Stochastic Optimal Strategies and Management of Electric Vehicles and Microgrids

Abstract

This study combines the Nash–Cournot competition model and the stochastic optimization model to examine the impact of electric vehicle (EV) quantity fluctuations on microgrid operations, aiming to optimize energy usage in a competitive electricity market. Integrating distributed energy resources and bidirectional charging, microgrids offer a novel approach for energy optimization, aiding in renewable energy generation, peak demand management, and emission reduction. Empirical evidence highlights benefits in Taiwan’s electricity market and net-zero emissions target by 2050, with a case study demonstrating enhanced local renewable energy generation due to EVs and microgrid integration. As the number of EVs increases, electricity sales from microgrids decline, but electricity purchases remain stable. The degree of electricity liberalization also influences the supply and demand dynamics of the electricity market. Microgrids selling electricity only to the main grid increases total power consumption by 65.55 million MWh, reducing the market share of the state-owned utility (Taipower). Conversely, allowing retailers to purchase from microgrids increases total consumption by 30.87 million MWh with a slight market share decrease for Taipower. This study contributes to providing an adaptable and flexible general model for future studies to modify and expand based on different scenarios and variables to shape energy and environmental policies.

1. Introduction

The integration of electric vehicles (EVs) and microgrids stands as a pivotal solution to current issues, harnessing local renewable energy resources to generate power while facilitating the charging of EVs [1,2,3,4]. Subsequently, surplus power derived from this process is directed back to the main grid. Distributed energy resource-based microgrids can effectively integrate renewable energy sources, EVs, and energy storage systems to improve the reliability and resilience of the main grid [5]. Notably, EVs also serve a dual purpose. Besides functioning as consumers in the power system, EVs can also act as energy storage units by injecting power back into the microgrid during periods of peak load. This capability helps mitigate power transmission losses and reduce greenhouse gas emissions [1,6,7]. This study endeavors to conduct simulations and measurements to assess the impact of EVs and microgrids in the electricity market, aiming to contribute novel insights to the scientific discourse in this domain.

In 2017, the Taiwanese government passed the Electricity Act Amendment, initiating the liberalization of the electricity market. Unlike the previous situation where the Taiwan Power Company (Taipower) held a monopoly on power generation, transmission, distribution, and retail, the amendment allows the private sector to participate in renewable energy generation and retail markets, while retaining state ownership of the transmission and distribution networks. Further legislative amendments are planned to open up traditional power wheeling and direct supply businesses, aiming to achieve full liberalization of the energy sector [8]. On 22 April 2021, Earth Day, the Taiwanese government declared its goal of achieving net-zero emissions by 2050, and in March 2022, it officially announced the “Taiwan’s Pathway to Net-Zero Emissions in 2050”, outlining the trajectory and action plan towards achieving net-zero emissions by 2050 [9]. With 90% of carbon emissions in Taiwan attributed to power consumption, it is crucial to ensure a stable power supply while achieving net-zero emissions in the energy sector.

In alignment with Taiwan’s 2050 net-zero transition goal, the Ministry of Transportation has introduced key strategic action plans for “Electrification and Decarbonization of Transportation”, aiming to achieve full electrification of bus transportation by 2030 and comprehensive electrification of new passenger cars and scooters by 2040 [10]. EVs represent a promising advancement in transportation technology, utilizing external energy sources to power vehicles equipped with rechargeable batteries for energy storage. This innovative electrical drive system effectively reduces reliance on petroleum combustion, thereby contributing to a reduction in air pollution emissions [11,12]. However, challenges such as high costs and insufficient public or private Electric Vehicle Supply Equipment (EVSE) persist, hindering the widespread adoption of EVs [12,13,14,15]. According to research findings, EVs are not anticipated to precipitate an energy demand boost, but they have the potential to reshape the load curve. A Monte Carlo analysis revealed that a 25 percent EV penetration in a typical residential feeder of 150 households would increase the peak load by around 30 percent. Operating public fast charging stations, which exhibit highly volatile and spiky load profiles, will necessitate additional efforts to maintain system balance. It is noteworthy that a single fast charging station may quickly surpass the peak-load capacity of a standard feeder-circuit transformer. While the investment required for EVSE may be minimal at low EV adoption rates, it will escalate rapidly as the number of EVs increases, eventually stabilizing at higher levels of adoption [16]. Managing the charging/discharging of distributed batteries has become a critical aspect for the successful integration of EVs and hybrid electric vehicles (HEVs) into power grids [16,17,18,19].

On a parallel front, by utilizing energy management technology and integrating energy storage systems, microgrids emerge as a notable solution to revolutionize renewable energy sources’ power generation dynamics [20,21,22,23,24], bolstering local renewable energy generation, mitigating power transmission losses, and curtailing greenhouse gas emissions within the energy sector [25,26,27,28].

The impact of EVs and microgrids on the competitive electricity market in this study is assessed through the application of Nash–Cournot and stochastic optimization models. Initially, the Nash–Cournot model is employed to analyze the interactive behavior of microgrids within the competitive electricity market. The solution to this model involves deriving the Karush–Kuhn–Tucker (KKT) optimality conditions, which are then combined to form a linear complementarity problem (LCP) model. The LCP model incorporates both complementarity and non-negativity conditions, aiming to ensure the optimal solution satisfies these constraints. Specifically, it seeks a real n-tuple vector variable, x, which must satisfy a perpendicular condition, x^T(Ax + B) = 0, along with two non-negativity conditions, (Ax + B) ≥ 0 and x ≥ 0. These conditions are represented as follows: [0 ≤ (Ax + B)] ⊥ [x ≥ 0], where A is a real n × n matrix, B is a real n-tuple vector, and the perpendicular relationship is denoted by the operator “⊥”. Furthermore, the LCP model imposes constraints on the multiplication of the ith element of (Ax + B) and the ith element of x equal to zero, as well as ensuring non-negativity for both (Ax + B)_i and x_i, ∀ i = 1, 2, …, n. This results in n complementarity conditions and 2n non-negativity constraints being combined into n equality equations, forming a system of equations with n variables and n equations. This square system of equations can then be solved.

