A Method for Calculating the Optimal Size of Energy Storage for a GENCO

Abstract

Market liberalization and the growth of renewable energy sources have enabled the rise of generation companies (GENCOs) managing diverse generation portfolios, creating a dynamic market environment that necessitates innovative energy management strategies to enhance operational efficiency and economic viability. Investing in the energy storage system (ESS), which, in addition to participating in the energy and ancillary services markets and in joint operations with other GENCO facilities, can mitigate the fluctuation level from renewables and increase profits. Besides the optimal operation and bidding strategy, determining the optimal size of the ESS aligned with the GENCO’s requirements is significant for its market success. The purpose of the ESS impacts both the sizing criteria and the sizing techniques. The proposed sizing method of ESS for a GENCO daily operation mode is based on the developed optimization operation model of GENCO with utility-scale energy storage and a cost-benefit analysis. A GENCO operates in a market-oriented power system with possible penalties for undelivered energy. The proposed method considers various stochastic phenomena; therefore, the optimization calculations analyze the GENCO operation over a long period to involve multiple potential combinations of uncertainties. Numerical results validate the competencies of the presented optimization model despite many unpredictable parameters. The results showed that both the battery storage system and the pumped storage hydropower plant yield a higher net income for a specific GENCO with a mixed portfolio, regardless of the penalty clause. Considering the investment costs, the optimal sizes for both types of ESS were obtained.

1. Introduction

The liberalization of the electricity market and the development of renewable energy technologies have enabled the rise of independent generation companies (GENCOs). In a restructured power system, GENCOs can participate and bid in both energy and ancillary services markets (ASM), such as regulation market, spinning reserve market (SRM), and others.

GENCOs are typically private companies managing portfolios that include one or more power plants. Nowadays, renewable energy sources are the most common types of generation facilities owned by GENCOs, with photovoltaic power stations (PVPSs), wind power stations (WPSs), as well as gas and biomass thermal power plants (TPPs) being the most prevalent. A GENCO’s investment policy and optimal operational strategy are heavily influenced by the regulatory framework established by the system operator to whom it offers services, as well as prevailing market trends. Investing in an energy storage system (ESS) in addition to participation in energy and ASM markets mitigates fluctuations associated with variable renewable energy sources (VRES), contributing to environmental sustainability and reducing carbon emissions. In recent years, there has been significant development in utility-scale battery storage systems (BSS), with lithium-ion, sodium-sulphur, lead-acid, and flow batteries as the most widely utilized technologies. The fast response and geographical independence of BSS enable a wide range of applications, ranging from power system stability enhancement and short-term power quality improvement to long-term energy management [1]. However, pumped storage hydropower plants (PSHP) are still the prevailing storage technology, accounting for over 90% of the world’s installed storage capacity and stored energy available for power network demands [2].

Numerous studies examine the joint operation of various generation facilities (especially PVPS or WPS) with various types of grid-scale storage. In [3,4,5,6,7,8,9], the combined operation of ESS with WPS was observed in various markets and under different conditions. In [3], the authors developed a deterministic mixed integer convex program model to optimize the operation of power plants in the day ahead market (DAM) and reserve market, considering various uncertainty parameters. The authors concluded that joint operation of the WPS and ESS may not be economically viable, but it remains economically feasible in the regulation market. In [4], the authors proposed a model that optimizes the operation mode of the electric power system to maximize the profit of the system in the spot market. Additionally, the optimal size of ESS was obtained considering the reduction in wind generation and economic losses resulting from wind forecast errors. The results showed that the ESS effectively mitigates wind curtailment and reduces the adverse impact of wind forecast errors. In [5], a model was developed to maximize the profit of a GENCO that participates in the DAM and ASM while accounting for the uncertainty in WPS production. The authors concluded that the joint operation of the WPS and PSHP reduced the uncertainty in WPS production and increased the GENCO’s profitability. In [6], the developed model ensures a constant energy supply from a WPS and PSHP system. The model is suitable for system operators in deciding between minimizing the energy fluctuations of the power system and decreasing the abandoned rate. The results showed that the obtained system operation strategy can maintain reliable production, increase output power stability, and reduce abandoned rate. In [7], the authors propose the market-based participation of an ESS to support large-scale renewable energy penetration. The results of the optimization problem show that the PSHPs can play a significant role in increasing the penetration of renewable energy for the procurement of energy and spinning reserve. In [8,9], the authors developed a model that maximizes the total profit of a GENCO, providing the optimal bidding strategy for a system consisting of the WPS and an air-based high-temperature heat and power storage (HTHPS). The results indicated that such a system combination is profitable in DAM, with profitability varying according to the different configurations of the HTHPS system.

In [10,11], the efficiency of energy storage operation with a TPP was proven, while in [12,13,14,15,16] the authors dealt with GENCOs that, in addition to TPP and ESS, also own WPS. In [12], a stochastic model was developed to optimize the daily profit of the aforementioned power system in the spot and regulation markets. In [13], the authors proposed a model that optimizes the operation mode of a power system, simultaneously minimizing TPP emissions and maximizing the profit in the DAM and SRM. The results showed the SRM is more profitable for ESS, as well as for the entire GENCO. In [14], a model was developed to maximize GENCO’s profit over a one-week period from the DAM and ASM (SRM and regulation market), while in [15], a model maximizes GENCO’s profit over one day from the DAM and spinning reserve market. Additionally, CO₂ emission reduction and wind power uncertainty are considered. The results showed that the maximum power of the WPS and PSHP, along with the price of CO₂ emissions, significantly influence the optimal bidding strategy of the system. In [16], a stochastic based method for the optimal bidding strategy of a GENCO participating in the DAM and spinning reserve markets is proposed. The considered ESS type is compressed air energy storage (CAES). The results indicated that CAES can generate higher profits in the energy and reserve markets due to their high ramp rates.

