In this work, I investigate the collaborative optimal scheduling of shared energy storage and electric vehicle charging stations. The rapid deployment of electric vehicles has created a strong demand for coordinated charging infrastructure, while the integration of distributed renewable generation has made the operation of charging stations more complex. I focus on the economic interactions among a shared energy storage operator, a group of electric vehicle charging stations, and the distribution grid. Using game theory, I model the hierarchical competition between the storage operator and the charging stations, as well as the cooperative behaviours among charging stations that exchange power with each other. The overall goal is to reduce the operation cost of electric vehicle charging infrastructure while preserving acceptable revenue for the storage operator.
The motivation of this study comes from the fact that a conventional independently owned battery system is often too expensive for a single charging station. By contrast, shared energy storage allows multiple electric vehicle charging stations to access a centrally operated battery through a paid service. This arrangement reduces the capital burden of individual stations and increases the utilisation rate of the battery. From an economic perspective, the relationship between the storage service provider and the charging stations is a natural leader–follower game. The storage operator first publishes its electricity price or leasing price, and each electric vehicle charging station responds by adjusting its purchase and storage plan. When several charging stations are allowed to trade energy with each other, an additional cooperative game emerges because the stations can reduce their total cost by exploiting complementary renewable generation profiles. I therefore combine a Stackelberg game with a cooperative game in a hybrid framework.
1. Research Background and Problem Statement
The electrification of road transport is one of the most important measures for reducing carbon emissions and fossil-fuel dependence. With the rapid growth of the electric vehicle market, the scale of charging infrastructure has expanded quickly. Chinese statistics show that the number of public charging piles has already exceeded three million, and the total number of registered electric vehicles has passed thirty million. This expansion creates both opportunities and risks. Charging stations introduce a large, time-varying electrical load, while photovoltaic and wind generation connected to charging stations add further uncertainty to the operation of distribution networks.
The main operational challenge is the peak-valley difference caused by uncontrolled electric vehicle charging. Most users start charging when they arrive at work in the morning or return home in the evening, so the charging demand is strongly correlated with commuting patterns. If all electric vehicle charging stations purchase electricity from the grid during these peak periods, the local transformer can become overloaded and the grid voltage may deviate from acceptable limits. Energy storage can shift this load from peak periods to valley periods, and it can also store surplus renewable energy during the middle of the day. However, the investment cost of energy storage remains relatively high for an individual charging station.
Shared energy storage offers a practical solution. Instead of buying an entire battery system, an electric vehicle charging station can pay a service fee to use a centrally managed storage facility. The storage operator benefits from the aggregation of diverse demand profiles, and the charging stations avoid the fixed investment and maintenance cost. In this model, the storage operator must decide the service price, while each electric vehicle charging station must decide how much energy to buy from the grid, how much power to exchange with the storage device, and how much energy to sell to other stations. These decisions are interdependent, which makes game theory a suitable modelling tool.
2. Analytical Framework and Game-Theoretic Preliminaries
I begin with the basic elements of shared energy storage. A shared storage system may be owned by a third-party operator, by the users jointly, or by individual users who share their spare battery capacity. In this thesis I focus on the model in which an independent storage service provider invests in and operates a central battery. The operator sells charging and discharging services to multiple electric vehicle charging stations and may also arbitrage against the grid by purchasing electricity at low prices and selling it at high prices.
The economic interactions in such a system fall into two broad categories:
Non-cooperative Stackelberg game. The storage operator is the leader. It announces the hourly price for energy exchange. The electric vehicle charging stations are followers. They observe this price and minimise their own costs. The leader can anticipate the response of the followers and chooses a price that maximises its own utility. The equilibrium of this game is the Stackelberg equilibrium.
Cooperative game. Electric vehicle charging stations may form a coalition and trade electricity among themselves. In the cooperative game, each station has a disagreement cost, which is the cost it would pay if it acted alone. The coalition seeks to minimise the total cost, and the resulting benefit must be distributed among the stations in a fair way. I use a Nash–Harsanyi bargaining solution to allocate the benefit, and I introduce an exponential mapping function to quantify the bargaining power of each station from its contribution to the inter-station energy transactions.
The combination of the two games is called a hybrid game. The upper level is a leader–follower game between the storage operator and the coalition of electric vehicle charging stations, while the lower level is a cooperative game among the stations.
