Stochastic analysis of residential micro combined heat and power system

Energy Conversion and Management 138 (2017) 190–198

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Stochastic analysis of residential micro combined heat and power system H. Karami a, M.J. Sanjari b,⇑, H.B. Gooi b, G.B. Gharehpetian a, J.M. Guerrero c a

Department of Electrical Engineering, Amirkabir University of Technology, Iran School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore c Department of Energy Technology, Aalborg University, Denmark b

a r t i c l e

i n f o

Article history: Received 4 October 2016 Received in revised form 21 January 2017 Accepted 28 January 2017

Keywords: Combined heat and power (CHP) Hybrid energy system (HES) Time of use (TOU) Monte Carlo simulation (MCS) Stochastic load Optimal scheduling Residential load sector Load demand uncertainty Stochastic programming

a b s t r a c t In this paper the combined heat and power functionality of a fuel-cell in a residential hybrid energy system, including a battery, is studied. The demand uncertainties are modeled by investigating the stochastic load behavior by applying Monte Carlo simulation. The colonial competitive algorithm is adopted to the hybrid energy system scheduling problem and different energy resources are optimally scheduled to have optimal operating cost of hybrid energy system. In order to show the effectiveness of the colonial competitive algorithm, the results are compared with the results of the harmony search algorithm. The optimized scheduling of different energy resources is listed in an efficient look-up table for all time intervals. The effects of time of use and the battery efficiency and its size are investigated on the operating cost of the hybrid energy system. The results of this paper are expected to be used effectively in a real hybrid energy system. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Distributed generation integration into micro-grids has gained more attention in the past few years, e.g. photovoltaic systems and wind turbines [1]. Using distributed generations in houses caters for the customers with electrification at a competitive price and economic use of electrical and thermal energies [2]. Use of distributed generations may have other benefits such as emissions decrease and security enhancement [3]. Thermal and electrical energies can be produced by combined heat and power (CHP) or combined cooling, heating and power systems as cogeneration systems [4]. Recently, investigation of CHP scheduling is an important issue in economic operation of systems [5]. Fuel cells (FCs) is one of the attractive devices in CHP systems, especially in houses [6]. The literature has shown that the operating costs (OCs) of FCs can be reduced by applying the optimal settings determined through a cost minimization procedure [7].

⇑ Corresponding author. E-mail addresses: [email protected] (H. Karami), [email protected] (M.J. Sanjari), [email protected] (H.B. Gooi), [email protected] (G.B. Gharehpetian), [email protected] (J.M. Guerrero). http://dx.doi.org/10.1016/j.enconman.2017.01.073 0196-8904/Ó 2017 Elsevier Ltd. All rights reserved.

In line with the acceptable performance of CHP functionality in the hybrid energy system (HES), some studies have been carried out to investigate the FC integration to the HES and its appropriate scheduling. An intelligent control scheme based on a modelpredictive strategy was applied to dispatch the micro-CHP systems in [8] and the effect of the proposed approach was evaluated in terms of energy cost reduction. In [9], it is shown that the energy optimization algorithms can play a major role at the residential level to achieve benefits. Operating cost minimization was formulated as a mixed-integer linear programming in [9]. The effects of energy storage device operation and applying of different time of use (TOU) rates on the system OC were studied in [10]. However, the OC related to the integrated energy system was not taken into consideration. In [11], the economic scheduling problem of a CHP system in the presence of wind power dynamics, PV power variations and time-varying load profile was discussed by applying chance-constrained programming. The stochastic load effect has not been considered in the abovementioned studies. This is a critical issue in the economic dispatch of energy resources (ERs) in HES because residential loads show sudden variations associated with the household inhabitant’s lifestyles [12]. Compared with [12], in this paper, the electrical demand is considered as a stochastic variable and its effect on

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191

Nomenclature Abbreviations CHP combined heat and power CCA colonial competitive algorithm ER energy resource FC fuel cell HES hybrid energy system OC operating cost OF objective function TOU time of use Overall parameter i indicates in the subscript of the variables and shows number of time intervals T time interval Utility and home parameters PeL electrical demand (kW) PhL thermal demand (kW) purchasing electricity tariff from utility ($/kW h) C Up C U;peak , C U;off
peak cost of electricity purchased from utility in the peak and off-peak periods ($/kW h) CU purchased electricity cost from utility during a day ($/day) PU utility electrical power (kW) Fuel-cell parameters DP FC;U , DPFC;D upper and lower limits of fuel cell power ramp rate (kW) PFC;max , PFC;min maximum and minimum limits of fuel cell generated power (kW) gFC fuel cell efficiency

