Probabilistic Expansion Planning of Energy Storage Systems Considering the Effect of Cycle Life

Abstract

Energy storage systems (ESSs) are the key elements to improve the operation of power systems. On the other hand, these elements challenge the power system planners. The difficulties arise as a result of the ESSs’ economic and technological features. The cycle life of ESSs is a critical aspect that influences the choices made during expansion planning processes. In this manuscript, we have focused on a new model for the expansion planning of ESSs considering the impacts of technical properties, such as the cycle life and depth of discharge. For this purpose, the proposed model consists of the hourly operation planning of ESSs in the sample days of year. A new indicator is proposed to determine the daily charging/discharging cycles of ESSs. The numerical results show the ability of the proposed model to determine the optimal technology and capacity of ESSs.

1. Introduction

Energy storage systems (ESSs) play a significant role in distributed power systems and utility power systems [1,2]. But the high investment cost is one of the main challenges of ESSs. As a result, expanding ESSs while taking their technical and economic aspects into account is a primary research area for power system experts [3].

In recent years, significant research has been conducted on the expansion planning of the ESSs. The main difference among these studies is how to model ESSs expansion planning problem and how to formulate the problem. In [4], the objective function is considered to minimize the hourly social cost and maximize the penetration of wind power resources.

The minimizing of total system costs is considered in [5] as the objective function. But, in this manuscript, the possibility of price arbitrage and profitability resulting from the voltage and frequency control is applied in the model.

In reference [6], along with the optimal expansion planning of ESSs, the wind production capacity was optimized. In the proposed model, ESS was used to reduce the need to invest in transmission systems to increase the penetration of wind generation resources.

A multi-objective optimization problem, which minimizes wind energy curtailment and total social cost and maximizes profit from the price arbitrage, is proposed in [7] for the ESS expansion problem. A new model for expansion planning of distributed ESSs is presented in [8]. Instead of using a centralized storage system, several distributed storage systems are used in the proposed method. In the model presented in this manuscript, a three-level planning method is used to determine the location and capacity of the ESS. In the first stage, the optimal location of ESS and its power capacity is separately determined for each day of year. The second stage determines ESS’s power and energy capacity at the sites selected in the previous phase. The third phase determines how to employ ESS to alleviate transmission line congestion based on the location and capacity of ESS indicated in the preceding stages.

In 2016, Kargarian et al. divided the ESS expansion planning problem into two different time horizons, hourly and inter-hourly [9]. In the inter-hour time horizon, the algorithm determines the optimal energy capacity of the ESS to provide the adequate ramping capability. At hourly intervals, the optimal power capacity of ESS is determined to have sufficient production capacity to supply the load.

A new model for the coordinated expansion planning of transmission systems and ESSs, taking into account transmission switching, is proposed in [10].

A three-level model for the simultaneous optimization of the transmission lines and ESSs is presented in [11]. In this model, the optimal location and capacity of the ESS are determined. The centralized development of the transmission lines were carried out. The suggested model’s high level optimizes the return on investment for investors. The model’s intermediate level optimizes the major choices associated with ESS expansion planning. The low-level simulation models market settlement.

In [12], the coordinated investment planning of the transmission lines and ESSs is presented from the perspective of a central planner. The proposed model aims to achieve the effective and coordinated development of transmission lines and ESSs to minimize the cost of investment with significant penetration of renewable energy sources.

A bi-level model is presented in [13] for expansion planning of the electrical energy resources considering the ESSs. The ESSs are modeled in the upper level, and social welfare is considered in the lower level. In [14], a two-stage robust optimization-based model is proposed for the expansion planning of active distribution systems coupled with urban transportation networks considering ESSs. The uncertainties of renewable energy, load, and traffic demand in the proposed model are jointly considered.

A new stochastic model is proposed in [15] to study the expansion planning problem of ESSs in microgrids, which contain renewable resources and responsive loads. The optimization problem is prepared as two-stage stochastic programming.

