Research On Multi-Entity Mixed-Integer Programming Model And Algorithm For Transmission-Distribution Co-optimization In Power Systems

With the wide connection of distributed energy resources to the distribution systems or end-users,these resources can not only be applied to solve the operation problems of distribution systems,but also provide flexible services to the transmission systems.Consequently,the coupling between the transmission system and distribution systems has become stronger,and the existing separated management mode of the integrated transmission and distribution system will face severe challenges.The domestic and overseas research shows that the coordinated optimization of the transmission system and distribution systems is conducive to dealing with the voltage support,line congestion and boundary bus congestion of transmission and distribution system,the balance of the global system,and so on.According to the existing management mode,the coordinated optimization problems of the integrated transmission and distribution system are essentially joint optimization problems covering multiple entities,such as the operator of the transmission system and operators of distribution systems.The distributed optimization methods and bi-level optimization methods are two effective ways to solve the joint optimization problems.In addition,there are a large number of continuous and discrete control devices in the transmission system and distribution systems,and the dispatch and control of these devices have a significant impact on the safe operation of the power system.Therefore,it is of great significance to study the mixed-integer programming modeling theory and algorithms for the coordinated optimization of the integrated transmission and distribution systems with multiple entities.However,most of the existing methods for solving multi-entity mixed-integer optimization problems cannot guarantee their convergence and optimality.This paper focuses on two kinds of multi-entity mixed-integer optimization problems,namely: the distributed optimization problems and bi-level optimization problems of the integrated transmission and distribution systems with continuous and discrete control,and makes in-depth studies on their modeling and algorithms.The main research contents and results are summarized as follows:First,a distributed mixed-integer linear programming(MILP)problem for the coordinated optimal scheduling of the integrated transmission and distribution system is established,and an improved Lipschitz dynamic programming-based distributed algorithm is proposed to solve it.Based on the theory of dynamic programming,the coordinated scheduling problem of the integrated transmission and distribution system is decomposed into a MILP subproblem of the transmission system and multiple MILP subproblems of distribution systems.Based on the theory of Lipschitz continuous function,the reverse norm cut is applied to approximate the value functions of the MILP subproblems of distribution systems during iteration,to solve the original model in a distributed manner.Based on the characteristic that the subproblem at each period of the distribution systems is only coupled with the ones at adjacent periods,two strategies for the introduction of reverse norm cuts are proposed to improve the convergence of the proposed distributed algorithm.A curve fitting method is applied to obtain the predefined lower bound of the value function of the subproblem at each period of each distribution system,to further accelerate the convergence of the proposed algorithm.Simulations on a constructed48-bus integrated transmission and distribution system and a practical integrated transmission and distribution system are conducted to verify the correctness and effectiveness of the proposed distributed algorithm.Second,considering the existence of distributed energy resources with multi-period coupling characteristics(such as energy storage systems)in distribution systems,a distributed MILP model for coordinated optimal scheduling of the integrated transmission and distribution system with energy storage systems is established,and a mixed-integer dual dynamic programming(MIDDP)algorithm is proposed to solve the model in a distributed manner.Based on the characteristic that the approximate value functions of convex optimization problems obtained by the dual dynamic programming method are the lower bounds of their exact value functions,the proposed algorithm forms an approximate value function of the original MILP subproblem of a distribution system by combining the approximate value functions of multiple linear programming(LP)subproblems of the distribution system with different fixed combinations of integer variables,to solve the proposed model in a distributed manner.In each outer iteration of the MIDDP algorithm,a new set of optimal combination of the integer variables in the MILP subproblem of a distribution system is retrieved by constructing and solving a bi-level optimization model,and the approximate value function of the LP subproblem of the distribution system after fixing this combination is obtained using the dual dynamic programming method.Simulations on a constructed 48-bus integrated transmission and distribution system and a practical integrated transmission and distribution system are conducted to verify the correctness and effectiveness of the proposed MIDDP algorithm.Then,for the bi-level optimization problem of the integrated transmission and distribution system with continuous and discrete control,two algorithms are proposed for solving the bilevel mixed-integer linear programming(BMILP)model with both continuous and discrete variables in both upper-and lower levels,namely: the relaxation-based bi-level reformulation and decomposition(RBRD)method and its corresponding improved accelerated relaxationbased bi-level reformulation and decomposition(A-RBRD)method.Based on the partial enumeration and the column-and-constraint generation method,the proposed two algorithms replace the original MILP lower-level model by using multiple LP lower-level models with different fixed combinations of integer variables,to solve the BMILP model.Nonnegative relaxation variables and penalty functions are introduced in the added LP lower-level problems to ensure the equivalence before and after the replacement of the original MILP lower-level model.In addition,by constructing a simple bi-level optimization model,the proposed ARBRD algorithm can identify and remove the inactive inequality constraints of the added new LP lower-level models in each iteration,to effectively reduce the complexity of solving the BMILP model.Next,considering the impact of the security check of the distribution system,a mixedinteger Stackelberg game model for a distributed energy aggregator to participate in the dayahead electricity market is established.To guarantee the operation security of the distribution system,the security check problems under three different scenarios are involved in the lower level of the proposed Stackelberg game model.Consequently,the strategic bidding problem of distributed energy aggregator as a price maker in the day-ahead energy and reserve markets is formulated as a bi-level mixed-integer nonlinear programming model with both LP and MILP lower-level models.Based on the Karush-Kuhn-Tucker-based reformulation method,the proposed bi-level optimization model is transformed into a BMILP model with only MILP lower-level models,and the proposed RBRD and A-RBRD algorithms are applied to solve the BMILP model.Simulations on a 48-bus constructed integrated transmission and distribution system and a practical integrated transmission and distribution system are conducted to analyze the impact of the security check of the distribution system on the profits and behaviours of the distributed energy resource aggregator in the day-ahead electricity markets and verify the effectiveness of the proposed RBRD and A-RBRD algorithms.Finally,a BMILP model for the load-redistribution attacks on reactive power optimization in the integrated high-and medium-voltage distribution system with a similar structure to the integrated transmission and distribution system is established to analyze the information security of reactive power optimization of high-and medium-voltage distribution system.In the proposed BMILP model,the attacker in the upper level aims to maximize the voltage deviation and load curtailment of the integrated high-and medium-voltage distribution system,while the high-voltage distribution system and medium-voltage distribution systems aim to minimize the voltage deviation in their regions in the lower level.By duplicating the coupling variables between the lower-level models in the upper-level model and introducing the corresponding coupling constraints,the lower-level models are decoupled,and then the proposed BMILP model is solved using the proposed RBRD and A-RBRD algorithms.Simulations on a constructed 40-bus integrated high-and medium-voltage distribution system and a practical integrated high-and medium-voltage distribution system are conducted to analyze the impact of load redistribution attack on reactive power optimization of integrated high-and medium-voltage distribution systems,and verify the correctness and effectiveness of the proposed RBRD and A-RBRD algorithm.