the OCs of the HES is considered. Electrical and thermal loads, a FC and a battery are integrated to the HES. In addition, natural gas resource and the electrical grid are also accessible. The economic operation of the HES is formulated by integrating the economic models of the devices. The colonial competitive algorithm (CCA) is applied to determine the optimal dispatch of ERs in the system. Random electrical load variations are modeled by applying Monte Carlo simulation as described in [13]. The effectiveness of the proposed method in HES management is shown by using a real load demand data. The results of the cost minimization problem of optimal ERs scheduling are presented as a look-up table. The amount of thermal and electrical powers produced by different ERs for each time interval is shown in this table. The minimum dispatch costs are achieved if different ERs supply thermal/electrical power according to the mentioned table in each time interval. A HES with the same elements combination as this paper was modeled in [14] and its optimal OC was achieved by applying the harmony search algorithm. In [14] deterministic load is considered and the time-varying nature of the residential load is not modeled. In [15] stochastic nature of the residential loads is considered through the power scheduling optimization, which is carried out by applying harmony search algorithm. However, the effects of battery capacity and efficiency were not investigated. In this paper, the HES economic model is studied and the presence of stochastic residential demand is considered. In order to determine the optimal scheduling of ERs, CCA is applied as a powerful optimization algorithm. The results of this study are more realistic compared with that presented in [14] since the stochastic behavior is considered for load demand. Compared to [15], a more accurate scheduling strategy is achieved in this paper by applying a more powerful optimization algorithm. Moreover, the effects of battery parame-

c1 , c2 Z startup and shutdown cost ($) C FC PhFC , PeFC r FC PLR

cost of generated power by fuel cell ($/day) heat and electrical power generated by fuel cell (kW) heat to electrical power ratio of fuel cell part load ratio

Battery parameters W energy stored in the battery (kW h) DW i battery energy change (kW h), DW i ¼ W i
W i
1 gch , gdch charging and discharging battery efficiency W max , W min maximum and minimum limits of stored energy in the battery (kW h) CB cost of operation of battery ($/day) C Bp operation and maintenance cost of battery per kW h ($/kW h) PBdch max , P Bch max maximum discharging and charging rates of the battery (kW) PB electrical power absorbed or supplied by battery (kW) Natural gas parameters purchased natural gas cost per kW h ($/kW h) C gasp C gas cost of purchasing gas during a day ($/day) directly produced heat power from gas (kW) Pgas Colonial Nimp Ncol n b

a

competitive algorithm parameters number of imperialists number of colonies role of colonies in empire total power determining weight factor of colonies movement maximum deviation angle from the original direction

ters such as efficiency and capacity on system operating cost are investigated and the minimum efficiency of the battery to participate in the system is determined. The rest of this paper is organized as follows. HES is introduced in Section 2 and the problem of ERs optimal scheduling is formulated. In Section 3, the CCA is explained. Section 4 discusses the simulation results and Section 5 concludes the article. 2. HES and optimization model Fig. 1 shows the interrelations among a battery, a FC, and electrical and thermal loads as a layout of HES. As shown in this figure, the electrical load is energized by the utility grid, the battery, or FC while either the resource of natural gas or the FC recovered heat can supply the thermal load. The energy management system determines the power supplied by different ERs in the HES at any time slot. The major part of the EMS function is to minimize the OC of supplying demand. In order to effectively use available ERs, optimal operation scheduling can be carried out for one day or more in advance. Minimization of one day-ahead OCs considering stochastic load demand is the aim of this paper. It is assumed that components integrated to the HES have been installed and installation costs are not considered, since the HES operation optimization is the aim of this paper. 2.1. Objective function This section is presented to define an appropriate objective function. Since the objective of this paper is minimization of the

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2.2. Constraints of power balance

Thermal Part

Gas Network

The following equations express the equality constraints related to the thermal and electrical powers balance with the assumption that no curtailment in electrical or thermal load is allowed.

Furnace

mCHP (Fuel Cell)

Thermal Load

Battery

Electrical Load

Electrical Network Natural Gas

ð6Þ

Pgas;i þ PhFC;i
PhL;i ¼ 0

ð7Þ

It should be noted that positive or negative values of PB,i indicate the battery discharging or charging modes of operation, respectively. 2.3. Constraints of devices The operation constraints for through the procedure of system energy is limited by its capacity. the constraint of the battery state

Electrical Part Electricity

Heat

PeFC;i þ PB;i þ PU;i
PeL;i ¼ 0

W min < W i < W max

all devices should be satisfied cost minimization. The battery The following inequality states of charge.

ð8Þ

Fig. 1. Integrated HES.

OC related to daily HES operation, the following objective function (OF) is considered to provide a quantitative index of HES daily OC.

X X X X OF ¼ min E C FC;i þ C gas;i þ C U;i þ C B;i i

i

i

!

