Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems
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Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.Keywords:
Karush–Kuhn–Tucker conditions
Bilevel optimization
Karush–Kuhn–Tucker conditions
Bilevel optimization
Theory of computation
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For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints, in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper.
Karush–Kuhn–Tucker conditions
Bilevel optimization
Complementarity (molecular biology)
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Karush–Kuhn–Tucker conditions
Bilevel optimization
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When the lower-level optimal solution set-valued mapping of a bilevel optimization problem is not single-valued, we are faced with an ill-posed problem, which gives rise to the optimistic and pessimistic bilevel optimization problems, as tractable algorithmic frameworks. However, solving the pessimistic bilevel optimization problem is far more challenging than the optimistic one; hence, the literature has mostly been dedicated to the latter class of the problem. The Scholtes relaxation has appeared to be one of the simplest and efficient way to solve the optimistic bilevel optimization problem in its Karush-Kuhn-Tucker (KKT) reformulation or the corresponding more general mathematical program with complementarity constraints (MPCC). Inspired by such a success, this paper studies the potential of the Scholtes relaxation in the context of the pessimistic bilevel optimization problem. To proceed, we consider a pessimistic bilevel optimization problem, where all the functions involved are at least continuously differentiable. Then assuming that the lower-level problem is convex, the KKT reformulation of the problem is considered under the Slater constraint qualification. Based on this KKT reformulation, we introduce the corresponding version of the Scholtes relaxation algorithm. We then construct theoretical results ensuring that a sequence of global/local optimal solutions (resp. stationarity points) of the aforementioned Scholtes relaxation converges to a global/local optimal solution (resp. stationarity point) of the KKT reformulation of the pessimistic bilevel optimization. The results are accompanied by technical results ensuring that the Scholtes relaxation algorithm is well-defined or the corresponding parametric optimization can easily be solved.
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In this paper,we consider some special bilevel programming problem.Replacing the lower-level problem with its KKT conditions leads to a one-level problem with a complementary constraint.Under certain conditions,we derive the first order optimal conditions by Fritz-John conditions.These conditions are easy to check and simple.They are different from the constraint qualification in [1].
Karush–Kuhn–Tucker conditions
Bilevel optimization
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In this paper we investigate a bilevel optimization problem by using the optimistic approach.Under a non smooth generalized Guignard constraint qualification, due the optimal value reformulation, the necessary optimality conditions in terms of convexificators and Karush-Kuhn-Tucker (KKT) multipliers are given.
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Approximate Karush-Kuhn-Tucker condition for multi-objective optimistic bilevel programming problems
The approximate Karush-Kuhn-Tucker (AKKT) condition is introduced being a necessary condition of the local weak efficient solution for optimistic bilevel optimization problems with multiple objectives in upper-level problems. We transform the multi-objective bilevel optimization problem into single-level multi-objective optimization problem by means of the value function transformation or the KKT transformation. We then prove that the AKKT condition is necessary for the point to be a local weak efficient solution without any constraint qualification for the transformed one-level problem. Besides, we give examples to show that the bilevel problem has no KKT point or the lower-level problem violates the Slater CQ, but may have an AKKT point, and we introduce some suitable constraint qualifications that can ensure that the AKKT condition implies the KKT condition. Finally, numerical results are given to show the AKKT conditions' necessity.
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Bilevel optimization
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Karush–Kuhn–Tucker conditions
Bilevel optimization
Calmness
Operator (biology)
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Bilevel optimization problems are often reduced to single level using Karush-Kuhn-Tucker (KKT) conditions; however, there are some inherent difficulties when it comes to satisfying the KKT constraints strictly. In this paper, we discuss single level reduction of a bilevel problem using approximate KKT conditions which have been recently found to be more useful than the original and strict KKT conditions. We embed the recently proposed KKT proximity measure idea within an evolutionary algorithm to solve bilevel optimization problems. The idea is tested on a number of test problems and comparison results have been provided against a recently proposed evolutionary algorithm for bilevel optimization. The proposed idea leads to significant savings in lower level function evaluations and shows promise in further use of KKT proximity measures in bilevel optimization algorithm development.
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Using the Karush–Kuhn–Tucker conditions for the convex lower level problem, the bilevel optimization problem is transformed into a single-level optimization problem (a mathematical program with complementarity constraints). A regularization approach for the latter problem is formulated which can be used to solve the bilevel optimization problem. This is verified if global or local optimal solutions of the auxiliary problems are computed. Stationary solutions of the auxiliary problems converge to C-stationary solutions of the mathematical program with complementarity constraints.
Karush–Kuhn–Tucker conditions
Bilevel optimization
Complementarity (molecular biology)
Regularization
Constrained optimization problem
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