In [17], the joint operation of a WPS, PVPS, PSHP, and a BSS in the DAM and ASM was observed over a one-day period. To model uncertainties related to WPS and PVPS production, as well as market prices, an autoregressive moving average model was used, and scenarios were generated using the Weibull distribution function. The results show that the joint operation of these power plants increases the system’s profitability and reduces risk compared to their uncoordinated operation.

In addition to deciding the optimal operation and bidding strategy, determining the optimal size of the ESS aligned with GENCO’s needs is significant for its market success and directly impacts technical and economic sustainability, reducing the risk of undelivered energy and enabling more efficient utilization of VRES and thereby contributing to the reduction of greenhouse gas emissions. An oversized ESS may result in excessive investment, as well as increased operation and maintenance (O&M) costs, while an undersized ESS may fail to fully leverage the potential of the existing generation portfolio. The purpose of the ESS significantly influences its size. The sizing criteria and sizing techniques for dimensioning utility-scale energy storage are significantly different from those used for sizing energy storage in small distributed renewable energy systems or microgrids operating in islanded mode [1,18]. Utility-scale ESS or grid-scale ESS is a large energy storage system designed to enable arbitrage in the power system, provide ancillary services, compensate for errors in forecasting the VRES production, perform peak shaving, decrease penalties for non-delivery bided energy, and mitigate renewable energy curtailment, among other functions. Several criteria concerning steady-state operation (with time horizons greater than 1 min) have been applied for sizing utility-scale ESSs in numerous studies. In [19], a stochastic model for optimal sizing of ESS considers the fluctuation of wind production. The objective function aims to minimize the expected generation fuel costs plus ESS amortized daily capital cost. Key factors that affect the system cost-benefit are ESS amortized daily capital cost, charging/discharging efficiency, and lifetime. The influence of energy and ancillary services price fluctuations on the profitability of investment and payback period in ESS was shown in [20]. In [21], the optimal ESS management and sizing problem, considering the production from renewable energy sources and consumed energy from the grid, was addressed. The results showed that the marginal value of storage decreases with storage size and therefore the optimal size under the optimal management policy can be computed efficiently. Furthermore, energy storage can provide significant value and savings by integrating renewable energy and reducing reliance on grid-supplied electricity. In [22], the impact of ESS capacity on power system reliability and congestion relief under N-1 contingency criteria were analyzed. The optimal ESS size is determined through a trade-off between investment cost and improvements in reliability metrics. In [23], the optimal sizing of an ESS in large-scale power systems to minimize total investment costs while mitigating renewable energy curtailment, considering the network power flow, was investigated. In [24], the authors presented the optimization problem of investing in an ESS (PSHP and hydrogen storage plant) for 30 years. A hydrogen storage plant uses electrolysis and fuel cells for the power-to-power process. A cost-benefit analysis of independent energy storage that performs energy arbitrage in market conditions was made. The results show that the PSHP is profitable under the observed conditions, while the hydrogen storage plant is not profitable without additional support. The sizing method for an integrated generation system (hydropower plant, PVPS, and PSHP), proposed in [25], considers PVPS generation uncertainty, spot price volatility, and the deviation between day-ahead and real-time load. The authors of [26] investigated a range of policies to reduce emissions and investment in renewable generation and BSS. The results show that, despite the decreasing price of BSS, a significant subsidy is still required to justify the investment in BSS. Additionally, modest investments in BSS can yield considerable benefits in terms of reducing system costs and CO₂ emissions. In [27], the economic evaluation of integrating an ESS (CAES and BSSs) with a combined cycle power plant in a price taker GENCO was examined. It introduces a method for determining optimal storage technology options and capacities through a stochastic price-based unit and storage commitment model. The approach identifies the optimal participation of the power plant and ESS in DAM, spinning reserve markets, and bilateral contracts. In [28], a stochastic model to determine the optimal sizes of gas turbines, natural gas storage, WPS, and BSS in a GENCO was proposed. The model aims to maximize the GENCO’s profit in electricity and natural gas markets while managing risks from market uncertainties and wind power generation. Technical constraints of the power transmission system, natural gas pipeline, and uncertainties in market prices and pipeline deliverability are incorporated.

This paper deals with the sizing of a utility-scale ESS suitable for a GENCO operating in a liberalized power system with possible penalties for undelivered energy. GENCO’s portfolio may include both renewable and non-renewable energy sources. The proposed method for calculating the optimal storage size, based on the daily optimization of GENCO operation, considers several stochastic phenomena such as uncertainty in the production of renewable energy sources (primarily PVPS and WPS), price fluctuations in energy markets, and fuel prices instability, among others. The problem is complex and extremely stochastic. None of the aforementioned references have addressed such a general approach to sizing a utility-scale ESS for a GENCO. The proposed model was developed and optimized in MATLAB R2022a using the Mixed Integer Linear Programming (MILP) optimization technique.

In contrast to the majority of previously described algorithms for calculating the optimal operation and storage size, which employ a probabilistic approach, we propose a deterministic method with numerous optimization calculations that use realized and recorded combinations of uncertain factors. Both approaches use the same historical data for predicting future trends. In addition to the well-known differences between probabilistic and deterministic approaches, it should be noted that the complexity of stochastic optimization increases significantly when multidimensional stochastic processes are considered. Therefore, the aforementioned references consider only a few uncertainties. Furthermore, none of these references account for correlations between uncertainty factors (e.g., electricity market price and production of renewables), but treat them as independent, which considerably simplifies calculation. The deterministic approach does not face the problem of the number of uncertain factors, as each calculation step is based on a new set of initial conditions. The interdependence of random factors is inherent for each realized set of initial conditions. Deterministic optimization is significantly simpler than stochastic optimization, allowing for the inclusion of additional conditions and constraints in the model of a GENCO operation. However, a notable disadvantage of the deterministic approach lies in managing a series of partial solutions and simultaneously handling a huge number of input data.