To solve the LCP, the General Algebraic Modeling System (GAMS) with the PATH solver is utilized, leveraging the GAMS’s capability to handle complex optimization problems. The GAMS serves as an algebraic language proficient in optimizing various problem types, including linear, nonlinear, integer, and complementarity problems [29]. Meanwhile, the PATH solver employs a Newton-based optimization algorithm, iterating towards an optimal solution until convergence is achieved [30].

Secondly, an optimization model is proposed for a multi-energy microgrid, aiming to minimize annual expenses to reach an optimal operation and performance under uncertainty. This mixed-integer linear optimization model determines an optimal mix of energy sources (battery, combined heat and power units, thermal energy storage, gas boiler, and photovoltaic systems), scale, and associated dispatch strategies. The energy management system optimizes total annual expenses while enhancing system resilience, considering hourly electrical and thermal load profiles [31]. Additionally, a stochastic programming model incorporating profit maximization is developed to simulate the behavior of EVs and microgrids [32,33,34,35,36,37]. This model enables the determination of stochastic optimal operating strategies, accounting for uncertain decision-making processes. Stochastic optimization involves multiple stages characterized by uncertainty, with each stage encompassing stochastic events and decision-making procedures. The objective function at each stage represents the expected profit over uncertain scenarios, and the summation of objective values across all stages constitutes the overall objective function. Subject to inequality, equality, and non-negativity constraints, this approach offers a robust framework for analyzing the dynamic interactions between EVs and microgrids within the competitive electricity market.

This study makes a significant contribution by proposing a general model for examining market equilibrium and conducting a stochastic optimization analysis of EVs and microgrids within a competitive electricity market, distinguishing it from related studies. The main aim of this study is to simulate the impact of fluctuations in EV numbers on the optimal stochastic operation decisions of the microgrid and the electricity market under the uncertainty of grid supply and demand with the development of Taiwan’s electricity market liberalization. This includes changes in electricity sales from the microgrids and overall electricity market supply and demand, providing policy and market management recommendations.

The paper is structured as follows: Section 2 introduces and discusses the methods employed, while Section 3 presents and discusses case studies involving EVs and microgrids in the Taiwanese electricity market. Finally, Section 4 presents the conclusions drawn from the study.

2. Methods

This section examines the impact of EVs and microgrids in the competitive electricity market. Initially, a Nash–Cournot model integrated with a stochastic Poisson Process to simulate the entry of EVs into microgrids and assess their impact. This framework enables the simulation of market equilibrium. Subsequently, a stochastic optimization model is formulated to analyze the optimal operation and performance of the microgrid. This comprehensive methodology offers a framework for evaluating the impact and optimal management of EVs and microgrids within the competitive electricity market under uncertain scenarios.

In a non-cooperative game, a Nash equilibrium is a strategy combination where each participant maximizes their expected payoff. It is an effective research method that can model uncertainty factors, environmental changes, and the interactions among economic entities, allowing for the detailed analysis of micro-level economic issues [38]. The Nash–Cournot model used in this study is frequently applied in oligopolistic competitive environments [39]. A stochastic Poisson Process is one of the most widely used counting processes. It is commonly applied in situations where it is necessary to count events that occur at a specific rate but entirely randomly (without a defined structure). By obtaining relevant statistical samples, it can be used to consider the occurrences of certain rare events over very short intervals [40]. Additionally, the linear complementarity problem (LCP) model can be adapted to different regions and situations by using key variables to explain the differences in occurrence rates. The stochastic optimization model is a mathematical programming method suitable for solving uncertainty problems, introducing random variables to explore the distribution of different scenario probabilities, making it more apt for solving real-life issues [41]. To solve the stochastic optimization problem, a number of scenarios can be generated based on real-world conditions to account for uncertainties [42]. This study combines these commonly used international theoretical tools to propose a general model for simulating the short-term electricity market in an oligopolistic competitive environment and rare EV charging events. The aim is to analyze and quantify the impact of EVs and microgrids, finding the optimal solution under market equilibrium. The model is designed to be adaptable and flexible, considering uncertainty factors, changing environmental factors, and interactions between economic entities. In the first stage of the model, the input variables include the demand curve of the electricity market, generator parameters, and microgrid uncertainties to simulate market equilibrium. Next, a stochastic optimization model is used to generate the optimal operation of microgrids under uncertainties, considering generator parameters and EV uncertainties. Figure 1 illustrates the flowchart of the methodology used in this study.

The Nash–Cournot competitive model for the electricity market is constructed within a framework where electricity retailer f generates power, fpr_f,n,t, purchases electricity from microgrids, fby_f,t,s, and sells it, fse_f,t, to users. The microgrids also distribute surplus power, sell_i,s,t, to other electricity retailers. CT_f,n represents the power generation cost for electricity retailer f, while CTM_f denotes the electricity selling price from the microgrids to electricity retailer, and BL_t signifies the number of hours at time block t. The electricity price is computed based on the inverse demand curve, represented as [PI_t − (PI_t/QI_t) × (∑_g fse_g,t)]; here, PI_t and QI_t represent the price intercept and the quantity intercept of the demand curve, respectively, and the revenue is calculated as the product of the price and the total sale volume. Equation (1) delineates the total profit for electricity retailer f, calculated as the difference between total revenue and generation costs ∑_n,t [(BL_t) × (CT_f,n) × (fpr_f,n,t)], along with the purchase cost from microgrids, ∑_t,s [(PROB_s) × (CTM_f) × (fby_f,t,s)]. This study addresses the uncertainty in renewable energy generation, which in turn affects electricity trading facilitated by microgrids to electricity retailers, with PROB_s representing the probability of stochastic scenarios. In this study, each case simulates the optimal solution under a combination of three scenarios, low uncertainty, medium uncertainty, and high uncertainty, each with a corresponding probability. Each scenario corresponds to a probability (PROBs) of 0.3333, 0.3334, and 0.3333, respectively, totaling a probability of 1.