3. Formulation of the Stackelberg Game with Uncertainty
In the first stage of this research, I consider a group of electric vehicle charging stations equipped with distributed renewable generation. The stations are served by a shared storage operator. I denote the set of stations by \(\mathcal{N}=\{1,2,\ldots,N\}\), and the dispatch horizon by \(\mathcal{T}=\{1,2,\ldots,T\}\). The daily expected load of station \(i\) in time interval \(t\) is \(\mu_{i,t}\). In practice, the real charging load is uncertain. I model the deviation as a zero-mean Gaussian random variable:
\[
P^{\text{load}}_{i,t} = \mu_{i,t} + \xi_{i,t},\qquad \xi_{i,t}\sim \mathcal{N}\left(0,\sigma_{i,t}^{2}\right)
\]
The positive and negative parts of the deviation are denoted by \(\xi_{i,t}^{+}\) and \(\xi_{i,t}^{-}\), respectively. The positive deviation means that the actual electric vehicle load is higher than the forecast, while the negative deviation means that the actual load is lower than the forecast. I define
\[
\xi_{i,t} = \xi_{i,t}^{+} – \xi_{i,t}^{-}
\]
The positive deviation is compensated by an increase in storage discharge and grid purchase, while the negative deviation is compensated by an increase in storage charge. The charging station does not sell electricity back to the grid in this model.
The storage service provider obtains revenue from three sources: leasing income from the electric vehicle charging stations, arbitrage income from trading with the grid, and possible cost reduction from peak shaving. The daily revenue can be written as
\[
R^{\text{SES}} = \sum_{t\in\mathcal{T}}\sum_{i\in\mathcal{N}}\left[
p_{t}\left(P^{\text{ess},c}_{i,t}+\xi^{-}_{i,t}+P^{\text{ess},d}_{i,t}+r_{i,t}\xi^{+}_{i,t}\right)
\right]\Delta t + R^{\text{arb}} – C^{\text{om}}
\]
where \(p_{t}\) is the leasing price, \(P^{\text{ess},c}_{i,t}\) and \(P^{\text{ess},d}_{i,t}\) are the planned charge and discharge power of the shared storage used by station \(i\), \(r_{i,t}\) is the fraction of positive deviation compensated by storage, \(R^{\text{arb}}\) is the arbitrage revenue, and \(C^{\text{om}}\) is the operation and maintenance cost.
Since the load deviation is random, the actual revenue of the storage operator is also random. I use a mean-variance utility to describe the trade-off between profit and risk. The leader’s objective is
\[
\max \quad U^{\text{SES}} =
\mathbb{E}\left[R^{\text{SES}}\right] – \lambda^{S}\,\mathrm{Var}\left[R^{\text{SES}}\right]
\]
where \(\lambda^{S}\) is the risk-aversion coefficient of the shared storage operator. A larger \(\lambda^{S}\) means that the operator is more conservative and penalises the fluctuation of revenue more heavily.
Similarly, the electric vehicle charging stations minimise their cost. The expected utility of the cluster is
\[
\max \quad U^{\text{EVCS}} =
\mathbb{E}\left[R^{\text{EVCS}}\right] – \lambda^{E}\,\mathrm{Var}\left[R^{\text{EVCS}}\right]
\]
where \(R^{\text{EVCS}}\) is the negative total cost of the cluster. The cost includes the grid purchase cost, the storage leasing fee, and the maintenance cost of renewable generation.
4. Chance Constraint Transformation
The random constraints in the charging station model involve the uncertain variables \(\xi^{+}_{i,t}\) and \(\xi^{-}_{i,t}\). Since \(\xi_{i,t}\) is normal, the expected value and variance of the positive and negative parts can be expressed in closed form. The storage power limit and the grid purchase limit cannot be guaranteed with probability one if the deviation is unbounded. Therefore I convert the original random constraints into chance constraints with a confidence level \(1-\varepsilon\).