In [16], a mixed-integer conic programming model (MICP) and a hybrid solution approach based on classical and heuristic optimization techniques, namely matheuristics to handle long-term distribution systems’ expansion planning problems is presented for sizing and allocation of dispatchable/renewable distributed generation (DG) and energy storage devices.

A bi-level formulation for the generation and transmission coordination problem considering the ESS expansion planning is presented in [17]. A new stochastic framework to deal with the expansion planning of large distribution networks in a smart grid context with high penetration of distributed renewable energy sources and ESSs considering the seasonal impact is proposed in [18].

In [19], a reliability-constrained optimal ESS sizing for a microgrid is proposed. The model presented in [20] for ESS expansion planning is based on cost minimization in the islanded operating mode of microgrid and profit maximization when it is operated in the grid-connected mode. In [21], pollutant emission costs are considered in the objective function of ESS expansion planning model. The joint optimization of hybrid ESS and generation capacity with renewable energy resources is carried out in [22].

An optimal sizing and control strategy of the isolated grid with wind power and ESS considering the compensation costs of the curtailed wind power and load shedding is presented in [23]. The proposed ESS sizing method in [24] is based on the discrete Fourier transform.

A simultaneous capacity optimization method for distributed generators (DGs) and ESSs considering the energy serving and annual losses costs is presented in [25]. Reference [26] proposes a novel model for planning ESS expansions based on the dynamic investment strategy. Reference [27] presents a frequency-based strategy for expanding ESSs in the power system using the Fourier transform. A new model for Optimal ESS expansion planning and load shedding to improve distribution system reliability is proposed in [28]. In the proposed method, the costs of ESS installation are optimized with respect to the reliability value expressed as customers’ willingness to pay in order to avoid power interruptions. Reference [29] uses a novel optimum methodology for designing a combined solar/battery/diesel system in Yarkant, Xinjiang Uyghur Autonomous Region of China.

This manuscript presents a new model for expansion planning of the ESSs. The proposed model is formulated as a probabilistic optimization problem from the perspective of system operator. The main novelty of the proposed model compared to the above-reviewed articles is modeling the cycling properties of the ESSs in the expansion planning problem. The depth of the discharge effect on the cycle life of the ESSs is formulated. In a number of articles, only the life cycle cost was examined, and exact modeling of the cycle life in the expansion planning of the ESSs has not been addressed. Cycle life is defined as the maximum number of ESS charge and discharge cycles during the ESS life span.

Indeed, the novelties of the presented model are as follows:

-

Proposing a new indicator to determine the daily charging/discharging cycles of the ESSs.

-

Modeling the cycle life in the expansion planning formulation.

-

Considering the depth of discharge of ESSs in the expansion planning model.

Thus, in this article, the expansion planning of the ESS considering the limitations of charge and discharge cycles has been discussed. For this purpose, an index has been defined for charge and discharge cycles, and the depth of discharge is also considered in this index.

This research is organized as follows. Section 2 introduces the proposed model for the ESS expansion planning. This section presents the formulation of the proposed model. The simulation results are described in Section 3. Finally, Section 4 provides the concluding remarks.

2. Model Description

2.1. ESS Expansion Planning

In this study, it is supposed that ESS expansion planning is carried out from the system planner’s point of view. To model the limitations of charging/discharging cycles of ESSs within ESS expansion planning problem, the operation problem is solved. The proposed operation problem deals with the participation of production units and ESSs in the day-ahead market. In the presented model, in addition to determining the optimal capacity of the ESSs, the optimal technology of the ESSs will also be determined.

The general structure of the proposed model is shown in Figure 1. In this model, the operation planning for the hours of the annual sample days is carried out inside the expansion planning problem. System operation during this target year is therefore modeled using representative days, each of which is given a weight proportional to the probability of occurrence of a similar day. At first, the historical data of wind power plants are used to generate the probabilistic scenarios of wind power generation. After generating probabilistic scenarios, the scenarios are reduced and considered inputs in ESS expansion planning problem. The technical and economic properties of power plants and ESSs are the other inputs of the ESS expansion planning problem.