To ensure that power generation remains within the installed capacity FCAP_f,n, the generation capacity constraint is delineated in Equation (2). Moreover, the total electricity purchased must not exceed the total power provided by the microgrids, leading to the formulation of the microgrid purchase constraint in Equation (3). Furthermore, Equations (4) and (5) articulate the power balance equation and non-negativity constraints, respectively. The related input data are shown in

Table A1. Related parameters regarding electricity demand.

t	BL(t)	PI(t)	QI(t)
1	2260	194.29	64,764.59
2	5000	173.34	57,779.86
3	1500	152.39	50,795.14

and

Table A2. Generator parameters.

Electricity Retailer (f)	Generator (n)	CT(f,n)	FCAP(f,n)
Taipower	1	87.42	2000
Taipower	2	27.53	600
Taipower	3	80.25	300
Taipower	4	80.245	2898
Taipower	5	27.53	5500
Taipower	6	80.25	280
Taipower	7	80.25	1785
Taipower	8	80.25	1118
Taipower	9	27.53	2100
Taipower	10	80.25	2226
Taipower	11	27.53	600
Taipower	12	87.42	750
Taipower	13	80.25	1050
Taipower	14	80.25	87
EverPower	1	80.25	900
KuoKuang	1	80.25	480
HsinTao	1	80.25	600
StarEnergy	1	80.25	490
MaiLao	1	27.53	1800
ChiaHui	1	80.25	670
SunBa	1	80.25	980
HoPing	1	27.53	1300

of Appendix A. The KKT optimality conditions are derived to solve the Nash–Cournot model for the profit-maximizing problem described in Equations (1)–(5). Decision variables and dual variables includes fse_f,t, fpr_f,n,t, fby_f,t,s, ρ_f,n,t, σ_f,t,s, and θ_f,t; then, the KKT conditions of the decision and dual variables are shown in Equations (6)–(16). Here, the dual variables, ρ_f,n,t, σ_f,t,s, and θ_f,t represent the generation capacity constraint of generator, the electricity sale constraint from the microgrids to electricity retailer, and the power balance constraint of the electricity retailer, respectively. The integration of these KKT conditions leads to the formulation of an LCP model.

Max           ∑_t [(BL_t) × (PI_t − (PI_t/QI_t) × (∑_g fse_g,t)) × (fse_f,t)]
−∑_n,t [(BL_t) × (CT_f,n) × (fpr_f,n,t)]
−∑_t,s [(PROB_s) × (CTM_f) × (fby_f,t,s)]

(1)

−FCAP_f,n + fpr_f,n,t ≤ 0                 (ρ_f,n,t)     ∀n, t

(2)

−∑_i sell_i,s,t + fby_f,t,s ≤ 0                 (σ_f,t,s)     ∀t, s

(3)

−∑_n fpr_f,n,t − fby_f,t,s + fse_f,t = 0                 (θ_f,t)     ∀t

(4)

fse_f,t, fpr_f,n,t, fby_f,t,s ≥ 0                 ∀n, t

(5)

For fse_f,t ≥ 0, KKT optimality conditions are

[−(BL_t) × (PI_t − (PI_t/QI_t) × (fse_f,t + ∑_g fse_g,t)) + θ_f,t] ≥ 0        ∀f, t

(6)

(fse_f,t) × [−(BL_t) × (PI_t − (PI_t/QI_t) × (fse_f,t + ∑_g fse_g,t)) + θ_f,t] = 0         ∀f, t

(7)

For fpr_f,n,t ≥ 0, KKT optimality conditions are

[(BL_t) × (CT_f,n) + ρ_f,n,t − θ_f,t] ≥ 0                 ∀f, n, t

(8)

(fpr_f,n,t) × [(BL_t) × (CT_f,n) + ρ_f,n,t − θ_f,t] = 0                 ∀f, n, t

(9)

For fby_f,t,s ≥ 0, KKT optimality conditions are

[(PROB_s) × (CTM_f) + σ_f,t,s − θ_f,t] ≥ 0                  ∀f, t, s

(10)

(fby_f,t,s) × [(PROB_s) × (CTM_f) + σ_f,t,s − θ_f,t] = 0                 ∀f, t, s

(11)

For ρ_f,n,t ≥ 0, KKT optimality conditions are

[FCAP_f,n − fpr_f,n,t] ≥ 0                 ∀f, n, t

(12)

(ρ_f,n,t) × [FCAP_f,t − fpr_f,n,t] = 0                 ∀f, n, t

(13)

For σ_f,t,s ≥ 0, KKT optimality conditions are

[∑_i sell_i,s,t − fby_f,t,s] ≥ 0                 ∀f, t, s

(14)

(σ_f,t,s) × [∑_I sell_i,s,t − fby_f,t,s] = 0                 ∀f, t, s

(15)

For θ_f,t, KKT optimality conditions is

[−∑_n fpr_f,n,t − fby_f,t,s + fse_f,t] = 0                 ∀f, t

(16)

EV entry is simulated using a stochastic Poisson Process, where random occurrences follow a defined average rate within a fixed time frame. This distribution quantifies the likelihood of specific events happening over a given period. The average arrival rate of EVs is denoted as λ car/day. For a given number of EVs, denoted as num, the Poisson probability of num vehicles entering in a day is determined by Equation (17). Consequently, this study employs the Poisson Process to simulate the entry of EVs into microgrids. It is important to note that num! represents the factorial of num, denoted as num! = (num) × (num − 1) × (num − 2) × … × (2) × (1).