\[
\Pr\left\{P^{\text{ess},c}_{i,t}+\xi^{-}_{i,t} \le v_{i,t}P^{\text{max},E}_{i}\right\} \ge 1-\varepsilon
\]
\[
\Pr\left\{P^{\text{ess},d}_{i,t}+r_{i,t}\xi^{+}_{i,t} \le (1-v_{i,t})P^{\text{max},E}_{i}\right\} \ge 1-\varepsilon
\]
Using the Cantelli inequality, which provides a one-sided probability bound for a random variable with known mean and variance, I transform these probabilistic constraints into deterministic linear constraints. For a random variable \(X\), the Cantelli inequality gives
\[
\Pr\{X-\mu \ge k\sigma\} \le \frac{1}{1+k^{2}},\quad k>0
\]
Setting \(1/(1+k^{2})=\varepsilon\) yields \(k=\sqrt{(1-\varepsilon)/\varepsilon}\). Consequently,
\[
\Pr\{X \le \mu + k\sigma\} \ge 1-\varepsilon
\]
For the Gaussian random variable \(\xi_{i,t}\), the upper one-sided quantile is
\[
Q_{1-\varepsilon}\left(\xi_{i,t}\right)
=
\mathbb{E}[\xi_{i,t}]
+
\sqrt{\frac{1-\varepsilon}{\varepsilon}}
\sqrt{\mathrm{Var}[\xi_{i,t}]}
\]
The positive and negative parts have equal expectation and variance:
\[
\mathbb{E}[\xi^{+}_{i,t}]
=
\mathbb{E}[\xi^{-}_{i,t}]
=
\sigma_{i,t}\phi(0)
=
\frac{\sigma_{i,t}}{\sqrt{2\pi}}
\]
\[
\mathrm{Var}[\xi^{+}_{i,t}]
=
\mathrm{Var}[\xi^{-}_{i,t}]
=
\sigma_{i,t}^{2}
\left[\Phi(0)-\phi(0)^{2}\right]
=
\sigma_{i,t}^{2}
\left(\frac{1}{2}-\frac{1}{2\pi}\right)
\]
After substitution, the deterministic equivalent of the chance constraints becomes:
\[
P^{\text{ess},c}_{i,t}
+
\frac{\sigma_{i,t}}{\sqrt{2\pi}}
+
\sqrt{\frac{1-\varepsilon}{\varepsilon}}
\,
\sigma_{i,t}
\sqrt{\frac{1}{2}-\frac{1}{2\pi}}
\le
v_{i,t}P^{\text{max},E}_{i}
\]
\[
P^{\text{ess},d}_{i,t}
+
r_{i,t}
\frac{\sigma_{i,t}}{\sqrt{2\pi}}
+
r_{i,t}
\sqrt{\frac{1-\varepsilon}{\varepsilon}}
\,
\sigma_{i,t}
\sqrt{\frac{1}{2}-\frac{1}{2\pi}}
\le
(1-v_{i,t})P^{\text{max},E}_{i}
\]
With this transformation, the stochastic scheduling problem is converted into a deterministic mixed-integer linear programming problem. I then use a co-evolutionary genetic algorithm to solve the Stackelberg game. The main population encodes the leasing price and grid interaction power of the shared storage service provider, while the sub-population encodes the charge and discharge power of each electric vehicle charging station. The two populations evolve in parallel and exchange information through the fitness function.
5. Numerical Results of the Stackelberg Model
I simulate a cluster of three electric vehicle charging stations. Station 1 has a photovoltaic system, station 2 has a wind turbine, and station 3 has both photovoltaic and wind generation. The grid uses a time-of-use tariff, which is summarised in the following table.
| Tariff category | Time interval | Grid selling price (RMB/kWh) | Grid purchase price (RMB/kWh) |
|---|---|---|---|
| Peak period | 09:00–14:00, 19:00–23:00 | 1.12 | 0.86 |
| Medium period | 08:00–09:00, 14:00–19:00, 23:00–24:00 | 0.72 | 0.53 |
| Valley period | 00:00–08:00 | 0.36 | 0.29 |
The shared storage device has a maximum charge and discharge power of 800 kW. The initial and final state of charge is 0.25, and the state-of-charge limits are 0.2 and 0.9. The charge and discharge efficiencies are both 0.95. The risk-aversion coefficient is 0.1 and the violation probability is 0.05.