Numerical results validate the competencies of the presented optimization model of GENCO operation under the given conditions, despite various uncertainties. The optimal sizes for both BSS and PSHP suitable for a specified GENCO with a mixed portfolio were obtained.

The main contributions of the paper are the development of a detailed model of GENCO operation with a diversified portfolio and energy storage under market conditions, as well as a method for determining the optimal storage size.

The rest of the paper is organized as follows: Section 2 presents the concept for calculating the optimal size of a GENCO energy storage with a description and mathematical formulation of the GENCO operation model to be optimized. Section 3 provides the simulations results for the optimal operation of a GENCO with a mixed portfolio, based on the proposed model, along with a cost-benefit analysis used to determine the optimal sizes for both BSS and PSHP storage systems. Conclusions are presented in Section 4.

2. A Method for Calculating the Optimal Size of Energy Storage for GENCO

2.1. Operating Conditions of GENCO in the Liberalized Power System

In the proposed method, the GENCO can own (or just manage) PVPSs, WPSs, TPPs, and ESSs. The GENCO can participate in both markets (DAM and ASM).

In a power system with a penalty clause, when the GENCO’s offer for a specified amount of electrical energy in the DAM is accepted, it enters into a contract with the system operator to deliver the agreed amount of energy. If the GENCO delivers less energy than contracted, the delivered energy is compensated at the market price, and penalties are imposed for each megawatt-hour (MWh) of undelivered energy. Conversely, if the GENCO delivers more energy than contracted, the surplus energy is remunerated at a degraded price. In contrast, in a power system without penalty clauses, all delivered energy is paid at the full market price.

In the ASM, generation facilities make an income through power capacity and energy components. A power plant selected in a competitive bidding process for the provision of an ancillary service in a certain hour receives compensation for reserved capacity (€/MW) and compensation for delivered energy according to the energy spot prices (€/MWh) if a unit is called to generate. Due to the prerequisites for the provision of ancillary services, it is assumed that VRES does not participate in ASM. Furthermore, since TPP is usually not profitable to operate in ASM due to fuel and start-up costs as well as efficiency reduction [29], in the proposed method, the ESS is the sole facility that generates income in this market. Thus, energy storage can generate income in the ASM and DAM, while all other power plants generate income exclusively in the DAM.

2.2. The Concept of Calculating the Optimal Size of GENCO’s Energy Storage

The proposed method for calculating the optimal size of the GENCO’s ESS for daily operation is based on calculating GENCO optimal operation under given conditions (e.g., energy prices, wind and solar generation, etc.) with the addition of virtual energy storage with certain characteristics. The optimal operation of the GENCO, which includes the production and losses of all its power plants, is calculated for each hour of the day. If the observation period is extended to one year, incorporating variations in uncertain factors and aggregating daily results, the potential annual income for the GENCO can be determined. By comparing the possible income in cases with and without virtual storage, the income of the virtual storage with certain characteristics can be extracted.

Approximately optimal storage characteristics (such as energy capacity, power capacity, round-trip efficiency, etc.) can be identified by repeating the above procedure multiple times while varying one or more of the characteristics. The variation in expected ESS income as a function of different storage characteristics over the observed period can be quantified and analyzed. This can be compared with the corresponding variation in investment costs associated with such storage over the same period. From the profit characteristics, the profitability of investing in a particular storage and the optimal characteristics of the storage can be concluded.

The modeling of the ESS operation in this research is based on a general energy storage operation model [30] that, with minor adjustments, can be adapted to model the operation of any energy storage that provides multiple services. All energy flows of energy storage can be expressed in a form suitable for specific storage technology, while O&M costs can be considered in several ways. This approach enables easier handling with different storage technologies. To reduce the amount of input data, all power plants of the same type are modeled as a single (equivalent) plant of that type. For instance, all PVPSs are represented as one equivalent PVPS. Since TPPs have non-negligible shutdown and restart costs, the equivalent TPP is modeled as a facility with multiple identical production units, each defined by certain maximum and minimum technical power outputs, as well as fuel and start-up costs.

Since the objective function optimizes the daily operation mode of a GENCO with VRES and energy storage, the observation period needed to obtain the optimal storage size must be extended over a sufficiently long period. This is necessary to capture a broad range of combinations of uncertainties. These random parameters are influenced by natural conditions, such as PVPS and WPS production, as well as energy and ancillary services market prices, which are affected by various factors, including seasonal variability. Determining the appropriate observation period from historical data is challenging due to the complexity and variability of these factors. However, it is generally recommended to consider a period spanning several years. The scope of historical data can be a limiting factor. The required stochastic data in the proposed method can be data measured and recorded in a selected previous period or data generated based on statistical processing of past data to produce representative scenarios.