ð1Þ

i

The expected value of their summation, E(.) is considered as the objective function in this paper. Four terms in (1) were explained in [14]. As described in [14], in each time interval, C U;i is calculated by multiplying the value of electricity price and purchased electric power together from the grid. Similarly, C gas;i can be calculated by multiplying gas price and the amount of purchased gas together. Therefore, the formulation of calculating C gas;i and C U;i can be expressed as follows:

C U;i ¼ C Up P U;i T

ð2Þ

C gas;i ¼ C gasp Pgas;i T

ð3Þ

The battery power is positive if the battery discharges its stored energy to the system and is negative if it is charged from the system. Therefore, the cost of battery operation is the absolute value of battery power to the cost of operation and maintenance cost of the battery, as shown in the following:

C B;i ¼ C Bp jP B;i jT

ð4Þ

It should be noted that in this system, it is assumed that the system is installed and the aim is to find optimal scheduling of the system. Therefore, the installation cost is not considered. The cost of fuel cells in each time interval is the cost of its input fuel, which is gas in this paper, to supply electrical and heat power. As the ratio of generated electric power of the fuel cell to its input is the efficiency of the fuel cell, the cost of fuel cell can be calculated as multiplying C gasb by ðP eFC;i =gFC;i Þ in each time interval. It should be noted that if the supplied electric power of fuel cell at an interval is under its lower bound, the fuel cell is shut down and the shutdown cost should be considered in the system OC. Similarly, if the fuel cell is started up at an interval, the startup cost should be added to the system OC. The formulation of the FC is as follows:

C FC;i ¼

8 > < C gasb TðPeFC;i =gFC;i Þ þ c1

if PeFC;i > 0; PeFC;i
1 ¼ 0

> :

else

c2

C gasb TðPeFC;i =gFC;i Þ

if PeFC;i
1 > 0; P eFC;i ¼ 0

ð5Þ

Assuming ideal battery, if it is discharged/charged with the specified power of PB,i, the battery energy is decreased/increased by PB,i  Dt. However, in practice the charging and discharging efficiencies should be taken into consideration. The following equations express the relation between the battery energy and the rate of charging/discharging, i.e. (9) and (10), respectively.

W i ¼ W i
1


PB;i  Dt

gdch

W i ¼ W i
1 þ PB;i  Dt  gch

ð9Þ ð10Þ

The following inequalities express the limitations of the battery discharging/charging rates, i.e. (11) and (12), respectively.

DW i < PBdchmax Dt

ð11Þ

DW i > PBchmax Dt

ð12Þ

The rate of variation in the FC output power is limited to its upper boundary, which is stated, in the following inequality [16].

PeFC;i
PeFC;i
1 < DP FC;U

ð13Þ

Similarly, the following inequality should be considered for lower boundary limitation of the variation in the FC output power rate:

PeFC;i
1
PeFC;i < DP FC;D

ð14Þ

It should be noted that the equivalence of DPFC;U and DP FC;D is not necessary. Another constraint related to the FC output power states that the FC cannot work properly if its output power is less than a lower threshold or greater than an upper bound. In other words, if P eFC;i is set to a lower value than PFC,min, the FC should be turned off. Similar to the lower bound, if the PeFC;i is set to a higher value than PFC,max, the FC output is forced to PFC,max.

PFC;min < PeFC;i < PFC;max

ð15Þ

The ratio of FC supplied electrical power to its power rating is part load ratio (PLR). FC efficiency and rFC are functions of PLR as shown in Fig. 2 [17] and the FC efficiency is decreased if it supplies more electrical power. The mathematical formulation of the understudy FC operation, known as proton exchange membrane fuel cell (PEMFC) is extracted from experimental measurement studies as described in [17]. The efficiency and ratio of the supplied electrical to thermal energy are functions of PLR as follows:

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1 0.9

(%)

0.8 0.7

Heat to electrical ratio Efficiency

0.6 0.5 0.4 0.3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PLR Fig. 2. Efficiency and heat to electrical ratio for PLR in FC.

PLRi < 0:05 :

gFC;i ¼ 0:2716; rFC;i ¼ 0:6816

ð16Þ

PLRi P 0:05 : gFC;i ¼ 0:9033PLR5i
2:9996PLR4i þ 3:6503PLR3i
2:0704PLR2i þ 0:4623PLRi þ 0:3747

ð17Þ

r FC;i ¼ 1:0785PLR4i
1:9739PLR3i þ 1:5005PLR2i
0:2817PLRi þ 0:6838

ð18Þ

2.4. Time of use pricing In order to moderate the behavior of consumers and peak shaving in the electrical power system network, TOU pricing is applied in the electricity market. The TOU rates can change at different time intervals. In order to show the effectiveness of our approach in daily TOU rate variations in this paper, three TOU rates are investigated as shown in Table 1 [14]. It should be noted that this TOU pricing scheme is an example and the proposed approach can be applied to any TOU pricing scheme. However, this example can show the advantages of the proposed approach. Different factors for three TOU rates are listed in the third column of Table 1. 3. Applying CCA to HES optimization CCA is an optimization method, which has been applied to solve different optimization problems [18]. This algorithm can be summarized as follows 3.1. Step 1: Generating set of initial empires CCA initiates with generating the initial population named ‘‘countries”. Equation (19) states the position of the j-th country in an N-dimensional optimization problem assuming that the total number of countries is Ncnt.