I compare four scenarios. Scenario A uses independent energy storage at each charging station under a deterministic load. Scenario B uses shared energy storage under a deterministic load. Scenario C uses shared energy storage under load uncertainty. Scenario D is the proposed Stackelberg game with uncertainty, mean-variance utility, and chance constraints.
| Indicator (RMB/day) | Scenario A | Scenario B | Scenario C | Scenario D |
|---|---|---|---|---|
| Cost of station 1 | 5583.59 | 3820.33 | 3463.62 | 3072.15 |
| Cost of station 2 | 5012.07 | 3799.95 | 3309.03 | 3170.72 |
| Cost of station 3 | 4324.50 | 3655.78 | 3221.88 | 2860.15 |
| Total cost of cluster | 14920.16 | 11276.06 | 9994.53 | 9103.02 |
| Revenue of shared storage | — | 4395.81 | 3863.27 | 4966.22 |
Compared with the independent storage case, the shared storage model reduces the total cost of electric vehicle charging stations from 14920.16 yuan to 11276.06 yuan. After introducing the Stackelberg game and the mean-variance utility, the total cost is further reduced to 9103.02 yuan. At the same time, the revenue of the shared storage operator increases from 3863.27 yuan to 4966.22 yuan. The results confirm that both the charging stations and the storage operator can benefit from the proposed coordination mechanism.
6. Hybrid Stackelberg-Cooperative Game Model
In the second part of this thesis, I extend the model by allowing electric vehicle charging stations to trade electricity with each other. The storage operator is still the leader, but the follower is now a coalition of charging stations. Inside the coalition, the stations participate in a cooperative game. This structure is represented by the following system architecture: the shared energy storage operator exchanges energy with the grid and with each station; the stations also have inter-station transmission lines for peer-to-peer energy exchange.
The price hierarchy is designed to create a reasonable incentive for all participants:
\[
\lambda^{\text{min}} < \lambda^{b}_{t} < \lambda^{ij}_{t} < \lambda^{s}_{t} < \lambda^{g}_{t}
\]
where \(\lambda^{b}_{t}\) is the price at which the storage operator buys from a charging station, \(\lambda^{s}_{t}\) is the price at which the storage operator sells to a charging station, \(\lambda^{ij}_{t}\) is the inter-station trading price, and \(\lambda^{g}_{t}\) is the grid retail price. This hierarchy encourages the stations to trade first with each other, then with the storage operator, and finally with the grid.
The energy exchange among stations is modelled by an antisymmetric flow variable \(P_{ij,t}\), which satisfies \(P_{ij,t}=-P_{ji,t}\). A positive value means that station \(i\) exports to station \(j\). This representation simplifies the model, because I only need to define one variable for each ordered pair.
The upper-level model maximises the revenue of the shared storage operator. The objective includes the income from selling electricity to the coalition, the expenditure for buying surplus electricity from the coalition, the arbitrage revenue from the grid, and the daily investment and maintenance cost of the storage device:
\[
\max
\quad
F^{\text{SESO}}
=
\sum_{t=1}^{T}\Delta t\left[
\lambda^{s}_{t}\sum_{i=1}^{n}P^{b}_{i,t}
–
\lambda^{b}_{t}\sum_{i=1}^{n}P^{s}_{i,t}
\right]
+
R^{\text{grid}}
–
C^{\text{storage}}
\]
The storage state-of-charge evolution is:
\[
E^{E}_{t}=E^{E}_{t-1}
+
\left(\eta_{c}P^{ch}_{t}
–
\frac{P^{dis}_{t}}{\eta_{d}}
\right)\Delta t
\]
The power balance relation for the storage operator is:
\[
P^{ch}_{t}+P^{G,s}_{t}+\sum_{i=1}^{n}P^{b}_{i,t}
=
P^{dis}_{t}+P^{G,b}_{t}+\sum_{i=1}^{n}P^{s}_{i,t}
\]
The lower-level model minimises the total cost of the electric vehicle charging station coalition:
\[
\min
\quad
F^{\text{CS}}
=
\sum_{i=1}^{n}
\sum_{t=1}^{T}
\Delta t
\left[
\lambda^{g}_{t}P^{G}_{i,t}
+
\lambda^{s}_{t}P^{b}_{i,t}
–
\lambda^{b}_{t}P^{s}_{i,t}
+
c_{PV}P^{PV}_{i,t}
+
c_{WT}P^{WT}_{i,t}
\right]
\]
The power balance constraint for station \(i\) is:
\[
P^{PV}_{i,t}+P^{WT}_{i,t}+P^{G}_{i,t}+P^{b}_{i,t}+\sum_{j\neq i}P_{ji,t}
=
P^{\text{load}}_{i,t}+P^{s}_{i,t}
\]
The inter-station power is bounded by the line capacity, and the price of inter-station energy must satisfy the hierarchy constraint above.