2.3. Optimization Model of GENCO Operation

2.3.1. Objective Function

The objective function (1), which is to be maximized, represents the sum of N daily incomes of the GENCO depending on the operation mode. That income consists of income from DAM and ASM considering production and maintenance costs, as well as penalties due to undelivered energy. The indexes n, t, and z in (1) denote day, hour, and ancillary market service, respectively. A comprehensive list of the nomenclature used is provided.

$$\begin{array}{l}{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^{24}\left\{E_{n,t}^1\cdot c_{n,t}^e+E_{n,t}^2\cdot c_{n,t}^e\cdot kc+{\displaystyle \sum _{z=1}^k\left[\left(c_{n,t,z}\pm s{b}_{n,t,z}\cdot \eta \cdot c_{n,t,z}^e\right)\cdot h_{n,t,z}\cdot e{k}_z\right]}\right.}}-k_1-\\ k_2\cdot t_{n,t}^r-k_3\cdot E_{n,t}-k_4\cdot t_{n,t}^{su}-k_5\cdot t_{n,t}^{sd}-terp{l}_{n,t}\cdot c_{terpl}\left.-k\_terp{l}_{n,t}\cdot c_{terpl}^{su}-E_{n,t}^{pen}\cdot c\_pe{n}_{n,t}\right\}\end{array}$$

(1)

The first two terms in (1) refer to GENCO income from DAM. The variables $E_{n,t}^1$ and $E_{n,t}^2$ denote the amount of energy produced by the GENCO up to the contracted amount and the amount of energy produced above the contracted amount, respectively. The sum of GENCO production is also defined by constraints (2) and (3), which are detailed later. The price traded in DAM was denoted by $c_{n,t}^e$. The coefficient k_c expresses the factor (ranging from 0 and 1) by which the price paid for $E_{n,t}^2$ is degraded.

The third term refers to GENCO income from k ancillary services (z = 1, 2, …, k). So, c_n,t,z is the power capacity price, while $c_{n,t,z}^e$ is the energy price (spot price) during executing ancillary services. The constant sb_n,t,z, calculated separately by the specific algorithm, denotes the hourly probabilities (ranging from 0 and 1) that energy storage will be engaged to participate in the ancillary services. The coefficient η refers to the efficiency of charge and discharge or round-trip efficiency (AC to AC) of energy storage. The variable h_n,t,z denotes the energy flow of the energy storage for providing ancillary services (Figure 1), expressed in the natural dimension for a certain energy storage technology, while ek_z denotes the corresponding energy coefficient. In the case of PSHP and compressed air energy storage, the variable h_n,t,z is the volume of fluid for each service in each hour (i.e., m³ of water or air) and ek_z is the constant that represents the coefficient of fluid flow conversion into energy, expressed in MWh/m³ for the mentioned two technologies. For batteries, the variable h_n,t,z is calculated in MWh, and the energy coefficients are dimensionless constants. Thus, the product c_n,t,z·h_n,t,z·ek_z refers to the power capacity component while the product $s{b}_{n,t,z}\cdot \eta \cdot c_{n,t,z}^e\cdot h_{n,t,z}\cdot e{k}_z$ refers to the energy component of the ancillary service z. Since the energy component is additionally paid for the execution of some ancillary service (e.g., reg-up service), while for some ancillary services the energy component is repaid (e.g., reg-down service), this product has a ± sign.

The following five terms consider O&M costs of ESS. The coefficient k₁ represents the fixed O&M cost, which was expressed on an hourly basis. The following product refers to the O&M cost dependent on operating time, where k₂ represents time dependent component of ESS cost, while the indicator of ESS operation is represented by $t_{n,t}^r$. This cost is considered only when ESS is in generating or charging (pumping) mode. The following term represents the ESS cost dependent on generated energy, where k₃ refers to O&M cost per MWh while E_n,t represents the total generated energy of ESS. The following two products refer to start-up and shut down ESS costs, where k₄ and k₅ are start-up and shut down costs per starting-up/shutting down of ESS, while $t_{n,t}^{su}$ and $t_{n,t}^{sd}$ are indicators of start-up and shut down of ESS, respectively.

The following two terms refer to the TPP fuel and start-up costs, respectively. The variable terpl_n,t denotes the energy generated by TPP, while c_terpl represents the unit fuel cost of TPP. The variable k_terpl_n,t denotes the number of TPP aggregates that have started in the nth day and the tth hour, while $c_{terpl}^{su}$ is the cost price of the TPP aggregate start-up.

The final term in (1) represents the profit loss due to penalties that are paid when the hourly energy production falls below the contracted amount between the GENCO and the system operator. The variable $E_{n,t}^{pen}$ is penalized energy, while c_pen_n,t is penalty price per MWh of penalized energy.

Thus, the objective function, in addition to encompassing all GENCO incomes, also incorporates costs influenced by ESS operation, including fuel costs, start-up of the TPP, penalties, and others. GENCO operating costs not related to ESS operation, such as maintenance costs of other plants, are not included in (1).

2.3.2. Constraints

The proposed optimization model also includes several constraints in the form of equalities and inequalities, given below, as well as limits on variables. Constraints (2)–(8), (14) and (15) relate for every hour in N days, while constraints (9)–(13) relate for every day.

• Constraints of GENCO’s energy flow in the DAM market

The equality constraints (2) and (3) apply to the total GENCO production (E_genco_n,t), while constraints (4) and (5) apply to the difference between the generated and delivered energy of the GENCO ($E_{n,t}^{diff}$), respectively. The variables h_n,t and p_n,t (both expressed in the natural dimension for a certain energy storage technology) denote the energy flow of the ESS for discharging (production for sale on DAM) and charging, respectively. The generated energies of PVPS, WPS, and TPP are denoted by fne_n,t, ve_n,t, and terpl_n,t, respectively. The contracted hourly energy between GENCO and the system operator is denoted by E_contr_n,t, while the penalized energy is expressed by $E_{n,t}^{pen}$.

$$E\_genc{o}_{n,t}=\eta \cdot h_{n,t}\cdot ek-p_{n,t}\cdot ek+fn{e}_{n,t}+v{e}_{n,t}+terp{l}_{n,t}$$

(2)

$$E\_genc{o}_{n,t}=E_{n,t}^1+E_{n,t}^2$$

(3)

$$E_{n,t}^{diff}=E\_genc{o}_{n,t}-E\_cont{r}_{n,t}$$

(4)

$$E_{n,t}^{diff}+E_{n,t}^{pen}\ge 0$$

(5)

• Constraints of energy storage

The model scheme of energy storage is similar to the PSHP schema with the upper and lower energy accumulations. As previously noted, this model can be adapted to represent other types of ESSs (e.g., for BSS modeling) by setting the energy inflows of the upper accumulation to zero and oversizing the lower accumulation capacity, which corresponds to “unlimited” possibilities of taking energy from the power network.