According to the structure presented in Figure 1, the problem of planning the development of storage is modeled as a one-level optimization problem. In this model, the minimization of the investment cost in the reservoir as well as the annual operating cost is considered as the objective function. Also, the characteristics of existing power plants and candidate storage are assumed as input to the model. In this model, in addition to the conventional constraints, the limitations of the charging and discharging cycles and the lifetime of the storage device are also considered as constraints in the problem. These issues are discussed in detail below.

In the following, at first, the general formulation of the problem without considering the limitations of charging/discharging cycles is presented. Then, the necessary relationships for modeling the cycle life of the ESSs are presented.

Social welfare is defined as the difference between producers’ profits and consumers’ costs. In the proposed model, the price elasticity of the load is neglected. Therefore, minimization of the total operating costs of the system and investment costs of the ESSs are considered the objective function of the optimization problem. The objective function of the proposed model is as follows:

$$\min {\displaystyle \sum _{s=1}^S{\lambda }_s}\cdot \left(\begin{array}{l}{\displaystyle \sum _{j=1}^{N_c}\left(\stackrel{1}{\overbrace{C_{sj}^c}}+\stackrel{2}{\overbrace{A{C}_j{}^{c,O\$M}}}\right)+{\displaystyle \sum _{m=1}^{N_w}\left(\stackrel{3}{\overbrace{C_{sm}^w}}+\stackrel{4}{\overbrace{C_m^{spil}}}+\stackrel{5}{\overbrace{A{C}_m{}^{w,O\$M}}}\right)}+}\\ {\displaystyle \sum _{n=1}^{N_s}\left(\stackrel{6}{\overbrace{C_n^s}}+\stackrel{7}{\overbrace{C_{sn}^{s,o}}}+\stackrel{8}{\overbrace{A{C}_n{}^{s,O\$M}}}\right)+\stackrel{9}{\overbrace{{\displaystyle \sum _{d=1}^D{\displaystyle \sum _{t=1}^{24}Loa{d}_{sdt}^{cut}·{\alpha }_{dt}^{cut}}}}}}\end{array}\right)$$

(1)

The first and second terms refer to the traditional power plants’ yearly operating and maintenance costs, respectively. The third term is used to predict the yearly running costs of wind generating installations. The fourth and fifth terms, respectively, define the yearly leakage cost and maintenance cost of wind power facilities. The sixth term is the annual cost of investing in ESSs. The annual operating cost and maintenance cost of ESSs are calculated in the seventh and eighth terms, respectively. The proposed formulation makes it possible to simultaneously determine the optimal technology and capacity of the ESSs. In fact, terms six, seven and eight terms can be calculated for different ESS technologies to determine the optimal technology. The ninth term determines the rewards paid to the flexible loads.

The annual operating cost of the conventional power plants is determined using the following equation.

$$C_{sj}^c={\displaystyle \sum _{d=1}^D\frac{365}{D}{\displaystyle \sum _{t=1}^{24}M{C}_j^c·P_{sjdt}^c+S{U}_{jdt}^c+S{D}_{jdt}^c}} \forall s,\forall j$$

(2)

The annual operating cost consists of the cost of power supply and the cost of turning on and off the power plants.

