1. What is the scheduling problem with energy storage?
Energy storage plays a vital role in the smart grid, enabling more efficient and flexible power system operations. By integrating energy storage into economic dispatch, it becomes possible to manage peak loads and facilitate the integration of renewable energy sources. The scheduling problem involving energy storage can typically be modeled mathematically, as illustrated in Figure 1 (assuming that the charging and discharging prices are input parameters). The goal is to minimize the total operating cost of the power grid, subject to constraints such as energy storage operation limits, unit operation rules, and network constraints. With the inclusion of energy storage, two additional cost components must be considered: the cost of charging the storage and the cost of discharging it.Regarding the charging cost, if the price is positive, the energy storage pays the grid for the charge. Conversely, if the price is negative, the grid compensates the storage for the charge. Similarly, for the discharge cost, a positive price means the grid pays the storage for the energy returned, while a negative price indicates the storage must pay the grid for the discharge.
If there is a long-term contract between the energy storage system and the grid, the operational cost of the storage may not be considered during dispatch. In this case, the charging and discharging prices can be set to zero.
2. What are the challenges of the energy storage scheduling problem?
Compared to traditional scheduling without energy storage, all problems involving storage face a common challenge: energy storage cannot charge and discharge simultaneously. This constraint must be incorporated into the optimization model. A simple approach might involve using the net power exchanged between the storage and the grid as a single variable—positive when charging and negative when discharging. However, this method is not feasible because the efficiency of charging and discharging differs, and the prices at these times vary. Additionally, the financial flow is different (e.g., the grid pays the storage during charging, and the storage pays the grid during discharging), making it impossible to capture these dynamics with just one continuous variable. Therefore, most research models energy storage using two independent variables: charging power and discharging power. This leads to the introduction of "complementary constraints," which ensure that at any given time, the product of the charging and discharging power is zero. These nonlinear constraints make the optimization problem highly non-convex and difficult to solve. Although some methods like penalty functions or binary variables can transform the problem, they often require solving multiple sub-problems, leading to increased computational time and reduced efficiency.3. How to deal with this challenge? - The strict relaxation method
The core of the problem lies in the complementary constraint. If this constraint were removed, the problem could be transformed into a standard convex optimization problem, which has well-established and efficient solution techniques. So, what if we deliberately relax this constraint during the optimization process? In other words, can we ignore the complementary constraint and still obtain an optimal solution that naturally satisfies it? This article investigates whether such a relaxation is "strict" under certain conditions. That is, does the simplified model, after relaxing the constraint, produce an optimal solution that automatically satisfies the condition that the product of the charging and discharging power equals zero? If such conditions can be identified and are generally valid, then many complex energy storage scheduling problems can be significantly simplified, greatly improving the efficiency of the solution process.4. What conditions can make the relaxation "strict"?
Mathematical analysis shows that the relaxation is "strict" if two conditions are met simultaneously:Condition 1: At any scheduled time, the discharge price for each energy storage must be greater than or equal to the charge price.
Condition 2: At any scheduled time, the charge price for each energy storage must be less than or equal to the node price.
From an economic perspective, these conditions imply that simultaneous charging and discharging is not optimal from the grid's point of view. Instead, the optimal solution will naturally satisfy the complementary constraint when both conditions are met. This makes the relaxed model both valid and efficient in practice.
5. How can I determine if these specific conditions can be established?
Checking Condition 1 is straightforward since the charging and discharging prices are inputs to the model. It is easy to verify whether this condition holds. For Condition 2, however, we need to predict the node price over the scheduling period using historical data. If the charging price is consistently lower than the minimum predicted node price, then Condition 2 is satisfied.6. Can these specific conditions be generally established in practice?
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