Equality conditions are given with the accumulation mass balance equations for upper (6) and lower accumulation (7) and by Equation (8), which refers to node 2 in the model scheme presented in Figure 1.

$$s_{n,t}=s_{n,(t-1)}-r_{n,t}-s\_lo{s}_{n,t}+i_{n,t}$$

(6)

$$s{d}_{n,t}=s{d}_{n,(t-1)}-r{d}_{n,t}-sd\_lo{s}_{n,t}+r_{n,t}+i{d}_{n,t}$$

(7)

$$r_{n,t}=-p_{n,t}+p{r}_{n,t}+h_{n,t}+s{b}_{n,t,1}\cdot h_{n,t,1}-s{b}_{n,t,2}\cdot h_{n,t,2}+s{b}_{n,t,3}\cdot h_{n,t,3}$$

(8)

In the above equations, the parameters i_n,t and s_los_n,t, as well as variables r_n,t and s_n,t, refer to the inflow and losses (evaporation and seepage for PSHP, self-discharge rate for BSS) of the ESS upper accumulation and release and stored energy in nth day and tth hour, respectively. The inflow, release, stored energy, and losses of the ESS lower accumulation in nth day and tth hour are denoted by id_n,t, rd_n,t, sd_n,t, and sd_los_n,t respectively. The variable pr_n,t represents the spills for PSHP.

There are conditions that the last states of upper and lower accumulation are equal to the initial accumulation states in the next day:

$$s_{(n+1),0}=s_{n,24}s{d}_{(n+1),0}=s{d}_{n,24}$$

(9)

The coefficients k_acc_u_n and k_acc_l_n prevent the upper and lower accumulation from being emptied in a short time, respectively.

$$k\_acc\_u_n=k\_p{r}_n\cdot k\_in\_u_n$$

(10)

$$k\_p{r}_n=median\left(c_{n+1}^e\right)/median\left(c_n^e\right)$$

(11)

$$k\_in\_u_n=i_{(n+1)}/i_n$$

(12)

$$k\_acc\_l_n=1/k\_acc\_u_n$$

(13)

In the case of energy storage systems that do not have energy inflow into the upper accumulation, the input parameter value of the k_in_u_n is set to 1, while for storage systems without a lower accumulation, the input parameter value of the k_acc_l_n is set to 1.

Additional conditions and limits on ESS energy flow that determine the ban of production under minimum generator power output, the ban of simultaneous operation of some services, the indicators (t_p_n,t, t_h_n,t, t_h_n,t,z, t_pr_n,t), executing the ancillary services and O&M costs, are consistent with the methodology outlined in [20,30].

• Constraints of thermal power plant and VRES

The hourly production of equivalent TPP is given by the variable terpl_n,t. The maximum production and technical minimum of one TPP aggregate is denoted by the variables terpl_max and terpl_min, respectively, while the z_terpl_n,t is a real type variable that participates in calculating the start-up costs of the thermal power plant. The number of new started aggregates of TPP (k_terpl_n,t), necessary for calculating start-up costs, is calculated as a difference of active aggregates (act_terpl_n,t) in the current and previous hour ((14) and (15), respectively).

$${\displaystyle \frac{terp{l}_{n,t}}{terpl\_\max }}+z\_terp{l}_{n,t}=act\_terp{l}_{n,t}$$

(14)

$$act\_terp{l}_{n,t}-act\_terp{l}_{n,(t-1)}\le k\_terp{l}_{n,t}$$

(15)

The hourly production of PVPS and WPS is given by the variables fne_n,t and ve_n,t. The modelling operation mode of PVPS and WPS does not consider O&M.

3. Numerical Examples

In numerical examples, the influence of energy storage on the optimal operation of GENCO, which has PVPS, WPS, and TPP in its portfolio, was examined. It was assumed that GENCO participates in the DAM and ASM (reg-up, reg-down, spinning reserve). Additionally, for the specific GENCO, the optimal size of BSS and PSHP energy storage was determined.

The selected simulation period in all presented cases was the year 2023. One year is the minimum period that includes the complete seasonal variability of consumption and complete seasonal variability of VRES production. The production data of PVPS and WPS, as well as the ESS inflow data in the case of PSHP, were based on actual measured data for the selected year [31]. Electrical market prices were chosen from real power exchange data for the same year [32]. The fuel and start-up prices of TPP were taken according to [10,14,16].

3.1. Input Data

The input data (variables limits, coefficient values, recorded renewables production, and DAM prices) used in all numerical examples are presented below.

The upper (superscript u) and lower (superscript l) limits of the variables, as well as the coefficient values used in the numerical examples for modelling the energy storage system, are given in

Table 1. The values of ESS variables limits and coefficients.