Countryj ¼ ½x1j ; x2j ; . . . ; xNj  ðj ¼ 1; 2; . . . ; Ncnt Þ

ð19Þ

Initially, the number of variables associated with the problem of ER optimal dispatch should be determined which is noted as the optimization problem dimension. Considering power balance equations

related to thermal and electrical energies, i.e. (6), (7) it should be noted that for each time interval this problem has two independent variables, i.e. PeFC,i and PB,i. According to (6), the electrical utility power can be calculated knowing the independent variables and the electrical load demand. After that the supplied heat by the FC, i.e. PhFC,i can be calculated by knowing PeFC,i and the value of electricity to heat ratio as shown in Fig. 2. Finally, Pgas,i is determined using (7), knowing PhL,i and PhFC,i. Therefore, by knowing PeFC,i and PB,i, all the electrical/thermal powers can be determined related to different components of the HES. The next step is calculating the OC according to (1). The aforementioned explanation shows that two independent variables for each time interval, i.e. PeFC,i and PB,i should be considered. Considering Dt = 1 h as time interval length, the HES is studied in 24 temporal horizons. Therefore, the optimization problem dimension is N = 24  2 = 48. Fig. 3 shows the way of forming a country by using the mentioned independent variables. It should be noted that the OC of the HES related to the j-th country (j = 1, 2, . . ., Ncnt) can be calculated according to (1) for the variables ðx1j ; x2j ; . . . ; xNj Þ. Initially there are Ncnt countries each including 48 variables as shown in Fig. 3. To form the initial empires, Nimp of the countries are selected as imperialists while the other countries will be their colonies. In other words, the Nimp of the most powerful countries form empires while the remaining Ncol countries are divided among them. It should be noted that the power of imperialists is inversely proportional to the objective function values of the optimization problem, i.e. the more powerful imperialist is the one with lower value of the objective function. The normalized costs of the empires are used to determine the initial number of colonies belonging to an empire, i.e. an imperialist with the lower costs will posses more colonies. This step is explained in detail in [18]. 3.2. Step 2: Movement of colonies toward their imperialists The colonies move toward their corresponding imperialists in the second step. This procedure is a random process which is explained in [19]. The old position and the region of probable position of a country after its movement toward the corresponding imperialist are shown in Fig. 4. A random amount of deviation, i.e. h is added to the direction of colonies movement to search for different points around the imperialist. Parameter h is selected as a random angle inside the interval (–a, a) which adjusts the deviation from the original direction, as shown in Fig. 4. During colonies movement, the values of PeFC,i and PB,i are varied. All constraints of the system, i.e. (8), (11)–(15) should be met at each time interval as stated in the following rules: Rule 1: In each time interval, if P eFC;i < P FC;min it is assumed to be turned off and its output is set to zero; if P eFC;i > PFC;max it is assumed to be PFC;max in that time interval. Rule 2: In each time interval, if PeFC;i < P eFC;i
1 and PeFC;i
PeFC;i
1 > DPFC;D , P eFC;i is equal to PeFC;i
1
DPFC;D in that time interval.

country Table 1 Different factors for different TOU rates [14].

var1

Label of TOU rate

Time range (h)

Factor for electricity purchase price (pu)

Off-peak Intermediate Peak

[1–8], [23–24] [13–16] [9–12], [17–22]

0.78 0.9 1

var2

…. var24

var25

var26 …. var48

PB,2

….

PeFC,1

PeFC,2

country PB,1

PB,24

Fig. 3. Variables that form a country.

….

PeFC,24

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4. Simulation results and discussion

Region of probable position of the colony after movement

If a colony during the colonies movement reaches lower cost than its imperialist, its positions is swapped with related imperialists.

Fig. 5 shows the dynamics of the normalized demand over one day, which illustrates an example of Latvian power and Riga district heating systems [20]. It should be noted that the proposed approach can be applied for each load demand with different variations and characteristics. This example is used only to show the powerfulness and applicability of the proposed approach on an example of real data. The electrical demand curve cannot be accurately determined. Therefore, the Gaussian distribution function is used in this paper to describe the electrical load uncertainty [21]. Fig. 6 shows the assumed deviation of load demand. The stochastic load varies according to the Gaussian probability distribution function. The mean value of this function is the real-time measured electrical demand which is shown in Fig. 5 with a blue line and its standard deviation (std) is 0.5 kW. The mean value plus and minus 0.5 kW are depicted by a black line and dashed red line, respectively, in Fig. 6. These bounds show that the probability of having load demand value between these lines is about 68% which is within the first standard deviation (std = 0.5). In this paper Monte Carlo simulation is applied for each time interval to model the electrical load uncertainty. As depicted in Fig. 6, random samples are generated according to the Gaussian probability distribution function. In this paper, 1000 samples are generated, i.e. for each time interval, 1000 samples are simulated which follow the Gaussian function with the aforementioned mean and standard deviation values. It should be noted that the Monte Carlo simulation generates the load demands in different time intervals independently. In other words, the load demands in different hours are assumed uncorrelated. In the next step, CCA is applied to the problem of objective function optimization to determine the optimal scheduling of the HES for each of 1000 scenarios. The CCA and system parameters under study are listed in Tables 2 and 3, respectively.

