CD (C), are made for the next day. The PSO algorithm is applied to determine the

Battery State Of Charge (SOC %)

80 Sc.2 Sc.3 Sc.4, gama=5

70 60 50 40 30 20 10 0

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (h) Fig. 9. The battery SOC.

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optimal amount of output power of each DG, the battery scheduling as well as the amount of exchanging energy with upstream network in each interval of operating period based on minimizing objective function. According to the proposed flowchart (Fig. 3), the PSO algorithm with regard to decision variables, finds the best answers. The first results of PSO will be saved and RO, which seeks the most robust solution, generates and transfers the other load demand scenarios to the PSO algorithm. 4. Simulation results The proposed algorithm is used in order to perform an optimal operation of a Low Voltage (LV) MG including renewables and conventional DGs as well as a battery bank. The MG includes two residential feeders and one commercial feeder as shown in Fig. 4. The MG data is extracted from Ref. [36]. In order to facilitate operating in islanding mode, two circuit breakers are improvised in both ends of feeders; they disconnect the faulty parts of MG from the rest of the MG. In addition, this MG is capable to switch into islanding mode with regard to the technical or economic aspects by help of a Static switch (SD) which is installed in Point of Common Coupling (PCC) [19]. According to Fig. 4, DGs are located just in residential busses. In order to assess the accuracy of the proposed optimal operation procedure, it is done for different scenarios and their result will be compared with each other. It is required to mention that in all scenarios the exact values of [SOC min ,SOC max ] and [V bus;min ,V bus;max ] are [4, 80]% and [0.95, 1.05] p.u. respectively. The forecasted load demand in 24 h of operation period and the energy price are shown in Fig. 5, which 1490

1470 1460 1450 1440

1448.9

1430

Γ =0

1471.4

1474.3

Γ =10

Γ =15

1483

1486.2

Γ =20

Γ =24

1457.8 Γ =5

Fig. 10. MG operational cost in Sc.4 for different CDs.

Scenario 1: None of DGs (Sc.1) In this scenario, we assume that none of DGs and batteries are installed, so MG should purchase all of its required energy from the upstream distribution network. In this scenario, the MG operating cost includes the cost of purchasing energy with regard to the prices, which are presented in Fig. 5, and the cost of ENS calculated by Eq. (5). MG reliability is low in this scenario and with occurring any failure event in feeders, the related part will be disconnected and it will become without supplier. Scenario 2: Deterministic Energy management (Base case (Sc.2)) In this scenario, all DGs and the battery are added to the MG, and all random variables (wind speed, solar irradiance and load demand) are considered as deterministic. Therefore, a deterministic optimization problem is created with regard to the aforementioned cost function and constraints. In Sc.2, load demand is the same as Fig. 5, and the mean value of wind power and solar power are illustrated in Fig. 6. In Fig. 6, the mean values of wind power and solar power are computed by Eqs. (19) and (23), also the availability of mean values of wind speed and solar irradiance for each hour respectively. Fig. 6 describes that the maximum of PV output occurs at 14:00 and the maximum of WT output occurs at 17:00. Scenario 3: Half probabilistic operation (Sc.3) In this scenario the output power of WT and PV are modeled probabilistically and load demand is modeled deterministically. PEM determines three estimated points for both wind speed and solar irradiance and the optimization problem will be solved subject to these points and their weighting factors. Scenario 4: full probabilistic operation (Sc.4) This scenario uses all parts of the proposed algorithm with regard to the Fig. 3, and finds the most robust and optimal solution. PEM method is applied to deal with the WT and PV uncertainties and RO optimization method is applied to deal with load demand uncertainty. This problem has been solved for different CDs in order to show the sensitivity of operation procedure and MG total cost over the CD (C).