Prob(num) = (λ^num) × exp(−λ)/(num!)

(17)

A two-stage stochastic optimization model is developed to assess the optimal management and impact of microgrids integrated with EVs in competitive electricity markets. The model constructed in this study aims to minimize the annual cost of microgrids. The microgrid setup comprises several items of equipment, such as EVs, fuel cells, gas turbines, photovoltaic systems, generators, and batteries. Among the equipment, the battery serves as an existing device with a capacity of 10 kW. Both the fixed costs and charging/discharging costs of the batteries are disregarded in this study.

In the initial stage of decision-making, investment decisions regarding the capacity of power generation are made. This involves determining the number of fuel cells (fcn_i), gas turbines (gtn_i), photovoltaic systems (pvn_i), and wind power generators (wdn_i). The model takes into account the uncertainty in electricity demand from EVs and the variability in the power supply from the photovoltaic system and wind power generators. Consequently, this study assumes that the EV model is the Nissan Leaf, with a battery capacity of 24 kWh [43]. A lower depth of discharge (DoD) can mitigate degradation [43]. The DoD is set to 80% for the relevant model parameter settings, resulting in an available battery capacity of 20 kWh in this study. The model also assumes the use of DC chargers for bidirectional charging, treating EVs entering the microgrid as batteries capable of bidirectional charging which regulates microgrid power generation and electricity purchase/sale. This study proposes employing the Poisson Process model to simulate the entry of EVs into microgrids. To simplify the related analysis, the average arrival rate of EVs in the microgrid is set to three scenarios (1–3 EVs). Subsequently, in the second stage of stochastic decision-making, the optimal operation of microgrids with EVs is ascertained under conditions of uncertainty. During periods of power shortage, the microgrid purchases electricity (buy_i,s,t), whereas it sells excess power (sell_i,s,t) during instances of power surplus.

In Equation (18), the total cost of the microgrid is computed, with the objective of minimizing this cost. The equation, Σ_i [(FCFC × fcn_i) + (GTFC × gtn_i) + (PVFC × pvn_i) + (WDFC × wdn_i)], Σ_i,s,t [(FCVC × BL_t × fcg_i,s,t) + (GTVC × BL_t × gtg_i,s,t)], and Σ_i,s,t [(FCRC × fcf_i,s,t) + (GTRC × gtf_i,s,t)], respectively, accounts for fixed costs, variable operating and maintenance costs, and fuel costs for each generator. Specifically, it includes the fixed costs of fuel cells (FCFC), gas turbines (GTFC), photovoltaic systems (PVFC), and wind power generators (WDFC); the variable costs of fuel cells (FCVC) and gas turbines (GTVC); and the fuel costs of fuel cells (FCRC) and gas turbines (GTRC). Here, fcg_i,s,t and gtg_i,s,t represent the power generation of fuel cells and the power generation of gas turbines, and fcf_i,s,t, and gtf_i,s,t, represent the fuel demand of fuel cells and fuel demand of gas turbines, respectively. In this study, we have considered the specific operational lifespans of installed equipment. For instance, the estimated operational lifespan of a PV module is around 30–35 years [44], while EV batteries typically have a warranty period of around 8–10 years [43]. Additionally, photovoltaic systems and batteries experience efficiency degradation over time. Photovoltaic systems degrade at a rate of less than 1% per year [45]. Across all vehicles, on average, EV batteries degrade at a rate of 2.3% per year [46]. Factors such as deeper discharge cycles and the use of fast charging can further accelerate battery aging. [43]. To optimize the investment portfolio and operational performance, the investment costs for the relevant equipment are spread over their operational lifespan as fixed costs.

Additionally, the equation includes the cost of electricity trading between the microgrid and the main grid, represented by [ELEP_t × BL_t × (buy_i,s,t − sell_i,s,t)]. Here, ELEP_t represents the electricity selling price of the main grid. Equation (19) defines the energy storage equation for EVs, where the energy storage at the next stage, evs_h,i,s,tt+1, is determined by the current stage’s energy storage, evs_h,i,s,tt, plus the power inflow, evi_h,i,s,tt, minus the power outflow, evo_h,i,s,tt. Equation (20) represents the battery capacity constraint, where EVCP represents the energy storage capacity of the electric vehicle, while Equations (21) and (22) establish boundary conditions for EVs. Similarly, Equation (23) calculates the storage status of the microgrid batteries, where the energy storage at next storage, bts_i,s,t+1, is equal to the current storage, bts_i,s,t, plus the power inflow, bti_i,s,t, minus the power outflow, bto_i,s,t. Equation (24) imposes a capacity constraint on microgrid batteries, where BTCP represents energy storage capacity of battery. Furthermore, Equations (25) and (26) set constraints on the power generation capacity of fuel cells and gas turbines, where FCCP and GTCP represent power generation capacity of fuel cells and of gas turbines, respectively. Moreover, in this study, microgrids target electricity sales exclusively to electricity retailers. Therefore, the power flow analysis primarily aims to optimize power flow under transmission capacity constraints. Equation (27) indicates that electricity demand, PODM_i,s,t, and the power inflow to the transmission system, pin_i,s,t, are met by the power generation of fuel cells, gas turbines, photovoltaic systems, wind power generators, batteries, and EV battery inflow/outflow, and the difference between electricity purchase and sale. Transmission capacity constraints are defined in Equations (28) and (29), ensuring that the power transmission (Σ_i [PTDF_ik × pin_i,s,t]) of interface k does not exceed the transmission upper bound (TCUP_k) or fall below the lower bound (TCLW_k). Equations (30) and (31) compute fuel consumption for fuel cells and gas turbines, denoted as (fcf_i,s,t) and (gtf_i,s,t), respectively, where FR^FC and FR^GT represent hydrogen for fuel cell consumption per kWh and natural gas for gas turbine consumption per kWh respectively. Greenhouse gas emissions for fuel cells and gas turbines are estimated in Equations (32) and (33) as (fce_i,s,t) and (gte_i,s,t), respectively, where ER^FC and ER^GT represent the carbon dioxide emission rate for fuel cells and for gas turbines, respectively. Integer variable constraints and non-negativity constraints are specified in Equations (34) and (35). The stochastic optimization model aims to minimize the objective function of total cost in Equation (18), subject to the constraints outlined in Equations (19)–(35).