3.4. Step 4: Calculating total power of empire

4.1. Scenario 1: Mode of basic operation

Mainly affected by the power of the imperialist, the total power of an empire also depends on the powers of its colonies, as stated in (20).

In the basic operation scenario, the battery OC is not considered and the fixed TOU rate is assumed equal to 0.13 $/kW h. The fixed electricity and gas prices are 0.13 $/kW h and 0.05 $/kW h, respectively. The stochastic model of the electrical load is used by applying Monte Carlo simulation with generating 1000 random samples. The mean of the stochastic load in each time interval is determined as shown in Fig. 5, being the standard deviation equal to 166.7 W. The optimal power dispatch in this scenario is achieved by applying CCA. In this scenario, the TOU rate remains constant and is higher than the FC electrical power cost. This explains why the

Imperial θ θ

β×d d Fig. 4. Randomly colony movement toward its imperialist.

Rule 3: In each time interval, if P eFC;i > PeFC;i
1 and PeFC;i
P eFC;i
1 > DP FC;U , then PeFC;i is set to P eFC;i
1 þ DP FC;U in that time interval. Rule 4: If PB,i < 0 (charge mode), the difference between PB,i and PB,i-1 should be lower than PBch,max; in other words, in this situation, PB,i is assumed to be PB,i-1 + PBch,max. Rule 5: If PB,i > 0 (discharge mode), the difference between P B;i and PB,i-1 should be lower than PBdch,max; in other words, in this situation, PB,i is assumed to be PB,i-1 + PBdch,max. Rule 6: If the maximum/minimum energy of the battery in charge/discharge mode exceeds limitations, the battery energy between these limitations can be fit by adjusting PB,i. 3.3. Step 3: Imperialists position updating

TP n ¼ costðimperialistÞ þ n  meanfcostðcolonies of empireÞg ð20Þ Being 0 < n < 1, it represents the role of colonies to determine the empire total power (TPn) 3.5. Step 5: Competition of imperialistics The weakest empires will lose their weakest colonies (usually one) while based on their total powers, all the other empires compete to take possession of these colonies. The greater chance of the mentioned colonies possession is for the more powerful empires. 3.6. Step 6: Powerless empires eliminating The powerless empires will collapse, when all of its colonies are lost in the competition. 3.7. Step 7: Convergence After some iterations, only the most powerful empire will exist which possesses all the countries. There is no meaningful difference among the colonies and their imperialist in this condition. In this condition, the algorithm ends.

1

Demand per peak

ny

lo Co

0.9 0.8 0.7 0.6 0.5

Thermal Demand Electrical Demand

0

5

10

15

Time (h) Fig. 5. Daily demand [20].

20

25

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H. Karami et al. / Energy Conversion and Management 138 (2017) 190–198

0

Fig. 6. Deviation of the load demand.

Table 2 CCA parameters. Number of imperialists, N imp Number of colonies, N col Role of colonies in empire total power determining, n Weight factor of colonies movement, b Maximum deviation angle from the original direction, a

20 1000 0.1 2 p/4

Table 3 System parameters. FC startup cost, c1 ($) Maximum limit of FC power, P FC;max (kW) Minimum limit of FC power, P FC;min (kW) Upper limit of ramp rate of FC, DP FC;U (kW) Lower limit of ramp rate of FC, DP FC;D (kW) FC shutdown cost, c1 ($) Initial energy in battery (kW h) Charging efficiency of battery, gch (pu.) Discharging efficiency of battery, gdch (pu.) Maximum energy in battery, W max (kW h) Minimum energy in battery, W min (kW h) Upper limit of battery charging rate, P Bch max (kW) Upper limit of battery discharging rate, P Bdch max (kW) Length of time interval, T (h) Operation and maintenance cost of battery, C Bp ($/kW) Cost of purchasing natural gas, C gasp ($/kW)

0.15 1.2 0.05 0.75 0.9 0 0 0.927 0.971 3 0
1.75 2.25 1 0 0.05

Maximum cost of purchasing electricity from utility, C U;peak ($/kW)

0.13

because FC cost is more than utility cost. Reduction of FC cost needs decreasing FC efficiency and according to Fig. 2, the generation of the FC should be reduced in this operating point. This means that the FC does not generate at its maximum capacity and decreasing the FC contribution is economic, since the generation cost of FC is equal to the cost of utility. As shown in Fig. 2, the FC efficiency is decreased if it supplies more electrical power and hence, this leads to higher generation costs more than the utility costs. The average generated power of the CHP and power grid for all time slots are depicted in Fig. 7. In this scenario, based on (1), the P total system OC is achieved as 6.10 $/day being CB,i = 0 and flat electricity tariff. The better performance of CCA is achieved here compared to the total OC of 6.11 ($/day) obtained in [15] by applying harmony search algorithm. Using the flat electricity tariff, the battery does not contribute to the energy balance in scenario 1 because this situation has the same costs of the electrical energy purchased from the main grid in the charging and discharging modes of the battery. 4.2. Scenario 2: Battery operation and different TOU rates consideration In this scenario, the battery operation and different TOU rates are taken into consideration. The gas price is equal to 0.05 $/ 1.4 Utility Power FC Power