2100 Sc.4 CD=15 Sc.2 Sc.4 CD=24 Sc.4 CD=5 Sc.4 CD=10 Sc.4 CD=20 Sc.3

2050 2000 1950 1900

operation cost ($)

Cost ($)

1480

are normalized and stated in percent. In Fig. 5, the maximum power of residential and commercial load demand for each bus are 60 kW and the maximum energy purchasing and selling price are 24 and 16 Cent/kW respectively. The MG elements characteristic presented in Appendix.

1850 1800 1750 1700 1650 1600 1550 1500 1450 1400

0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

Iteration Fig. 11. Convergence characteristics of scenarios in the case of cost objective.

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The numerical results of MG optimal optimization in Sc.2 to Sc.4 are shown in Table 1. According to the Table 1, the diesel generator due to producing pollution is only used in peak time of load demand which the price of purchasing energy from the upstream distribution network is high, and in low price hours this DG is not used or is used less than 25 % of its rated power. In addition, the MT optimal output power should be always in its rated power due to the load level and the cost of start-up considered for it. Moreover, the battery optimal charge and discharge power follow the authorized boundary and the rest of related constraints. Battery starts to charge in low energy price and off-peak hours and then starts to discharge in high energy price and on-peak hours in order to decrease MG costs. The optimal schedule of purchasing energy from the upstream distribution network is stated as follows: when the price of electricity is high, MG does not purchase energy or purchases a little energy or even it can sell energy to the distribution network. Applying the proposed optimal operation scheduling, the bus voltages will become as Fig. 7. According to the Fig. 7, it can be realized that the authorized bus voltage boundary is observed in all scenarios. This constraint is a considerable issue that should be observed in operation of distribution systems. Moreover, Fig. 7 demonstrates that when there is not any installed DG in MG (Fig. 7.a) the bus voltage drops in direct relation to its distance from the slack bus. Adding DG units to the busses of the MG (Fig. 7.b to d) results in increasing the bus voltage, when the installed DG is generating power, in comparison to that not having any DG unit. The total operational cost of MG in Sc.1–Sc.4 is depicted in Fig. 8. It is noticeable that applying DGs as well as battery unit in a MG system will decrease total operational cost, as in by comparing total cost in Sc.1 and Sc.3 we can find out that the total cost in Sc.3 decreased about 896 dollars whereas the MG reliability increased. The Optimal Charge and discharge of the battery has an important effect on decreasing MG total cost. Fig. 9 shows the SOC of battery in operation during in Sc.2, Sc.3 and Sc.4. In (Sc.4), the deviations of DPtLoad is bounded into a specified boundary [4% +5%]. Selecting C equals to zero means that the load demand profile of 24 h of the day is completely deterministic which was analyzed in (Sc.3) and the MG total operational cost was 1448.9 dollars. The optimal operation is applied on MG for different CDs 5, 10, 15, 20, 24 which the results for C ¼ 5 are shown in Table 1, and for the other CDs just the total operational cost has been extracted in this paper. The sensitivity of MG operational cost over CD is depicted in Fig. 10. According to Fig. 10, it is obvious that increasing CD results in rising MG operational cost while the operation will be more robust against load demand uncertainty. Fig. 11 also indicates the convergence characteristics of all scenarios in the case of cost objective.

5. Conclusion In this paper, an optimal strategy for operating a MG with the capability of operating in islanding mode is proposed. The uncertainty related to WT output power; PV output power and load demand were modeled successfully by PEM method and RO respectively, and decreased the operation risk. The numerical results showed that applying the proposed procedure decreased the MG operation cost significantly. Moreover, by increasing C, the system conservation and the operation cost increased while its risk decreased. In addition, both the islanding capability of MG and the presence of DGs decreased the amount of MG energy not supplied and increased the reliability of system.

Appendix Table Ap. 1 Installed DG source. ID

Type

Min

Max

1 2 3 4 5

Micro-turbine Diesel generator Battery PV WT

0 0 235.6 (kW h) 0 0

300 (kW) 400 (kW) +235.6 (kW h) 92 (kW) 250 (kW)

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