7. Cooperative Benefit Allocation Method
After the coalition minimises its total cost, the next question is how to allocate the cooperative benefit. If the total cooperative cost is \(C^{*}_{\text{coalition}}\), then the set of individual costs \(C^{*}_{i}\) must sum to this value:
\[
\sum_{i=1}^{n}C^{*}_{i}=C^{*}_{\text{coalition}}
\]
Each station compares its cooperative cost with its disagreement cost \(C^{NC}_{i}\). The disagreement cost is the cost that station \(i\) would pay if it did not participate in the coalition and only traded with the shared storage operator and the grid. The saved cost for station \(i\) is therefore
\[
C^{NC}_{i}-C^{*}_{i}\ge 0
\]
I compute the supplied energy and received energy of each charging station from the cooperative dispatch solution:
\[
E^{\text{sup}}_{i}
=
\sum_{t}\Delta t
\sum_{j\neq i}
\max(0,P^{*}_{ij,t})
\]
\[
E^{\text{rec}}_{i}
=
-\sum_{t}\Delta t
\sum_{j\neq i}
\min(0,P^{*}_{ij,t})
\]
I then calculate the bargaining weight of each station using an exponential mapping function:
\[
\omega_{i}
=
\exp\left(\frac{E^{\text{sup}}_{i}}{E^{\text{sup}}_{\max}}\right)
–
\exp\left(-\frac{E^{\text{rec}}_{i}}{E^{\text{rec}}_{\max}}\right)
\]
where \(E^{\text{sup}}_{\max}\) and \(E^{\text{rec}}_{\max}\) are the maximum supplied and received energy among all stations. The weight is normalised as
\[
w_{i}
=
\frac{\omega_{i}}{\sum_{j=1}^{n}\omega_{j}}
\]
The weighted Nash–Harsanyi bargaining solution is obtained by solving
\[
\max
\quad
\prod_{i=1}^{n}
\left(C^{NC}_{i}-C^{*}_{i}\right)^{w_{i}}
\]
Because the product function is difficult to handle numerically, I take the logarithm and solve the equivalent convex problem:
\[
\min
\quad
\sum_{i=1}^{n}
-w_{i}
\ln
\left(C^{NC}_{i}-C^{*}_{i}\right)
\]
The inter-station price must satisfy the bilateral consistency constraint \(\lambda_{ij,t}=\lambda_{ji,t}\). I solve the cooperative problem with the alternating direction method of multipliers. The augmented Lagrangian is:
\[
\mathcal{L}
=
\sum_{i=1}^{n}
-w_{i}
\ln
\left(C^{NC}_{i}-C^{*}_{i}\right)
+
\sum_{i<j}\sum_{t} +="" <p="" \]="" \frac{\rho}{2}="" \lambda^{admm}_{ij,t}="" \left(\lambda_{ij,t}-\lambda_{ji,t}\right)="" \left\|\lambda_{ij,t}-\lambda_{ji,t}\right\|_{2}^{2}="" \sum_{iThe ADMM updates decompose the problem into independent subproblems for each charging station, which is important for protecting the privacy of individual stations.
8. System Parameters for the Hybrid Model
In the hybrid game simulation, I use the following system parameters. The shared storage device has a larger capacity than in the first model, because it now serves the aggregated coalition and provides more grid arbitrage.
| Component | Parameter | Value |
|---|---|---|
| Shared storage | Capacity | 5000 kWh |
| Shared storage | Maximum power | 1000 kW |
| Shared storage | Charging/discharging efficiency | 0.95 / 0.95 |
| Shared storage | State-of-charge lower/upper limit | 0.2 / 0.9 |
| Shared storage | Investment cost per unit | 1897 yuan/kWh |
| Shared storage | Lifetime | 8 years |
| Shared storage | Operation cost | 0.15 yuan/kWh |
| Grid interaction | Maximum exchange power | 500 kW |
| Charging station | Maximum grid purchase | 1000 kW |
| Charging station | Maximum exchange with storage | 800 kW |
| Inter-station link | Maximum flow | 500 kW |
| Renewable generation | PV/wind operation cost | 0.15 yuan/kWh |
9. Numerical Results of the Hybrid Model
I compare two scenarios. In the non-cooperative scenario, the electric vehicle charging stations do not exchange energy with each other. In the cooperative scenario, the stations form a coalition and trade energy through the inter-station links. The following table reports the daily cost of each station and the revenue of the shared storage operator.