			PSHP	BSS
Variables limits	sn,t	$s_{n,t}^u$	2 × 106 m3	1000 MWh
		$s_{n,t}^l$	0.1 × 106 m3	0 MWh
	sdn,t	$s{d}_{n,t}^u$	1 × 106 m3	oversized
		$s{d}_{n,t}^l$	0.1 × 106 m3	oversized
	prn,t	$p{r}_{n,t}^u$	1 × 106 m3	1 MWh
		$p{r}_{n,t}^l$	0 m3	0 MWh
	rdn,t	$r{d}_{n,t}^u$	1 × 106 m3	10 MWh
		$r{d}_{n,t}^l$	0.001 × 106 m3	0 MWh
	k_spr_g	k_spr_gu	1.05	1.05
		k_spr_gl	0.95	0.95
	$h_{n,t,1}^u$		0.03 × 106 m3	0.03 MWh
	$h_{n,t,2}^u$		0.02 × 106 m3	0.02 MWh
	$h_{n,t,3}^u$		0.01 × 106 m3	0.01 MWh
	$h_{n,t}^u$		0.11 × 106 m3	0.1 MWh
	$p_{n,t}^u$		0.1 × 106 m3	0.1 MWh
Coefficients	s_los		0 m3	0 MWh
	sd_los		0 m3	0 MWh
	ek1		1000 MWh/(106 m3)	1000
	ek2		1000 MWh/(106 m3)	1000
	ek3		1000 MWh/(106 m3)	1000
	ek		1000 MWh/(106 m3)	1000
	kdss		0.022 × 106 m3	0.02 MWh
	η		0.8	0.85
	k1		30 €	180 €
	k2		30 €	10 €
	k3		3.42 €/MWh	9 €/MWh
	k4		300 €/start-up	100 €/start-up
	k5		0 €/shut down	0 €/shut down

.

Figure 2 shows the inflow in the upper basin of PSHP in the observed period according to the measured daily data [31]. It was assumed that the hourly inflows in the upper and lower basin are equal over 24 h.

The upper and lower limits of the variable terpl_n,t, as well as the coefficients values used in all numerical cases for modelling thermal power plant, are shown in

Table 2. TPP variables limits and coefficients.

Variables limits	terpln,t	$terp{l}_{n,t}^u$	100 MWh
		$terp{l}_{n,t}^l$	0 MWh
Coefficients	terpl_max		50 MWh
	terpl_min		10 MWh
	cterpl		75 €/MWh
	$c_{terpl}^{su}$		5000 €/start-up

.

Since it would be illegible to show the measured production of the PVPS and WPS for each hour of the year according to [31], Figure 3 and Figure 4 show the production of one week in the 1st, 4th, 7th, and 10th months. By selecting these specified months, the weekly production of the mentioned generation sources in four different seasons is shown.

Figure 5 shows the realized prices at DAM in the observed year ($c_{n,t}^e$), while Figure 6 shows the aforementioned DAM prices in the same weeks as in Figure 3 and Figure 4 according to [32]. Both figures confirm the unpredictability of DAM prices and the occurrence of negative prices.

The capacity prices of ancillary services (c_n,t,1 − c_n,t,3) are assumed to be 40% higher than prices from DAM ($c_{n,t}^e$), while the energy prices of ancillary services ($c_{n,t,1}^e-c_{n,t,3}^e$) are taken to be equal to 8 €/MWh, 6 €/MWh, and 8 €/MWh for reg-up, reg-down, and spinning reserve, respectively.

In the examples with included penalties, it was assumed that the GENCO must constantly deliver 300 MW of power (E_contr_n,t = 300 MWh). Delivered energy up to E_contr_n,t is paid at the full market price, delivered energy greater than the contracted amount is paid for at 70% of the market spot price (kc = 0.7), while the GENCO pays a penalty of € 100 for each undelivered MWh (c_pen_n,t = 100 €/MWh).

3.2. Numerical Results

3.2.1. The Influence of ESS and the Penalty Clause on the Optimal Operation of the GENCO

Six cases of annual operation of the GENCO with previously defined parameters were simulated. Case 1, case 2, and case 3 refer to the operation of the GENCO without a storage system, with BSS, and with PSHP, respectively. In all these cases, the penalty clause is ignored, while in case 1p, case 2p, and case 3p, the penalty clause is considered. Cases 2 and 2p consider a battery with a charge and discharge power of 100 MW and with an efficiency coefficient of η = 0.85. Cases 3 and 3p consider a PSHP with 100 MW pumping power, 110 MW generating power, and η = 0.8. When choosing the parameters of both energy storage facilities, the intention was that the facilities have approximately the same, but realistic, characteristics. If equal η and zero inflow to the upper accumulation were used for both facilities, similar results would be obtained. However, the aim of the research is not to compare storage technologies but to consider the impact of both types of energy storage system on the optimal operation of the GENCO in a market-oriented power system.

Table 3. GENCO’s energy production and placement.

		Energies (GWh)
		Case 1	Case 1p	Case 2	Case 2p	Case 3	Case 3p
DAM	TPP	721	555	723	557	722	523
	PVPS	1674	1695	1685	1694	1684	1690
	WPS	2516	2533	2526	2533	2521	2532
	ESS production	-	-	163	167	401	428
	ESS consumption	-	-	−239	−237	−39	−41
	Total energy	4911	4783	4857	4713	5289	5132
	E1	4911	2358	4857	2426	5289	2487
	E2	-	2425	-	2287	-	2645
ASM	ESS reg-up	-	-	83	78	82	74
	ESS reg-down	-	-	−44	−44	−41	−41
	ESS spin	-	-	1	1	2	2

and

Table 4. GENCO’s operating hours.