The annual operating cost and spillage cost of the wind power plants are determined using (3) and (4), respectively.

$$C_{sm}^w={\displaystyle \sum _{d=1}^D\frac{365}{D}{\displaystyle \sum _{t=1}^{24}M{C}_m^w·P_{smdt}^w}} \forall s,\forall m$$

(3)

$$C_{sm}^{spil}={\displaystyle \sum _{d=1}^D\frac{365}{D}{\displaystyle \sum _{t=1}^{24}(S{P}_{smdt}^w·C^{w,spill}}}) \forall s,\forall m$$

(4)

The investment cost and operation cost of ESSs are calculated in (5) and (6), respectively.

$$C_n^s=P_n^s·A{C}_n{}^{p,s}+E_n^s·A{C}_n{}^{e,s}+E_n^s·A{C}_n{}^{r,s} \forall n$$

(5)

$$C_{sn}^{s,o}={\displaystyle \sum _{d=1}^D\frac{365}{D}{\displaystyle \sum _{t=1}^{24}(P_{sndt}^{s,ch}·M{C}_n{}^{s,ch}+P_{sndt}^{s,dis}·M{C}_n{}^{s,dis}}}) \forall s, \forall n$$

(6)

The constraints of the problem are presented below [22,30]. Minimum and maximum production of conventional power plants:

$$\underset{¯}{P_j^c}·I_{jdt}\le P_{{}^{sjdt}}^c\le \overline{P_j^c}·I_{jdt} \forall s,\forall j, \forall d, \forall t$$

(7)

Ramping up capability of conventional power plants:

$$P_{sjdt}^c-P_{sjdt-1}^c\le R{U}_j^c·I_{jdt} \forall s,\forall j, \forall d, \forall t$$

(8)

Ramping down capability of conventional power plants:

$$P_{sjdt-1}^c-P_{sjdt}^c\le R{D}_j^c·I_{jdt-1} \forall s,\forall j, \forall d, \forall t$$

(9)

Startup cost of the conventional power plants:

$$S{U}_{jdt}^c\ge K_j^c·(I_{jdt}-I_{jdt-1}) \forall j,\forall t, \forall d$$

(10)

Shut down cost of conventional power plants:

$$S{D}_{jdt}^c\ge J_j^c·(I_{jdt-1}-I_{jdt}) \forall j, \forall t, \forall d$$

(11)

Minimum up time of conventional power plants:

$${\displaystyle \sum _{t^{\prime }=t}^{t+T_j^{on}-1}{I}_{jd{t}^{\prime }}\ge T_j^{on}·(I_{jdt}-I_{jdt-1}) \forall j, \forall t\in \left\{1,\dots ,25-T_j^{on}\right\} , \forall d}$$

(12)

Minimum down time of conventional power plants:

$${\displaystyle \sum _{t^{\prime }=t}^{t+T_j^{off}-1}(1-I_{jd{t}^{\prime }})\ge T_j^{off}·(I_{jdt-1}-I_{jdt}) \forall j,\forall t\in \left\{1,\dots ,25-T_j^{off}\right\}, \forall d }$$

(13)

Dispatched production of wind power plants:

$$P_{smdt}^{w,d}=P_{smdt}^w-S{P}_{smdt}^w \forall m, \forall t, \forall d, \forall s$$

(14)

Spillage of the wind power plants:

$$0\le S{P}_{smdt}^w\le P_{smdt}^w \forall m,\forall t, \forall d, \forall s$$

(15)

Power balance of ESSs:

$${\displaystyle \sum _{t=1}^{24}{\eta }_{{}_n}^{ch}·(P_{sndt}^{s,ch})={\displaystyle \sum _{t=1}^{24}\frac{1}{{\eta }_{{}_n}^{dis}}}}·(P_{sndt}^{s,dis}) \forall n, \forall s, \forall d$$

(16)

State of charge of ESSs:

$$E_{sndt}=E_{sndt-1}+{\eta }_n^{ch}·(P_{sndt}^{s,ch}) -\frac{1}{{\eta }_n^{dis}}·(P_{sndt}^{s,dis}) \forall n, \forall t\in [2,24], \forall d, \forall s$$

(17)

Energy limitation of ESSs:

$$(1-Do{D}_n)·E_n^s \le E_{sndt}\le Do{D}_n^{}·E_n^s \forall n, \forall t, \forall d, \forall s$$