| Market participant | Non-cooperative scenario (RMB) | Cooperative scenario (RMB) |
|---|---|---|
| Station 1 | 2156.65 | 2012.26 |
| Station 2 | 2152.80 | 1990.46 |
| Station 3 | 771.26 | 588.27 |
| Total coalition cost | 5080.71 | 4590.99 |
| Storage operator revenue | 1555.57 | 1065.85 |
The total cost of the electric vehicle charging station coalition is reduced from about 5081 yuan to 4591 yuan, which is a reduction of approximately 9.6 percent. The shared storage operator still earns a positive revenue of 1065.85 yuan per day, confirming that the business remains viable when inter-station trading is allowed. The reduction in storage revenue is expected, because the charging stations can satisfy some of their needs internally. However, the storage operator still benefits from providing balancing service and grid arbitrage.
The inter-station trading prices in the cooperative equilibrium always remain between the purchase and sale prices of the storage operator. In the valley period, when the grid electricity price is low, the inter-station trading prices are close to the lower bound. In the peak period, the trading prices move towards the upper bound. This price pattern is consistent with the rational behaviour of market participants.
10. Comparison of Bargaining Methods
I compare the proposed weighted Nash–Harsanyi bargaining method with the standard Nash bargaining method. In the standard method, all electric vehicle charging stations are treated equally. In the proposed method, the bargaining power of each station is derived from its contribution to the inter-station energy transaction. The following table reports the daily cost allocation of each station under the two methods.
| Charging station | Non-cooperative cost (RMB) | Standard Nash bargaining (RMB) | Proposed weighted Nash–Harsanyi (RMB) |
|---|---|---|---|
| Station 1 | 2156.65 | 1993.41 | 2012.26 |
| Station 2 | 2152.80 | 1989.56 | 1990.46 |
| Station 3 | 771.26 | 608.02 | 588.27 |
The total coalition cost is the same for both bargaining methods, but the individual allocation is different. Station 3 receives a larger cost reduction in the proposed method, which is consistent with its high level of participation in the energy trading. Because the exponential mapping function is sensitive to the relative contribution of each station, it rewards the station that supplies and receives more energy. The fairness principle embodied in this method is “more contribution leads to more benefit”. In contrast, the standard Nash method is more conservative because it treats all participants as symmetric although their contributions are not equal.
11. Convergence and Algorithmic Properties
In the Stackelberg game, the proposed solution algorithm converges in about 30 iterations in the hybrid model and about 176 iterations in the first model. The convergence curve of the shared storage operator shows a clear stabilisation process. In the cooperative ADMM algorithm, each charging station updates its local variable in parallel and then exchanges the boundary price with its neighbours. The cost of each station converges within about 40 iterations. The independence of the local update steps makes the algorithm suitable for a distributed implementation environment where the charging stations do not want to reveal their private information.
The quadratic penalty parameter in ADMM is chosen carefully. A small penalty may cause slow convergence, while a very large penalty may make the station cost excessively sensitive to price changes. In numerical experiments, I use a moderate value \(\rho^{P2P}=1.0\) and a convergence tolerance of \(10^{-4}\). The algorithm converges reliably without parameter tuning in all tested cases.
12. Sensitivity Analysis and Risk-Benefit Balance
The risk-aversion coefficient plays an important role in the optimisation. If the coefficient is small, the operator mainly focuses on the expected revenue and ignores revenue fluctuation. If the coefficient is large, the operator tends to raise the leasing price in order to compensate for the higher risk. I observe that increasing the risk-aversion coefficient of the shared storage operator increases its utility, while the total cost of the electric vehicle charging stations only fluctuates in a narrow range. A similar conclusion holds for the charging station cluster.
