## 4 Stackelberg Equilibrium  
Stackelberg equilibrium is a refinement of Nash equilibrium, which belongs to a kind of sub-game perfect equilibrium [25]. For the same strategy of the leader, the follower may have more than one optimal response strategy, that is to say, the follower's payoff is the same under different strategies. However, the payoff is different for the leader. In [26], [PERSON] extends the Stackelberg equilibrium and proposed the concept of generalized Stackelberg equilibrium, which can guarantee the leader's minimum profit. On this basis, [PERSON] et al. divided Stackelberg equilibrium into strong Stackelberg equilibrium and weak Stackelberg equilibrium according to the different preference of the follower strategy [27]. When the follower has multiple optimal response strategies, in the strong Stackelberg equilibrium, the follower will choose the strategy that is most favorable to the leader; and in the weak Stackelberg equilibrium, the follower will choose the strategy that is the most unfavorable to the leader. At the strong Stackelberg equilibrium point, if the BSS refuses to choose the strategy most favorable to the RE company, then the payoff of the RE company will be reduced. In other words, the strategy at the strong Stackelberg equilibrium is an optimistic decision for the RE company, because there is a risk of less than its expected payoff. It should be noted that in the Stackelberg game model proposed in this paper, the RE company and BSS belong to different stakeholders, and there is no subordinate relationship between them. Although the RE company is in the leading status in the game, he cannot directly control the behavior of the BSS. As a rational decision-maker, the BSS has no incentive to choose the strategy which is more favorable to the RE company when BSS's payoff is the same under a variety of strategies. In order to guarantee the RE company's minimum revenue, the weak Stackelberg equilibrium is more suitable to the game model in this paper. At present, the common methods to solve Stackelberg equilibrium include KKT conditions method [28], penalty function method [29], intelligent algorithm [30] and so on. In the strong Stackelberg equilibrium, when BSS has multiple optimal response strategies, it will choose the strategy that is most favorable to the RE company. In other words, the BSS and the RE company have the same optimization goal. As a result, the bi-level programming model can be converted to a single layer programming model to be solved directly. However, for the weak Stackelberg equilibrium solution, the BSS will choose the strategy that is most unfavorable to the RE company. In this case, the above method is no longer applicable. In this paper, an intelligent algorithm based on the backward induction method is used to solve the weak Stackelberg equilibrium. The backward induction method starts from the last stage of a dynamic game, and gradually deduces the optimal strategy of the decision-makers in each stage. In the last game stage, for a given charge/discharge price strategy \(\left\{\pi^{t},t=1,2,\ldots,N_{t}\right\}\) of RE company, the follower's model is a linear programming model, and the optimal strategy of BSS and its maximum payoff\(f_{\text{BSS}}\) can be obtained by solving the model. \(f_{\text{BSS}}\) can be expressed as Equation (15):  
\[\sum_{t=1}^{N_{t}}\left(P_{\text{d}}^{t}-P_{\text{c}}^{t}\right)\rho^{t}\Delta t =f_{\text{BSS}} \tag{15}\]  
Adding Equation (15) as an equality constraint, a new BSS model is proposed, which is expressed as Equation (16). In this model, BSS could minimize the RE company's payoff while ensuring that its own payoff remain the same. \[\begin{array}{l}\min\sum_{t=1}^{N_{t}}\left(P_{\text{c}}^{t}-P_{\text{d}}^{t }\right)\rho^{t}\Delta t-\sum_{t=1}^{N_{t}}\left(P_{\text{c}}^{t}-P_{\text{d}} ^{t}\right)\rho_{RE}^{t}\Delta t\\ \text{s.t.