		Operating Hours (h)
		Case 1	Case 1p	Case 2	Case 2p	Case 3	Case 3p
DAM	TPP	7694	6881	7738	6891	7694	6578
	PVPS	4600	4676	4635	4678	4621	4672
	WPS	8620	8730	8724	8731	8655	8730
	ESS production	-	-	5579	5344	8162	8181
	ESS consumption	-	-	3173	3409	501	537
ASM	ESS reg-up	-	-	8353	8065	8595	7909
	ESS reg-down	-	-	8321	8368	8365	8390
	ESS spin	-	-	5579	5309	8158	7414

show the optimal energy production and the optimal operating hours of the specified GENCO for DAM and ASM participation, respectively. Production in Table 3 along with operating hours in Table 4 are listed by facilities for DAM and by services for ASM. In addition, in Table 3, the total production of the GENCO is divided into production up to the contracted amount (E¹) and above the contracted amount (E²). From these tables, it can be concluded that:

• Significant differences in the results obtained of BSS and PSHP production for DAM, as well as consumption (for pumping or charging) from DAM, are mostly the consequence of the PSHP upper basin inflow;
• A comparison between the cases with and without the penalty clause reveals that the production of TPP is significantly lower in the penalty clause cases, because production at degraded energy prices is not profitable for TPP;
• The negative energy prices, recorded during several hours of the observed year (Figure 5 and Figure 6), contribute to the small differences observed in the produced energy and operating hours of VRES, as presented in Table 3 and Table 4. In all cases involving a penalty clause, the GENCO will suspend production during periods of negative energy prices;
• The total energy produced by the GENCO in Case 2p is lower than in Case 1p due to charge/discharge energy losses of BSS, but the placement of energy at the nominal price (E¹) is still higher;
• As expected, the energies for providing ancillary services, along with the operating hours, are approximately identical for BSS and PSHP. The duration of providing ancillary services is high in all cases. It is obvious that the GENCO tends to participate in the ASM more than in the DAM.

The undelivered (penalized) energy and the penalty cost for optimal GENCO operation are shown in

Table 5. Undelivered energy and penalty cost.

	Case 1	Case 1p	Case 2	Case 2p	Case 3	Case 3p
Penalized energy (GWh)	-	0.247	-	0.174	-	0.114
Penalty cost (M€)	-	24.781	-	17.471	-	11.403

. As expected, both values are lower in cases with ESS.

Table 6. Annual incomes of GENCO.

		Incomes of GENCO (M€)
		Case 1	Case 1p	Case 2	Case 2p	Case 3	Case 3p
DAM	DAM income	505	415	509	418	547	447
	E1	505	246	509	255	547	260
	E2	-	169	-	163	-	187
ASM	ESS reg-up capacity	-	-	38	37	38	36
	ESS reg-up energy	-	-	0.7	0.6	0.7	0.6
	ESS reg-down capacity	-	-	25	25	25	25
	ESS reg-down energy	-	-	−0.3	−0.3	−0.2	−0.2
	ESS spin capacity	-	-	10	9	12	11
	ESS spin energy	-	-	0.01	0.01	0.02	0.01
	GENCO net income	450	347	521	424	566	466

presents the structure of the GENCO’s annual income. DAM and ASM incomes represent the revenue generated from participation in these markets. GENCO’s net income is the result of the optimization calculation given by the objective function (1), i.e., it is the sum of DAM and ASM income minus the fuel and start-up costs of the TPP, storage O&M costs, and penalty costs. The results clearly demonstrate that the GENCO with both types of ESS generates higher net incomes regardless of the penalty clause.

The annual net income of the considered ESS, shown in

Table 7. ESS net income.

	BSS		PSHP
	PS without penalties	PS with penalties	PS without penalties	PS with penalties
ESS net income	71 M€	77 M€	116 M€	119 M€

, can be extracted from the annual net income of GENCO with the given portfolio (last row in Table 6). For example, the net income of the BSS in the power system (PS) without penalty clause is the difference of the net income in case 2 and the net income in case 1, while the net income of the BSS in the system with penalty clause is obtained by subtracting the net income in cases 1p from that in case 2p. The net incomes for PSHP were determined analogously. As expected, the contribution of both ESS types is slightly higher in the PS with the penalty clause.

3.2.2. Optimal Size of ESS

The optimal size of the ESS for the GENCO with the above production portfolio (PVPS, WPS, TPP) was analyzed through a series of simulation calculations of the optimal GENCO operation in which the storage size was varied. The storage power capacity of both BSS and PSHP types was varied, falling in the range from 50 to 500 MW with a step of 50 MW. Subsequently, upon analyzing the results, it was observed that the optimal BSS size was in the smaller range, so additional simulations were conducted with BSS power capacity of 25 MW and 75 MW. Simultaneously, with the increase in the storage power capacity, some storage parameters, such as energy capacity, O&M costs, etc., have also increased linearly. The limits of all variables and coefficient values for both storage types varied in the calculations are shown in

Table 8. Varying ESS variables limits and coefficients.

	PSHP	BSS
Power capacity	50, 100, …, 500 MW	25, 50, …, 500 MW
$s_{n,t}^u$	(1.5, 2, …, 6) × 106 m3	250, 500, …, 5000 MWh
$s{d}_{n,t}^u$	(0.75, 1, …, 3) × 106 m3	oversized
s1,0	(0.75, 1, …, 3) × 106 m3	125, 250, …, 2500 MWh
sd1,0	(0.375, 0.5, …, 1.5) × 106 m3	oversized
$h_{n,t}^u$	(0.055, 0.11, …, 0.55) × 106 m3	0.025, 0.05, …, 0.5 MWh
$p_{n,t}^u$	(0.05, 0.1, …, 0.5) × 106 m3	0.025, 0.05, …, 0.5 MWh
k1	15, 30, …, 150 €	45, 90, …, 900 €
k2	15, 30, …, 150 €	2.5, 5, …, 50 €
k4	150, 300, …, 1500 €/start-up	25, 50, …, 500 €/start-up

. The other constants of ESSs are equal to the initial values already shown in Table 1. In the calculations, BSS maximum discharge power was equal to the maximum charging power, while in the cases of PSHP, the maximum production power was always 10% higher than the maximum pumping power.