Min  z = Σ_i [(FCFC × fcn_i) + (GTFC × gtn_i) + (PVFC × pvn_i) + (WDFC × wdn_i)]
+Σ_s {PROB_s × Σ_i,s,t [(FCVC × BL_t × fcg_i,s,t)
+ (FCRC × fcf_i,s,t) +(GTVC × BL_t × gtg_i,s,t) + (GTRC × gtf_i,s,t)
+ELEP_t × BL_t × (buy_i,s,t − sell_i,s,t)]}

(18)

Subjective to

evs_h,i,s,tt + evi_h,i,s,tt − evo_h,i,s,tt = evs_h,i,s,tt+1                 ∀ h, i, s, tt

(19)

EVCP_h,i − evs_h,i,s,tt ≥ 0                 ∀ h, i, s, tt

(20)

evs_h,i,s,tt = 0                 ∀h, i, s, tt = 1

(21)

evs_h,i,s,tt = EVCP_h,i                 ∀ h, i, s, tt = TT

(22)

bts_i,s,t + bti_i,s,t − bto_i,s,t = bts_i,s,t+1                 ∀ i, s, t

(23)

BTCP_i × btn_i − bts_i,s,t ≥ 0                 ∀ i, s, t

(24)

FCCP_i × fcn_i − fcg_i,s,t ≥ 0                 ∀ i, s, t

(25)

GTCP_i × gtn_i − gtg_i,s,t ≥ 0                 ∀ i, s, t

(26)

pin_i,s,t + PODM_i,s,t = (∑_h evo_h,i,s,t − ∑_h evi_h,i,s,t) + (bto_i,s,t − bti_i,s,t)
+(buy_i,s,t − sell_i,s,t) + fcg_i,s,t + gtg_i,s,t + PVGN_i,s,t × pvn_i + WDGN_i,s,t × wdn_i    ∀ i, s, t

(27)

Σ_i [PTDF_ik × pin_i,s,t] ≤ TCUP_k                 ∀k, s, t

(28)

−Σ_i [PTDF_ik × pin_i,s,t] ≤ TCLW_k                 ∀k, s, t

(29)

fcf_i,s,t − FR^FC × BL_t × fcg_i,s,t = 0                 ∀ i, s, t

(30)

gtf_i,s,t − FR^GT × BL_t × gtg_i,s,t = 0                 ∀ i, s, t

(31)

fce_i,s,t − ER^FC × BL_t × fcg_i,s,t = 0                 ∀ i, s, t

(32)

gte_i,s,t − ER^GT × BL_t × gtg_i,s,t = 0                 ∀ i, s, t

(33)

gtn_i, fcn_i, pvn_i, wdn_i are integer variables

(34)

fcg_i,s,t, fcf_i,s,t, fce_i,s,t, gtg_i,s,t, gtf_i,s,t, gte_i,s,t ≥ 0                 ∀ i, s, t
bts_i,s,t, bti_i,s,t, bto_i,s,t, evs_i,s,t, evi_i,s,t, evo_i,s,t ≥ 0                 ∀ i, s, t
buy_i,s,t, sell_i,s,t, pin_i,s,t ≥ 0                             ∀ i, s, t

(35)

The LCP model outlined in Equations (6)–(16) is formulated and solved using the GAMS, followed by determining the equilibrium solution using the PATH solver [29]. Additionally, the stochastic optimization of microgrid management described in Equations (18)–(35) is analyzed using the GAMS and the Commercial Planning and Linear Programming Expert (CPLEX) solver. The GAMS code of this study is included in Supplementary Materials. Through these computational processes, the impact of EVs and microgrids can be accurately assessed.

3. Results and Discussion

The following section explores the optimal stochastic operation of microgrids and EVs under demand and generation uncertainties. This study conducts the simulation of three case studies of optimal stochastic microgrid operation with 1–3 EV(s) under demand and generation uncertainties.

Table 1. Optimal stochastic microgrid operation with a single EV under demand and generation uncertainties.

	Battery			Electricity Trading
Uncertainty 1	Total Storage (kWh)	Total Input (kWh)	Total Output (kWh)	Total Sale (kWh)	Total Purchase (kWh)	Total Demand (kWh)
Low Uncertainty	20	10	10	160.43	17.55	1233.88
Medium Uncertainty	10	10	0	340.85	35.10	2467.75
High Uncertainty	0	0	10	521.28	52.65	3701.63

¹ Total cost of the microgrid is New Taiwan Dollar (NTD) 4,998,283.

,

Table 2. Optimal stochastic microgrid operation with two EVs under demand and generation uncertainties.

	Battery			Electricity Trading
Uncertainty 1	Total Storage (kWh)	Total Input (kWh)	Total Output (kWh)	Total Sale (kWh)	Total Purchase (kWh)	Total Demand (kWh)
Low Uncertainty	70	10	10	140.43	17.55	1233.88
Medium Uncertainty	10	10	10	320.85	35.10	2467.75
High Uncertainty	0	0	0	501.28	52.65	3701.63

¹ Total cost of the microgrid is NTD 5,020,183.

and

Table 3. Optimal stochastic microgrid operation with three EVs under demand and generation uncertainties.