(18)

Power limitation of ESSs:

$$0\le P_{sndt}^{s,ch},P_{sndt}^{s,dis}\le P_n^s \forall n, \forall t, \forall d, \forall s$$

(19)

Power constraint for the installation of ESS technologies:

$$P_{{}^n}^s\le SC{P}_n^{\max } \forall n$$

(20)

Energy constraint for the installation of ESS technologies:

$$E_{{}^n}^s\le SC{E}_n^{\max } \forall n$$

(21)

Equality of supply and demand in the energy market:

$${\displaystyle \sum _{j=1}^{Nc}{P}_{sjdt}^c+{\displaystyle \sum _{m=1}^{Nw}\left(P_{smdt}^w-S{P}_{smdt}^w \right)+}{\displaystyle \sum _{n=1}^{Ns}{P}_{sndt}^{s,dis}}=}Loa{d}_{sdt}^{total}-Loa{d}_{sdt}^{cut}+{\displaystyle \sum _{n=1}^{Ns}{P}_{sndt}^{s,ch}} \forall t, \forall d, \forall s$$

(22)

Constraint of the flexible loads:

$$0\le Loa{d}_{sdt}^{cut}\le Loa{d}_{sdt}^{\mathit{var}} \forall t, \forall d, \forall s$$

(23)

Ten minute spinning reserve constraint of the system:

$${\displaystyle \sum _{j=1}^{Nc}{R}_{sjdt}^c+{\displaystyle \sum _{n=1}^{Ns}\left(R_{sndt}^{s,ch}+R_{sndt}^{s,dis}\right)}}\ge SRM \forall t, \forall d, \forall s$$

(24)

Reserve constraint of the system:

$${\displaystyle \sum _{j=1}^{Nc}\overline{P_j^c}+{\displaystyle \sum _{m=1}^{Nw}{P}_{smdt}^w+}{\displaystyle \sum _{n=1}^{Ns}\left(R_{sjdt}^{s,ch}+R_{sjdt}^{s,dis}\right)}-\left(Loa{d}_{sdt}^{total}-Loa{d}_{sdt}^{cut}\right)\ge RM \forall t, \forall d, \forall s}$$

(25)

Capability of the conventional power plants for spinning reserve provision:

$$0\le R_{sjdt}^c\le 10·MS{R}_j^c \forall j, \forall t, \forall d, \forall s$$

(26)

$$R_{sjdt}^c\le \overline{P_j^c }-P_{sjdt}^c \forall j, \forall t, \forall d, \forall s$$

(27)

Capability of the ESSs for spinning reserve provision:

$$R_{sndt}^{s,dis}+P_{sndt}^{s,dis}\le {\eta }_{dis}^s·E_{sndt-1} \forall n, \forall t, \forall d, \forall s$$

(28)

$$0\le R_{sndt}^{s,dis}\le 10·MS{R}_n^s \forall n,\forall t, \forall d, \forall s$$

(29)

$$0\le R_{sndt}^{s,dis}\le P_n^s \forall n,\forall t, \forall d, \forall s$$

(30)

$$0\le R_{sndt}^{s,ch}\le P_{sndt}^{s,ch} \forall n,\forall t, \forall d, \forall s$$

(31)

2.2. Modeling the Cycle Life of the ESS

One of the ESSs’ technical features is their restricted number of charging/discharging cycles. In fact, in addition to having a lifetime limit in terms of years, the ESS also has a limit on the number of charge and discharge cycles. This means that during the operation period of the ESS, even if the ESS is charged and discharged as much as the cycle life before the end of the useful life, it must be replaced.