1.2 1

P (kW)

FC operates in 24 time intervals at almost its maximum limit of power. On the other hand, the costs of the system are C Up ($/kW h) and C gasp =gFC;i ($/kW h) for utility and FC electrical generation, respectively. The FC efficiency is related to its electric supplied power according to Fig. 2. For example, if the generation of FC is 1.02 kW, the FC efficiency is about 33%. Therefore, the utility electricity cost (0.15 $/kW h) and the FC cost (0:05=0:33 $/kW h) are approximately equal. It should be noted that increasing the gas price leads to increasing the FC generation cost compared with the utility. In order to make the problem of optimal scheduling in this mode clearer, assume that FC generates at its maximum capacity. In this situation, the FC generation cost is about 0:05=0:32 $/kW h (which is equal to 0.156 $/kW h) and so, the FC generation is not economic

0.8 0.6 0.4 0.2 0

0

5

10

15

20

Time (h) Fig. 7. Optimized operation schedule for scenario 1.

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4.2.1. Real battery charging/discharging consideration The efficiencies related to the discharging and charging modes of battery operation are the most important parameters in this scenario. To determine the lowest values of the charging and discharging efficiencies, that guarantees the battery cooperation in the HES, the following case is presented. Assume that the battery state of charge is increased by absorbing W (kW h) electrical energy from the grid, i.e. it charges with the efficiency of gch in a time interval when CU,ch is the cost of purchasing electricity from the utility. The battery state of charge increases by W  gch with the cost of W  C U;ch . The state of charge can be decreased with the efficiency of gdch , i.e., it can supply energy when the cost of purchasing electricity from the utility is CU,dch. In this condition, the charged battery supplies ðW  gch Þ  gdch of electrical energy while the utility electricity price is CU,dch. The mentioned scenario is economic if the following inequality is satisfied:

W  gch  gdch  C U;dch P W  C U;ch

ð21Þ

Which can be simplified as (21)

gch  gdch P

C U;ch C U;dch

5.96 5.94 5.92 5.9 5.88 5.86 5.84 0.75

0.8

0.85

0.9

0.95

1

Battery Efficiency (%) Fig. 8. Effect of battery efficiency variations on system cost considering stochastic load and battery capacity of 3 kW h.

5.94 5.92 5.9 5.88 5.86 5.84 5.82

ð22Þ

As the worst condition of (22), assume that the battery is charged with the efficiency of gch in the off-peak period of utility grid while C U;off
peak is the cost of purchasing electricity from the utility. The efficiency of supplied energy in the peak period is gdch while C U;peak is the cost of purchasing electricity from the utility. If the absorbed energy from the utility by the battery is W (kW h) in the off-peak period, the battery state of charge is increased by W  gch the cost of which is W  C U;off
peak . In the peak period, the supply of the charged battery is ðW  gch Þ  gdch of energy and so, the cost of purchasing this amount of energy from the utility is W  gch  gdch  C U;peak which is more than the cost of supplied energy by the battery. This leads to participation of battery in the system, if the following inequality is satisfied:

gch  gdch P C U;off
peak =C U;peak

5.98

Cost ($/day)

kW h and the electricity price of peak period is 0.13 $/kW h. According to Table 1, the prices of intermediate and off-peak periods are 0.117 $/kW h and 0.1014 $/kW h, respectively. The optimal scheduling of ERs as well as the battery operation in major reduction in system OC are shown.

Cost ($/day)

196

ð23Þ

If the inequality stated in (23) is not met, the battery will not cooperate in the HES energy dispatching and its supplied or absorbed power is zero at all intervals as expressed in scenario 1. In this paper, gch  gdch is denoted by the term ‘‘battery efficiency”. In [22], the effect of battery efficiency was studied without considering stochastic load. In this part, the battery efficiency and stochastic load effects on the energy routing in the HES are considered simultaneously. It is assumed that the battery capacity is 3 kW h. As shown in Fig. 8, by increasing the battery efficiency, the system cost decreases. In this figure, for each battery efficiency, the stochastic load is modeled by using Monte Carlo simulation with 1000 iterations and is optimized by using CCA. It should be noted that as listed in Table 1, according to the peak and off-peak electricity prices, if the battery efficiency is less than 78%, the battery cannot participate in energy supply or saving. The threshold of battery efficiency is calculated based on (23). 4.2.2. Effect of battery capacity variations Battery as a storage device can cooperate in the HES energy dispatch and assist in reducing the OCs. By increasing battery capacity, it is anticipated that system cost decreases. As shown in Fig. 9, the system was simulated with different battery capacities consid-