	Battery			Electricity Trading
Uncertainty 1	Total Storage (kWh)	Total Input (kWh)	Total Output (kWh)	Total Sale (kWh)	Total Purchase (kWh)	Total Demand (kWh)
Low Uncertainty	10	10	10	120.43	17.55	1233.88
Medium Uncertainty	10	10	10	300.85	35.10	2467.75
High Uncertainty	0	0	0	481.28	52.65	3701.63

¹ Total cost of the microgrid is NTD 5,042,083.

provide comparisons across different numbers of EVs plugged into the microgrid. Table 1 illustrates the case of a single EV, indicating total electricity sales of 160.43 kWh, 340.85 kWh, and 521.28 kWh for low, medium, and high uncertain scenarios, respectively. Table 2 and Table 3 depict similar analyses for two and three EVs, with total sales ranging from 140.43 kWh to 501.28 kWh and 120.43 kWh to 481.28 kWh, respectively.

These findings indicate that, while other electricity demands remain constant, there is a decrease in electricity sales from the microgrid to the main grid as the number of EVs increases. However, this decrease does not result in an increase in electricity purchases from the main grid. The introduction of EVs primarily affects the timing and frequency of battery charging and discharging to enhance local energy generation. This study simulated three scenarios—low, medium, and high uncertainty—demonstrating optimal stochastic microgrid 24 h operations with one to three EVs. Detailed data on renewable energy generation, electricity demand, and battery charge/discharge are presented in

Table A3. Optimal stochastic microgrid operation with a single EV under low-, medium-, and high- uncertainty scenarios.

Source	Scenario	Time (hour)
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
Photovoltaic	1									10	20	30	40	40	40	20	10
	2									20	40	60	80	80	80	40	20
	3									30	60	90	120	120	120	60	30
Wind power generator 1	1	9	37.5	39	36	45	103.5	97.5	105	90	9.75			25.5	36	22.5	90	112.5	90	81	72	9	9		9
	2	18	75	78	72	90	207	195	210	180	19.5			51	72	45	180	225	180	162	144	18	18		18
	3	27	112.5	117	108	135	310.5	292.5	315	270	29.25			76.5	108	67.5	270	337.5	270	243	216	27	27		27
Wind power generator 2	1	0.5	1.5	2	1.5	2.5	3.5	4.5	5	5.5	0.5			1.5	2	1.25	5	6.25	5	4.5	4	0.5	0.5		0.5
	2	1	3	4	3	5	7	9	10	11	1			3	4	2.5	10	12.5	10	9	8	1	1		1
	3	1.5	4.5	6	4.5	7.5	10.5	13.5	15	16.5	1.5			4.5	6	3.75	15	18.75	15	13.5	12	1.5	1.5		1.5
Battery storage	1																	10	10
	2																					10
	3
Battery inflow	1																	10
	2																					10
	3
Battery outflow	1																			10
	2
	3																						10
Electricity sale	1																				160.4
	2																				340.9
	3																				521.3
Electricity purchase	1						17.55
	2									35.1
	3						52.65
Energy demand	1	8.4	11.7	8.7	7.1	5.5	5.4	7.5	29.6	99.2	121.5	119.1	119.2	117.6	113.6	109.5	127.1	96.8	30.6	17.1	16.8	16.7	16.0	14.7	14.9
	2	16.7	23.4	17.5	14.1	11.1	10.8	15.0	59.1	198.4	243.1	238.2	238.5	235.2	227.3	219.1	254.2	193.6	61.2	34.1	33.5	33.4	32.0	29.3	29.7
	3	25.1	35.1	26.2	21.2	16.6	16.1	22.4	88.7	297.5	364.6	357.2	357.7	352.7	340.9	328.6	381.2	290.3	91.7	51.2	50.3	50.0	48.0	44.0	44.6

,

Table A4. Optimal stochastic microgrid operation with two EVs under low-, medium-, and high- uncertainty scenarios.

Source	Scenario	Time (hour)
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
Photovoltaic	1									10	20	30	40	40	40	20	10
	2									20	40	60	80	80	80	40	20
	3									30	60	90	120	120	120	60	30
Wind power generator 1	1	9	37.5	39	36	45	103.5	97.5	105	90	9.75			25.5	36	22.5	90	112.5	90	81	72	9	9		9
	2	18	75	78	72	90	207	195	210	180	19.5			51	72	45	180	225	180	162	144	18	18		18
	3	27	112.5	117	108	135	310.5	292.5	315	270	29.25			76.5	108	67.5	270	337.5	270	243	216	27	27		27
Wind power generator 2	1	0.5	1.5	2	1.5	2.5	3.5	4.5	5	5.5	0.5			1.5	2	1.25	5	6.25	5	4.5	4	0.5	0.5		0.5
	2	1	3	4	3	5	7	9	10	11	1			3	4	2.5	10	12.5	10	9	8	1	1		1
	3	1.5	4.5	6	4.5	7.5	10.5	13.5	15	16.5	1.5			4.5	6	3.75	15	18.75	15	13.5	12	1.5	1.5		1.5
Battery storage	1						10	10	10	10	10	10	10
	2																				10
	3
Battery inflow	1						10
	2																				10
	3
Battery outflow	1													10
	2																					10
	3
Electricity sale	1																				140.4
	2																				320.9
	3																					501.3
Electricity purchase	1						17.55
	2		35.1
	3						52.65
Energy demand	1	8.4	11.7	8.7	7.1	5.5	5.4	7.5	29.6	99.2	121.5	119.1	119.2	117.6	113.6	109.5	127.1	96.8	30.6	17.1	16.8	16.7	16.0	14.7	14.9
	2	16.7	23.4	17.5	14.1	11.1	10.8	15.0	59.1	198.4	243.1	238.2	238.5	235.2	227.3	219.1	254.2	193.6	61.2	34.1	33.5	33.4	32.0	29.3	29.7
	3	25.1	35.1	26.2	21.2	16.6	16.1	22.4	88.7	297.5	364.6	357.2	357.7	352.7	340.9	328.6	381.2	290.3	91.7	51.2	50.3	50.0	48.0	44.0	44.6

and

Table A5. Optimal stochastic microgrid operation with three EVs under low-, medium-, and high- uncertainty scenarios.