In this manuscript, a new index is proposed to evaluate the cycle life of the ESS. In the proposed method to determine this index, the number of charge and discharge cycles is calculated for different operating scenarios every day. For this purpose, the daily charging/discharging cycles are determined using the following equation:

$$N_{nsd}=\frac{{\displaystyle \sum _t^{}\left({\eta }_{{}_n}^{ch}\times P_{nsdt}^{ch}+\frac{1}{{\eta }_{{}_n}^{dis}}\times P_{nsdt}^{dis}\right)}}{2\times E_n^s} \forall n,\forall d, \forall s$$

(32)

According to the above relationship to determine the cycle life, the total energy charged and discharged is calculated during the day.

In this index, the number of ESS charging cycles per day is determined based on the amount of energy stored in it and the energy capacity. This calculation is performed for all wind generation scenarios. In this way, all conditions of the system will be considered. In fact, the difference between the scenarios will be the difference in the production of the wind unit and therefore the amount of energy stored daily. With a good approximation, the number of charge and discharge cycles can be determined by dividing the obtained energy by 2 times the energy storage capacity.

The yearly cycle life of ESS is restricted using the following equation.

$${\displaystyle \sum _d^{}{N}_{nsd}\le N_n^{\max } \forall n, \forall s}$$

(33)

Based on the above relationship, the number of charge and discharge cycles on the days of operation of the ESS in each year should not exceed the allowed charge and discharge capacity per year. Another important parameter affecting the performance of the ESSs is the depth of discharge, which has not usually been considered in the expansion planning studies of the ESSs. But studies show that the depth of discharge has a significant effect on the number of allowed cycles of the ESSs. So, not taking this into account takes the expansion planning outputs away from what actually happens in operation. For this purpose, in this article, the effect of the depth of discharge on the lifetime of the ESS is modeled, and to model this phenomenon in the planning problem, constraints have been added to the proposed formulation.

By examining the technical characteristics of the ESSs, it can be seen that the depth of discharge of the ESSs is inversely related to the cycle life.

In the proposed model in this manuscript, to model the effect of the depth of discharge on the cycle life, the curve of the maximum number of the cycle life is modeled as the piecewise linear approximation according to Figure 2. The approximation is formulated as follows.

$$N=N\max +{\displaystyle \sum _{y=1}^Y{\mathsf{\mu }}_y\times DO{D}^y}$$

(34)

Depth of discharge is restricted as follows.

$$\overline{DO{D}^{y-1}}\le DO{D}^y\le \overline{DO{D}^y}$$

(35)

If depth of discharge is considered as a variable, then (18) will be nonlinear. To linearize this relationship, the product of two variables, $Do{D}_n·E_n^s$, is defined as a new variable using the following equation:

$$D{E}_n^s=Do{D}_n·E_n^s$$

(36)

Thus (18), will be as follows:

$$E_n^s-D{E}_n^s \le E_{sndt}\le D{E}_n^s \forall n, \forall t, \forall d, \forall s$$

(37)

And the following constraints are added to the formulation.

$$\underset{¯}{D{E}_n^s}\le D{E}_n^s \le \overline{D{E}_n^s}$$

(38)

$$\underset{¯}{Do{D}_n}\le Do{D}_n\le \overline{Do{D}_n}$$

(39)

3. Results

To illustrate the capability of the proposed model, the structure presented in Figure 3, is considered to be the case study. The system consists of three conventional power plants with properties displayed in

Table 1. Conventional power plants properties [31].

MC ($/MWh)	RU (MW/h)	RD (MW/h)	$\overline{\mathit{P}}$(MW)	$\underset{¯}{\mathit{P}}$(MW)	Number of Power Plant	MSR	K ($)	J ($)	ACO$M ($/MW/Year)	Ton (h)	Toff (h)	K ($)
20	200	200	400	50	1	5	150	100	20,000	6	6	150
30	60	60	300	40	2	4	100	50	9000	4	4	100
35	80	80	200	20	3	4	50	30	8000	2	2	50

. A wind power plant introduced in

Table 2. Wind power plant properties [31].