5.8

2

3

4

5

6

7

8

Battery Capacity (kWh) Fig. 9. Battery capacity variations effect on system cost in the battery efficiency of 90%.

ering stochastic load with the battery efficiency of 90% and the results showed that the operating cost of the system could be decreased by using batteries with greater capacity. It should be noted that in this paper, installation costs of devices have not been considered. In other words, it is assumed that the system is installed and the objective is to properly minimize the operating cost of the system. In this paper, only a day-ahead (short term scheduling) is investigated. Yet the installation cost can be considered in long term studies. It can be our future work to investigate the effect of installation cost of devices in long term optimization studies. 4.2.3. Optimal scheduling for a specific battery size and efficiency In this case, the optimal energy dispatch regarding a specific case, i.e. 3 kW h battery with an efficiency of 90% is presented. Stochastic load behavior is modeled by generating 1000 samples through the Monte Carlo simulation. Then, CCA is applied to optimize the energy scheduling in the HES. Fig. 10 shows different ERs generation and the battery discharging/charging power associated with the optimal operation of the HES. Recall that the charge and discharge modes of the battery are identified by negative and positive values of PB,i, respectively. It should be noted that at the first time interval, there is no stored energy in the battery. All the stored energy in the battery has been discharged until the end of the last day and there is not any stored energy in the battery for today because it is not economic to store energy and does not discharge it completely to the system. However, as shown in Fig. 10, the battery should reach its maximum state of charge at 8-th time interval and so, if the battery has any state of charge at the beginning of the day, it should be completely charged before 9-th time interval. The battery consumes power to increase the state of charge in the first eight time intervals, by the power purchased from the

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3 Utility Power FC Power Battery Power

P (kW)

2 1 0 -1 -2

0

5

10

15

20

25

Time (h) Fig. 10. Optimized scheduling for scenario 2.

utility. This issue is reasonable because the utility price is at the lowest level. In each time interval, the FC generates the amount of power that leads to operating cost equal or less than the utility electricity cost in that period. As depicted in Fig. 2, once the optimal operating point of FC is determined through the minimization procedure, it should not be violated to achieve lower cost, because if the FC generates more/less power than the optimally determined one, the OC of the HES will be increased. Therefore, it is economic that FC does not work at its maximum efficiency to have FC generation cost less than utility. After the eighth time intervals, the battery is fully charged. During the peak periods, i.e. ninth to twelfth and seventeenth to twenty second time intervals, the utility price is at its greatest values, thus, the battery supplies power and depletes all of its stored energy and as described in Scenario 1, the generation of FC is at near-maximum limits. Along the thirteenth to sixteenth time intervals, the charging of the battery is carried out again and at the sixteenth interval, the battery reaches its maximum available energy. Table 4 lists the average power generation of ER, which is achieved through OC minimization. In this scenario, the total OC of the HES is achieved as $5.89 according to (1). The mentioned OC is lower than that achieved in [15] as 5.91 ($/day). Comparing

the OCs related to scenarios 1 and 2, i.e. 6.10 ($/day) and 5.89 ($/day), it can be seen that participation of the battery has remarkable benefit in the case of considering different TOU rates. The operating cost of the system considering TOU rates without battery is about 5.96 ($/day). Therefore, the reduction of cost by using the battery and considering TOU rates is about 0.2$ (about 4%) for a day compared with the system without battery and TOU rates. Compared with the system with TOU rates and without battery, considering battery leads to cost reduction of 0.07$ (about 1.2%) for a day. In addition, as shown in this paper in Figs. 8 and 9, by increasing the efficiency or the size of the battery, the system operating cost is reduced. Therefore, higher battery size or battery efficiency leads to more reduction of cost during a year. The proposed approach of this paper can be used as a look-up table by the EMS to dispatch the electrical and thermal powers in each time interval. In other words, if there is a different load profile or seasonal variations, the approach is similar and can be applied to generate a table similar to Table 4 for each HES with different data. The aforementioned explanations are listed in Table 5, in which the system without battery and TOU rates is considered as the base case and the other scenarios, i.e. the system considering TOU rates and without battery as case 2 and the system considering TOU rates and battery as case 3, are compared with that to show the percent of decrease in operation cost. it should be noted again that as mentioned in Section 4.1., the battery does not contribute in system without considering TOU rates. The power generation of ERs is determined based on Table 4 by EMS to reach the optimal HES operation. Moreover, minimum OC has been obtained by adjusting the battery state of charge and discharge for each time interval. It should be noted that Table 4 is the mean of scheduling results for all 1000 scenarios generated by the Monte Carlo simulation. In other words, for each time interval, the results of 1000 scenarios are averaged and set as the results of the resources generation scheduling. In order to show the advantage of considering stochastic load, the results of scheduling using the proposed approach in this paper are compared with the results of using the proposed approach in [14]. The system can be scheduled using [14] without considering the stochastic nature of the load. However, it is clear that the load during the day has some differences with one, which is used to