According to

Table 9. The ESS investment unit prices.

Unit Prices	PSHP	BSS
Cost per unit of power rating	2.11 M€/MW	2.03 M€/MW
Cost per unit of storage capacity	168.01 M€/(106 m3)	0.31 M€/MWh

, which shows the unit cost of power rating and the unit cost of storage capacity for both types of storage, the investment cost of the storage systems was calculated.

The investment cost of the storage systems was calculated based on Table 9, which shows the unit cost of power rating and the unit cost of storage capacity for both types of storage. Unit prices of energy storage systems were taken from [33]. These prices were increased by 50% due to inflation in the case of PSHP, while in the case of BSS, they were decreased by 10% due to the advancement of technology resulting in lower prices.

The results of the optimal storage size calculation for GENCO with the above defined portfolio in the power system without and with penalties for the BSS, as well as for the PSHP, are graphically presented in Figure 7, Figure 8, Figure 9 and Figure 10, respectively. In the presented diagrams:

• The investment cost is the sum of the ESS cost per power and the ESS cost per stored capacity divided by the storage life in years. The lifetime for BSS was estimated at 20 years and for PSHP at 40 years. So, the investment cost represents the annual ESS investment cost. As the size of the ESS increases, the investment costs rise linearly;
• The ESS net income represents the annual contribution of ESS operation, calculated as the difference between the operation of the GENCO with and without storage. In the same way as the net incomes of the 100 MW ESS were calculated in Table 7, the net incomes of the EES of both types were calculated for the specified power range. As the size of the ESS increases, the net income also rises, initially at a rapid rate, and subsequently at a significantly slower pace;
• Profit refers to the annual profit of EES, calculated as the difference between its annual net income and its annual investment costs. The profit curves are concave with a determinable maximum. For an extremely oversized ESS, the profit may even become negative.

The results of ESS annual profit calculations around the optimal storage size are additionally presented in

Table 10. The BSS optimal size analysis in 25–150 MW range of charging power.

	Charging power (MW)	25	50	75	100	150
	Investment cost (M€)	6.4	12.8	19.2	25.6	38.4
No penalty clause	Net income (M€)	49.8	64.0	69.3	70.9	73.6
	Profit (M€)	43.4	51.2	50.1	45.3	35.2
Penalty clause	Net income (M€)	50.8	66.5	73.4	76.9	82.7
	Profit (M€)	44.4	53.7	54.2	51.3	44.3

and

Table 11. The PSHP optimal size analysis in 150–350 MW range of pumping power.

	Pumping power (MW)	150	200	250	300	350
	Investment cost (M€)	24.5	30.5	36.6	42.6	48.7
No penalty clause	Net income (M€)	132.5	142.8	150.4	156.3	161.5
	Profit (M€)	108.0	112.3	113.8	113.7	112.8
Penalty clause	Net income (M€)	137.1	147.8	154.1	157.7	160.5
	Profit (M€)	112.6	117.3	117.5	115.1	111.8

for BSS and PSHP, respectively. Figure 7 and Figure 8, as well as Table 10, show that the optimal size of the BSS of GENCO with a given portfolio in the power system without the penalty clause is around 50 MW, and with the penalty clause in the power range from 50 to 75 MW. Analogously, it can be seen from Figure 9 and Figure 10, as well as Table 11, that the optimal size of PSHP of the same GENCO in the power system without the penalty clause is in the power range of 250 to 300 MW, and with the penalty clause in the power range of 200 to 250 MW. It should be noted that the obtained values of the optimal storage size are highly dependent on the assumed ESS unit investment costs, as well as on the ESS estimated lifetime.

4. Conclusions

The sizing of a utility scale energy storage suitable for a mixed-portfolio GENCO operating in a market-oriented power system with high penetration of renewables is important for its market success. The proposed method considers several uncertain parameters, such as the production of renewable sources, spot market energy prices, ancillary services prices, and fuel prices, among others. It may also consider a penalty to the GENCO for non-delivered energy. By efficiently managing these uncertainties, the approach contributes to economic and environmental sustainability, supporting the transition towards cleaner and more resilient energy systems.

Numerical examples showed that the profit curves, calculated as the difference between the ESS annual net income and the EES annual investment costs, are concave in all study cases. Hence, the maximum of the profit curve or the optimal EES size for GENCO operation under the given conditions can be determined. Further investment in larger storage only reduces the profit, which may even become negative if ESS is extremely oversized. The influence of the ESS characteristics, as well as the power system operation rules on the optimal operation of a given GENCO and the optimal ESS size, are presented in several tables and diagrams.

The proposed GENCO optimal operation model and the proposed ESS sizing method can be used to calculate the profitability of GENCO investments in energy storage technologies, determine the optimal size and characteristics of a certain type of energy storage, compare different ESSs or variant solutions of an ESS, optimize the joint operation of production units and ESS, and more. The optimal storage size for a GENCO in the coming years can be investigated by simulating assumed future trends in market prices of energy, ancillary services, fuels, etc. This approach not only enhances the economic viability of VRES integration but also promotes sustainable development by optimizing resource utilization and reducing greenhouse gas emissions.