MC ($/MWh)	SP ($/MWh)	ACO$M ($/MW/Year)	Properties
3	10	25,000	Quantity

is assumed in the case study. Two ESSs with properties presented in

Table 3. ESSs properties [32].

Investment Cost		Efficiency (%)	Life Time (years)	Cycle Life	ACO$M ($/MW/Year)	Number of ESS
Power Cost ($/kW)	Energy Cost ($/kWh)
200	250	95	20	10,000	5000	1
75	150	90	15	10,000	4000	2

are evaluated for allocating in the system.

In the proposed model, the sample daily demands of the year are considered the representative of each season. Demands are divided into two categories: fixed and flexible. The seasonal sample fixed demands are shown in Figure 4. The daily flexible demands are depicted in Figure 5.

In the presented method, the stochastic nature of wind power plant production is modeled by the scenario-based simulation. In this article, time series based on past information are used to determine the probable scenarios of wind unit production. For this purpose, the time series of observations should be determined based on historical data. Observations for wind unit production are considered 24 h. After generating the time series of observations, in the second step, the time series should be converted into a stationary series. After obtaining the stationary process, in order to determine the type of time series model, autocorrelation functions (ACF) and partial correlation (PACF) related to the time series of the remaining Mana observations are used. In this way, the degrees of the ARIMA model governing the observations can be determined. For this purpose, ACF and PACF of the manufacturing process are determined by MATLAB software. Then, 10,000 scenarios are generated using the Monte Carlo simulation. The 10,000 scenarios are then reduced to 10 scenarios using the fast forward scenario reduction technique [3]. The MATLAB 9.6R2019a software is employed to determine the ARIMA model degrees and coefficients, and scenario generation and reduction. The interruption cost for flexible demand is considered 100 $/MW, and the thresholds of spinning and non-spinning reserves are defined 100 and 200 MW, respectively.

The following four simulations were performed to investigate the impacts of the ESS specifications on the expansion planning results.

Simulation 1:

-

The depth of discharge is considered fixed and equal to 0.9.

-

The ESSs presented in Table 3 are considered in the studies.

Simulation 2:

-

The depth of discharge is considered fixed and equal to 0.9.

-

The efficiency of the second ESS in Table 3 is considered equal to 80%

Simulation 3:

-

The depth of discharge is considered fixed and equal to 1.

-

The efficiency of the second ESS in Table 3 is considered equal to 80%.

Simulation 4:

-

The curve presented in Figure 6 is considered to model the relation between the cycle life and depth of discharge.

-

The ESSs presented in Table 3 are considered in the studies.

Simulation 1: The results of the ESSs expansion planning for simulation 1 are presented in

Table 4. ESS expansion planning results in Simulation 1.

The Proposed Number of the ESS	Energy Capacity (MWh)	Power Capacity (MW)
2	100	70

. In this case, only the second ESS is proposed for development due to the lower cost. The simulation results show that in this case, the demand interruption and the spillage of wind output will be zero. The number of charging/discharging cycles according to the equation defined in Equation (32) will be equal to 1.225. If the ESS is operated in all days of the year, the annual cycles are about 447 cycles. Thus, during the life of this ESS (15 years), the number of required cycles will be about 6700 cycles. Therefore, there will be no challenge regarding the limitation of the charging/discharging cycles. On the other hand, using an ESS with the same specifications as the second ESS, but with fewer charging/discharging cycles, will be a good option in terms of the lower possible investment cost.

Simulation 2: In this case, the efficiency of the second ESS is considered to be 80%. The output of the proposed model is shown in

Table 5. ESS expansion planning results in Simulation 2.

The Proposed Number of the ESS	Energy Capacity (MWh)	Power Capacity (MW)
1	155	54
2	22	15

. As can be seen, in this case, ESS expansion for both ESSs is recommended. Because the first ESS is more efficient despite the higher investment cost. The simulations reveal that in this situation, the first and second ESSs have the same number of charging/discharging cycles (1.1 and 0.9, respectively). Thus, for the first and second ESSs, the total number of charging/discharging cycles needed is 8030 and 4927 cycles, respectively.