Table 4 Source generations and electrical and thermal demands. Interval (h)

PeL,i (kW)

PeFC,i (kW)

PB,i (kW)

PU,i (kW)

PhL,i (kW)

Pgas,i (kW)

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

1.1160 1.0890 1.0710 1.0800 1.0980 1.1970 1.3770 1.5480 1.6560 1.7100 1.7280 1.6920 1.6740 1.6560 1.6380 1.6596 1.8000 1.7820 1.7550 1.6560 1.6380 1.5300 1.3860 1.2600

0.6321 0.6443 0.5794 0.6594 0.6634 0.6664 0.6634 0.5870 1.0356 1.0299 1.0216 1.0157 0.9247 0.9310 0.9337 0.9304 1.0387 1.0372 1.0406 1.0374 1.0396 1.0336 0.6630 0.6486

0 0 0 -0.430 -0.26 -0.099 -0.849 -1.526 0.5439 0.6804 0.7124 0.6486 -0.101 -0.174 -0.924 -1.598 0.5733 0.7465 0.7098 0.5907 0.2733 0.0189 0 0

0.4809 0.4412 0.4903 0.8527 0.6959 0.6282 1.5684 2.4900 0.0741 0 0 0.0171 0.8532 0.9029 1.6285 2.3261 0.1822 0 0 0.0264 0.3244 0.4809 0.7216 0.6114

1.96 1.93 1.9 1.87 1.84 1.82 1.8 1.83 1.84 1.56 1.58 1.72 1.76 1.78 1.78 1.78 1.78 1.79 1.81 1.83 1.92 1.96 2 1.96

1.4883 1.4473 1.4747 1.3736 1.3399 1.3171 1.2998 1.3981 0.9206 0.6490 0.6809 0.8293 0.9881 1.0004 0.9971 1.0012 0.8561 0.8684 0.8833 0.9080 0.9948 1.0435 1.5002 1.4734

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Table 5 Operating cost reduction in different cases. Case number

Is TOU pricing scheme applied?

Is battery installed?

Decrease in system operating cost (with respect to base case, i.e. case 1)

1 2 3

NO YES YES

NO NO YES

Base case 1.2% 4%

cost was achieved by applying colonial competitive algorithm than other optimization methods. The optimization results of the operating cost can be compiled into a look-up table based on which the energy management center schedules power generation by different energy resources to reach the optimal operating cost. Hybrid energy system optimization by considering the stochastic characteristic of the electrical load is the most important goal of this paper. Acknowledgment

Table 6 System OCs with stochastic and deterministic loads. Scenario

1

2

Stochastic loads Deterministic loads [22] Percent of difference (%)

6.10 6.38 4.4

5.897 6.01 1.9

schedule the system by [14]. For a comparison, a day-ahead 24-h load demand is assumed which has stochastic nature. By using the approach proposed in [14], the system cost without considering the stochastic nature of load is 5.91 ($/day). However, because of its stochastic nature, the load during the day is not exactly equal to the load which is used to schedule the system and so, by considering stochastic nature of load, the system cost is 5.6631 ($/day) using [14]. On the other hand, Table 4, which gives the results of the proposed approach in this paper, is used in the mentioned system with stochastic load. The system cost is 5.3346 ($/day) for the proposed approach. Therefore, stochastic load consideration leads to a better scheduling of the system and reduction of system cost. This reduction of cost because of considering stochastic nature of load is about 0.33$ for a sample day in the above example, compared with using deterministic load for resource generation scheduling according to [14]. This cost reduction is about 120$ for a year. Without considering the stochastic nature of the residential load, the model could be simulated more easily and quickly. The comparison of OCs between two HES with stochastic and deterministic load characteristics is shown in Table 6. As shown in this table, the optimal OCs of HES with stochastic and deterministic load characteristics are almost equal. The deterministic approach can be used to ascertain the OC of HES. However, stochastic load changes should be considered if the goal is determination of a power generation look-up table. 5. Conclusion Integration of a FC and a battery to the hybrid energy system was studied from an economic point of view in this paper. The mentioned system can be energized by the main grid and/or gas as two energy resources. Energy scheduling of this energy system were optimized by applying the powerful optimization method colonial competitive algorithm. The variable behavior of the load was modeled by applying Monte Carlo simulation and its effect on the system operating costs was studied. Moreover, the effects of flat and variable TOU rates on the system cost were investigated and minimum battery efficiency for cooperation in the system was discussed. Using real data of load demand and component parameters, HES simulations show the advantages of considering the stochastic nature of load in the reduction of system operating cost compared with previous approaches in which the stochastic nature of load has not been considered. The effect of battery capacity and efficiency variations are investigated on system operating cost considering stochastic electrical load behavior. Moreover, less operating

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