Source	Scenario	Time (hour)
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
Photovoltaic	1									10	20	30	40	40	40	20	10
	2									20	40	60	80	80	80	40	20
	3									30	60	90	120	120	120	60	30
Wind power generator 1	1	9	37.5	39	36	45	103.5	97.5	105	90	9.75			25.5	36	22.5	90	112.5	90	81	72	9	9		9
	2	18	75	78	72	90	207	195	210	180	19.5			51	72	45	180	225	180	162	144	18	18		18
	3	27	112.5	117	108	135	310.5	292.5	315	270	29.25			76.5	108	67.5	270	337.5	270	243	216	27	27		27
Wind power generator 2	1	0.5	1.5	2	1.5	2.5	3.5	4.5	5	5.5	0.5			1.5	2	1.25	5	6.25	5	4.5	4	0.5	0.5		0.5
	2	1	3	4	3	5	7	9	10	11	1			3	4	2.5	10	12.5	10	9	8	1	1		1
	3	1.5	4.5	6	4.5	7.5	10.5	13.5	15	16.5	1.5			4.5	6	3.75	15	18.75	15	13.5	12	1.5	1.5		1.5
Battery storage	1															10
	2																				10
	3
Battery inflow	1															10
	2																				10
	3
Battery outflow	1																10
	2																					10
	3
Electricity sale	1																				120.4
	2																				300.9
	3																					481.3
Electricity purchase	1						17.55
	2									35.1
	3						52.65
Energy demand	1	8.4	11.7	8.7	7.1	5.5	5.4	7.5	29.6	99.2	121.5	119.1	119.2	117.6	113.6	109.5	127.1	96.8	30.6	17.1	16.8	16.7	16.0	14.7	14.9
	2	16.7	23.4	17.5	14.1	11.1	10.8	15.0	59.1	198.4	243.1	238.2	238.5	235.2	227.3	219.1	254.2	193.6	61.2	34.1	33.5	33.4	32.0	29.3	29.7
	3	25.1	35.1	26.2	21.2	16.6	16.1	22.4	88.7	297.5	364.6	357.2	357.7	352.7	340.9	328.6	381.2	290.3	91.7	51.2	50.3	50.0	48.0	44.0	44.6

of Appendix A. Additionally, charge/discharge curves for 24 h microgrid operations under these scenarios with three EVs are depicted in Figure 2, Figure 3 and Figure 4.

To mitigate the decline in electricity sales, first the upper and lower limits of EV charging numbers that can be supported by the microgrid composition while maintaining a certain level of profitability should be evaluated, and sensitivity analyses conducted to examine the impact of various variables. Secondly, this evaluation can be expanded to include a broader range of EV models and diverse charging behaviors. Furthermore, integrating bi-directional charging infrastructures equipped with 4-Quadrant P-Q control methods can enhance both the power quality and financial returns of the microgrid by delivering active and/or reactive power to provide different electricity commodities to trade in different electricity markets. This could bring technical and financial benefits to the microgrid.

Moreover, to examine the impact of electricity sales from the microgrid to the main grid under varying degrees of electricity market liberalization, this study primarily focuses on simulating the Taiwanese electricity market.

Table 4. Simulation of electricity trading from microgrids to Taipower in the Taiwanese electricity market.

Scenarios	Electricity Retailer	Peak Load (MWh)	Medium Load (MWh)	Base Load (MWh)	Total Load (MWh)
0% scenario	Taipower	42,110,580	93,165,000	26,991,671	162,267,251
	Total	58,427,780	129,265,000	35,436,452	223,129,232
5% scenario	Taipower	44,216,109	97,823,250	26,991,671	169,031,030
	Total	60,533,309	133,923,250	35,436,452	229,893,011
10% scenario	Taipower	46,321,638	102,481,500	26,991,671	175,794,809
	Total	62,638,838	138,581,500	35,436,452	236,656,790
15% scenario	Taipower	48,427,167	103,458,039	26,991,671	178,876,877
	Total	64,744,367	139,558,039	35,436,452	239,738,858
20% scenario	Taipower	50,532,696	103,458,039	26,991,671	180,982,406
	Total	75,272,012	156,801,045	39,935,064	272,008,122
30% scenario	Taipower	54,655,552	103,458,039	26,991,671	185,105,262
	Total	83,585,572	163,432,971	41,665,300	288,683,843

outlines a scenario where Taipower is the sole entity with contracts to purchase electricity from local microgrids. In contrast,

Table 5. Simulation of electricity trading from microgrids to all electricity retailers in the Taiwanese electricity market.

Scenarios	Electricity Retailer	Peak Load (MWh)	Medium Load (MWh)	Base Load (MWh)	Total Load (MWh)
0% scenario	Taipower	42,110,580	93,165,000	26,991,671	162,267,251
	Total	58,427,780	129,265,000	35,436,452	223,129,232
5% scenario	Taipower	44,216,109	97,823,250	26,962,609	169,001,968
	Total	61,349,169	135,728,250	35,465,515	232,542,934
10% scenario	Taipower	46,321,638	101,653,039	26,933,546	174,908,223
	Total	64,270,558	141,363,039	35,494,577	241,128,174
15% scenario	Taipower	48,427,167	100,750,539	26,904,484	176,082,190
	Total	67,191,947	142,265,539	35,523,640	244,981,126
20% scenario	Taipower	50,532,696	99,848,039	26,875,421	177,256,156
	Total	70,113,336	143,168,039	35,552,702	248,834,077
30% scenario	Taipower	52,207,972	98,043,039	26,817,296	177,068,307
	Total	73,420,332	144,973,039	35,610,827	254,004,199

examines a scenario in which all electricity retailers are allowed to purchase electricity from microgrids.

