1. What is the scheduling problem with energy storage?
Energy storage plays a vital role in smart grids, enabling more efficient energy management and supporting the integration of renewable sources. When energy storage is incorporated into economic dispatch, it helps reduce peak loads and enhance grid flexibility. The scheduling problem involving energy storage can typically be represented by a mathematical model (as shown in Figure 1, assuming that the charging and discharging prices are given inputs). The goal is to minimize the total operating cost of the power grid, subject to constraints such as energy storage operation rules, unit generation limits, and network flow conditions. With the inclusion of energy storage, two key factors influence the total cost: the cost of charging the storage and the cost of discharging it.
The charging cost depends on the price at which energy is stored. If the charging price is positive, the energy storage system pays the grid for the electricity it charges. Conversely, if the price is negative, the grid compensates the storage system for the energy it receives. Similarly, for discharging, if the price is positive, the grid pays the storage for the electricity it uses, while a negative price means the storage pays the grid for the energy it releases.
In cases where a long-term contract exists between the energy storage system and the grid, the operational cost of the storage may not be considered during dispatch. In such scenarios, the charging and discharging prices can be set to zero, simplifying the model.
2. What are the challenges of the energy storage scheduling problem?
Compared to traditional scheduling without energy storage, all energy storage-related problems face a common challenge: energy storage systems cannot charge and discharge simultaneously. This constraint must be properly modeled in the optimization framework. A simple approach might involve using a single variable representing the net power exchange between the storage and the grid—positive for charging, negative for discharging. However, this method is not accurate because the efficiency of charging and discharging differs, and the associated costs or revenues vary depending on the time of use. Additionally, the financial flows differ: when storing energy, the grid pays the storage, and when discharging, the storage pays the grid. Therefore, modeling this with just one continuous variable is insufficient.
To address this, most studies use two independent variables—charging power and discharging power—to represent the energy storage. This introduces “complementary constraints,†meaning that at any given time, the product of the charging and discharging power must equal zero. These nonlinear constraints make the optimization problem non-convex and harder to solve. While methods like penalty functions or binary variables can help transform the problem, they often require solving multiple subproblems, increasing computational time and reducing efficiency.
3. How to deal with this challenge? – The strict relaxation approach
The core issue lies in the complementary constraint. If it were removed, the problem would become convex and solvable using well-established techniques. So, could we relax this constraint and still obtain an optimal solution that naturally satisfies the original condition? This paper explores whether such a relaxation is "strict" under certain conditions. That is, if the relaxed model yields an optimal solution that also satisfies the constraint that charging power multiplied by discharging power equals zero, then the relaxation is valid. Finding such conditions would allow us to simplify complex energy storage models significantly, improving computational efficiency.
4. What conditions make the relaxation "strict"?
Mathematical analysis shows that the relaxation is strict if the following two conditions are met simultaneously:
Condition 1: At any scheduled time, the discharge price for each storage unit is greater than or equal to its charging price.
Condition 2: At any scheduled time, the charging price for each storage unit is less than or equal to the node price.
These conditions have clear economic interpretations. From an economic perspective, if both are satisfied, simultaneous charging and discharging becomes suboptimal. The optimal solution will naturally satisfy the complementary constraint, making the relaxation valid and effective.
5. How can we verify if these conditions hold?
Condition 1 can be easily checked since the charging and discharging prices are input parameters. For Condition 2, historical data is used to predict the node price during the scheduling period. If the predicted node price falls within a range where the charging price is lower than the minimum value of that range, Condition 2 is satisfied.
6. Can these conditions generally hold in real-world applications?
Under normal market conditions and with appropriate pricing strategies, these conditions can often be met. This makes the strict relaxation approach a practical and efficient tool for solving energy storage scheduling problems, especially in large-scale power systems where computational speed and accuracy are critical.
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