Simulation 3: In this case, all assumptions are considered similar to the second simulation, but the depth of discharge is assumed to be 1. It means that the full capacity of the ESSs is used.

Table 6. ESS expansion planning results in Simulation 3.

The Proposed Number of the ESS	Energy Capacity (MWh)	Power Capacity (MW)
1	141	55
2	17	14

presents the simulation results in this case. As can be seen in this case, the capacities of ESSs are reduced. In this case, the numbers of cycles required per day for the first and second ESSs will be equal to 1.24 and 1.04, respectively. Therefore, the numbers of cycles required during the life of the first and second ESSs are equal to 9052 and 5694 cycles, respectively.

Simulation 4: In this case, the goal is to determine the optimal depth of discharge of the ESSs based on the equations of Section 2.2. The curve presented in Figure 6 is considered the discharge depth/number of cycles curve for ESSs. The simulation results show that in this case the proposed depth of discharge is equal to 100%. This is because in this case, even with a discharge depth of 100%, the required number of charging/discharging cycles will be less than the limit of charging/discharging cycles. This indicates the correct operation of the proposed model.

If the relation N_max = −5000 × DOD + 10,000 is considered for modeling the discharge depth/number of cycles curve, the proposed depth of discharge will be 62.3%. Because with 100% discharge depth, the maximum charging/discharging cycles will be 5000, which does not meet the required number of cycles during the ESS life.

By comparing the first and second simulations, it can be concluded that the change in ESS’s efficiency has a significant effect on the composition of the developed ESSs. Comparing the second and third simulations shows that with a 10% increase in depth of discharge, the energy capacity of the first and second ESSs decreased by 10 and 22%, respectively. On the other hand, the required power capacity is slightly reduced. The results of the fourth simulation also show that the optimal optimization of the discharge depth can lead to more optimal decisions for ESSs development.

In order to demonstrate the capability of the proposed method compared to the articles reviewed in the introduction, development planning has been performed once without considering the charge and discharge limits and discharge depth modeling, and then the proposed capacities have been included in the exploitation problem and the amount. The performance of the storage devices has been measured. The results show that the amount of the proposed storage device is equal to 94 megawatt hours and 59 megawatts of the second type of storage device. However, the results of the evaluation of storage devices show that the proposed storage device will be unused in less than 80% of its useful life, taking into account the operating limit of the discharge depth of 0.9.

The main limitation of the proposed model in this paper is not considering the power network, which is among the authors’ goals as a new research topic.

4. Conclusions

In this research, a model is presented for expansion planning of ESSs considering the cycle life. This model determines the optimal technology and capacity of the ESSs. Simulations were performed for different modes, and ESS expansion planning results are presented. In these studies, the impact of changing the efficiency of ESSs and the depth of discharges was studied. In each case, the number of daily charging/discharging cycles and the minimum number of charging/discharging cycles required during the life of ESSs are determined. Studies were performed for two modes of the constant discharge depth and linear modeling of the discharge depth.

The simulation results show that a reduction in the ESS efficiency reduces the energy capacity of the ESS. Efficiency significantly affects the number of daily charging/discharging cycles. The degree of impact also depends on the specifications of other ESSs. The results show that a 10% increase in the ESS depth of the discharge reduces the ESS energy capacity required by about 20%. Furthermore, the number of daily charging/discharge cycles increases by about 14%, which indicates an increase in ESS participation in the energy supply.

One of the most important functions of this project is to make the results of energy storage development studies more operational and practical by investors as well as power system planners. Without considering the limitations of charge and discharge cycles and the depth of discharge of the ESSs, the development results of the ESSs will be in a way that cannot be achieved in practice and therefore will affect the profitability of them.