The simulations encompass scenarios ranging from 0% to 30% of the electricity import ratio from microgrids to electricity retailers. In Table 4, the results indicate that Taipower sells an additional 22.84 million MWh (equivalent to 185.11 million MWh minus 162.27 million MWh) of electricity when it has contracts for purchasing electricity from microgrids. This outcome stems from Taipower’s contractual obligations to purchase electricity from microgrids. However, despite the increase in total power consumption by 65.55 million MWh (from 223.13 million MWh to 288.68 million MWh), the ratio of Taipower’s sales to the total market sales decreases from 72.72% to 64.12%. Similarly, the ratios of Taipower’s sales to the total market sales decrease by 6.68%, 8.77%, and 11.39% for peak, medium, and base loading, respectively.

Table 5 simulates the scenario where every electricity retailer has the opportunity to purchase electricity from microgrids. With electricity trading enabled, each electricity retailer experiences an increase in electricity sales, resulting in a total power consumption rise of 30.87 million MWh (from 223.13 million MWh to 254.00 million MWh). The change in the ratio of Taipower’s sales to the total market sales is calculated as 3.01% (equal to 72.72% minus 69.71%), attributable to every electricity retailer purchasing electricity from the microgrids. Consequently, this also leads to slight reductions in the ratios of Taipower’s sales to the total market for peak, medium, and base loading, decreasing by 0.96%, 4.44%, and 0.86%, respectively.

4. Conclusions

EVs and microgrids represent advanced, energy-efficient, and environmentally friendly energy solutions. This study makes a substantial contribution by presenting a comprehensive framework for analyzing and quantifying the impact of EVs and microgrids. This study formulates Nash–Cournot competitive models to assess their influence in the electricity markets and establishes a stochastic optimization model to effectively manage these systems. The results illustrate a significant shift in power dynamics and market behaviors as EV penetration increases. Furthermore, it analyzes and explores the differences in and impacts of electricity sales from the microgrid to the main grid under varying degrees of electricity market liberalization in Taiwan, shedding light on the future of energy policies and market dynamics in the region. This study found that, with other electricity demands remaining constant, there is a decrease in electricity sales from the microgrid to the main grid as the number of EVs increases. However, this does not lead to an increase in electricity purchases from the main grid. The introduction of EVs only affects the timing and frequency of battery charging and discharging to enhance local energy generation.

The research limitations in this study involve mainly discussing and simulating case studies related to Taiwan’s electricity market and its regulatory framework. Another limitation is that it examines a scenario analysis of the short-term and uncertain charging demand for EVs, as EVs are still not prevalent in Taiwan. Additionally, in this study, microgrids target electricity sales exclusively to electricity retailers, as mandated by Taiwan’s amended electricity law, which requires electricity transmission and distribution enterprises to handle power dispatch operations. They must ensure the security and stability of the grid and provide necessary ancillary services according to dispatch requirements [8]. Therefore, the analysis of power flow and EV charging control in this study focuses solely on active power, without considering the objective of maintaining power quality. Furthermore, the Nash–Cournot equilibrium analysis method used in this study is primarily applicable to oligopoly markets. However, this study still makes the following contributions. The general model and research framework proposed in this study, based on the Nash–Cournot model, stochastic Poisson Process, and two-stage stochastic programming, endow it with adaptability to various uncertainties, changing environmental factors, and interactions between economic entities [38]. Therefore, it can be extended and applied to evaluate investment and subsidy strategies for EVs and microgrids, influencing energy and environmental policies.

Suggestions for future studies include conducting sensitivity analyses to examine the impact of various variables. To mitigate the decline in electricity sales as the number of EVs increases, future studies could first evaluate the upper and lower limits of EV charging numbers that can be supported by the microgrid composition while maintaining a certain level of profitability. Secondly, expanding the evaluation to include how a larger-scale and diverse EV models could help the microgrid enhance profitability by assessing the impact of and changes in electricity purchasing and sales resulting from different combinations of charging behaviors. Furthermore, despite the initial investment in additional converters for 4-quadrant chargers [47], implementing a bi-directional EV charging infrastructure with 4-quadrant P-Q control and integrating it with the microgrid could enable the delivery of bi-directional active and/or reactive power [48]. Future studies could further evaluate the changes in profitability under the aforementioned scenarios. Future studies could also comprehensively simulate various charging and discharging control strategies for EVs to assess the technical and financial impacts and the benefits on microgrids. This includes participation in fully competitive markets or provision of different electricity commodities, such as ancillary services (e.g., reactive power regulation and frequency regulation) for trading in various electricity markets. Additionally, studies could explore accommodating different market regulations, such as whether electricity retailers are permitted to own major generation equipment.

In addition to analyzing the energy technology and financial benefits, this study also examines how decarbonization trends in renewable energy and EVs, driven by challenges in mitigating climate change, impact the energy and transportation sectors. This study utilizes simulation models to explore how integrating and implementing microgrids and energy management systems can mitigate potential impacts and challenges posed by uncertainties in generation and load profiles, proposing optimization analyses. Building on this foundation, further exploration could include multi-objective and uncertain analyses of renewable energy generation, simulation of greenhouse gas emission policies, and demand-side management incorporating EVs and microgrids.