The confidence level \(1-\varepsilon\) also affects the result. When the allowable violation probability \(\varepsilon\) is small, the charging station has to reserve more capacity for the rare event of a large load deviation. This requirement causes the station to purchase more energy from the shared storage and the grid, which increases the total cost. Conversely, when \(\varepsilon\) is large, the system accepts a higher probability of violating the power limit and the cost decreases. This trade-off is an important engineering insight: the operator must choose a reasonable balance between the economic benefit and the system security.
13. Renewable Energy Accommodation
One of the main advantages of the proposed scheduling scheme is the improved utilisation of renewable energy. Because the storage operator can charge the battery when photovoltaic and wind generation exceeds the charging load, and because the charging stations can sell their surplus renewable electricity to each other, the curtailment of renewable energy is effectively reduced. The simulation results show that station 3, which has both photovoltaic and wind resources, is able to transfer a significant amount of energy to stations 1 and 2 during the middle of the day. In the night period, station 2 returns energy to station 1 through the inter-station link. This complementary exchange pattern is beneficial to the entire coalition and contributes to the absorption of local renewable energy.
From the perspective of electric vehicle users, the optimised charging schedule reduces the effective charging cost because the operator can procure cheaper electricity during valley periods and utilise renewable energy more efficiently. The service quality remains unchanged, because the total charging demand of each electric vehicle is fixed in the model. The game-theoretic framework only changes the source and timing of the electricity procurement, not the final charging service provided to the vehicles.
14. Practical Implications for Shared Energy Storage Operation
The practical contribution of this work is twofold. First, the Stackelberg model provides a price-setting strategy for the shared storage operator. Instead of using a fixed price, the operator can adaptively adjust the leasing price according to the grid tariff and the expected response of electric vehicle charging stations. Second, the cooperative model gives the stations a self-organising market mechanism. The stations do not need a central coordinator to determine the inter-station trading price; they can reach an agreement through the weighted Nash–Harsanyi bargaining process implemented by ADMM.
The main model can be extended from three stations to a larger number of stations without changing the underlying structure. The antisymmetric flow representation reduces the number of variables from \(n(n-1)\) to \(n(n-1)/2\). The cooperative bargaining problem is solved in a distributed manner, so the computational burden increases only linearly with the number of stations when the number of inter-station links is limited.
15. Conclusion
In this research, I have developed two game-based models for the collaborative optimal scheduling of shared energy storage and electric vehicle charging stations. The first model considers a cluster of charging stations that only trade with a shared storage operator and the grid. I handle the uncertainty of electric vehicle charging loads by using a zero-mean Gaussian distribution and convert the random constraints into deterministic chance constraints by using the Cantelli inequality. A mean-variance utility is employed to represent the risk attitude of the participants. A co-evolutionary genetic algorithm solves the resulting Stackelberg game.
The second model adds an additional layer of cooperation among electric vehicle charging stations. The charging stations can trade electricity with each other through inter-station links. The hybrid game contains a Stackelberg game between the storage operator and the station coalition, and a cooperative game inside the coalition. I use the antisymmetric flow variable to describe the inter-station energy exchange and apply the weighted Nash–Harsanyi bargaining solution for fair benefit allocation. The proposed exponential mapping function determines the bargaining weight from the actual energy contribution of each station. A two-stage algorithm, which combines the bisection method with CPLEX in the first stage and ADMM in the second stage, solves the hybrid game efficiently.
The numerical results show that shared energy storage can significantly reduce the operation cost of electric vehicle charging stations compared with independent energy storage. The introduction of a Stackelberg game with risk control further improves the economic benefit of both the charging stations and the storage operator. In the hybrid model, inter-station cooperation reduces the total cost of the charging station coalition while keeping the shared storage operator profitable. The proposed bargaining method achieves a fairer benefit distribution than the standard Nash bargaining method, because it reflects the diversified contribution of each electric vehicle charging station to the cooperative transaction.
Future work can extend this research in several directions. The model can incorporate the mobile storage capacity of idle electric vehicle batteries through vehicle-to-grid technology. The uncertainty distribution can be replaced by a distributionally robust formulation that only depends on the mean and covariance information. Finally, a multi-timescale scheduling framework can be developed to connect the day-ahead dispatch with the real-time control of the shared storage and electric vehicle charging stations. These extensions will make the scheduling model more realistic and more applicable to the future intelligent transportation and power systems.