Superlinear Convergence of a Smooth Approximation Method for Mathematical Programs with Nonlinear Complementarity Constraints

Mathematical programs with complementarity constraints(MPCC) is an important subclass of MPEC.It is a natural way to solve MPCC by constructing a suitable approximation of the primal problem.In this paper,we propose a new smoothing method for MPCC by using the aggregation technique.A new SQP algorithm for solving the MPCC problem is presented.At each iteration,the master direction is computed by solving a quadratic program,and the revised direction for avoiding the Maratos effect is generated by an explicit formula.As the non-degeneracy condition holds and the smoothing parameter tends to zero,the proposed SQP algorithm converges globally to an S-stationary point of the MPEC problem,its convergence rate is superlinear.Some preliminary numerical results are reported.

Extended Sufficient Conditions for Exact Relaxation of the Complementarity Constraints in Storage-concerned Economic Dispatch

Storage is widely considered in economic dispatch(ED)problems.To prevent simultaneous charging and discharging of a storage device,a storage-concerned ED problem should involve complementarity constraints for every storage device to make the problem strongly non-convex.In this case,the conventional Karush-Kuhn-Tucker optimality conditions are unsuitable,and the methods that are normally effective are also invalid.In our recent paper,we proposed a new exact relaxation method that directly removes the complementarity constraints from a storageconcerned ED model to make it convex and easy to solve.This paper extends the previous study by presenting and analyzing two new groups of sufficient conditions that guarantee exact relaxation.Different application conditions of these groups of sufficient conditions are discussed.Numerical tests are performed to show the benefit of using the exact relaxation method and the different suitable application conditions of these groups of sufficient conditions.This paper contributes to a wide application of exact relaxation in storage-concerned ED problems.

A linear complementarity model for multibody systems with frictional unilateral and bilateral constraints

The Lagrange-I equations and measure differential equations for multibody systems with unilateral and bilateral constraints are constructed. For bilateral constraints, frictional forces and their impulses contain the products of the filled-in relay function induced by Coulomb friction and the absolute values of normal constraint reactions. With the time-stepping impulse-velocity scheme, the measure differential equations are discretized. The equations of horizontal linear complementarity problems （HLCPs）, which are used to compute the impulses, are constructed by decomposing the absolute function and the filled-in relay function. These HLCP equations degenerate into equations of LCPs for frictional unilateral constraints, or HLCPs for frictional bilateral constraints. Finally, a numerical simulation for multibody systems with both unilateral and bilateral constraints is presented.

A new smoothing technique for mathematical programs with equilibrium constraints

A kind of mathematical programs with equilibrium constraints （MPEC） is studied. By using the idea of successive approximation, a smoothing nonlinear programming, which is equivalent to the MPEC problem, is proposed. Thereby, it is ensured that some classical optimization methods can be applied for the MPEC problem. In the end, two algorithm models are proposed with the detail analysis of the global convergence.

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.

EXPONENTIAL CONVERGENCE OF SAMPLE AVERAGE APPROXIMATION METHODS FOR A CLASS OF STOCHASTIC MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

In this paper, we propose a Sample Average Approximation （SAA） method for a class of Stochastic Mathematical Programs with Complementarity Constraints （SMPCC） recently considered by Birbil, G/irkan and Listes [3]. We study the statistical properties of obtained SAA estimators. In particular we show that under moderate conditions a sequence of weak stationary points of SAA programs converge to a weak stationary point of the true problem with probability approaching one at exponential rate as the sample size tends to infinity. To implement the SAA method more efficiently, we incorporate the method with some techniques such as Scholtes＇ regularization method and the well known smoothing NCP method. Some preliminary numerical results are reported.

An Improved Power Flow Method to Cope with Non-smooth Constraints of Integrated Energy Systems

Integrated energy system applications can significantly improve energy efficiency.In this paper,we establish an integrated energy system containing heat,electricity and gas.The existing power flow(PF)calculation method applied to integrated energy systems(IESs)does not consider non-smooth constraints,such as the piecewise pipeline friction coefficient and generator buses reactive power limits,etc.Mixed integer nonlinear programming(MINLP)is conventionally used to deal with piecewise pipeline friction coefficients in gas network parts,but it is both complex and inefficient.Hence,we develop a piecewise linear function-based fitting method that can reduce the number of integer variables and enhanced the computational efficiency.In the electric network part,if the reactive power of the PV bus violates limits,it will be converted into a PQ bus,which is a non-differentiable and non-smooth constraint.Mixed complementarity problems are conventionally introduced to represent the PV-PQ buses type switching relationship and are addressed by the Newton-Raphson(NR)method.However,the above method is sensitive to the initial point.Here,we introduce a robust projected Levenberg-Marquardt(PLM)algorithm to cope with this issue.We demonstrate the advantages of our method and validate it both in a small-scale system and largescale network test cases.

Frequency aware robust economic dispatch

This paper proposes a novel frequency aware robust economic dispatch (FARED) approach to exploit the synergistic capability of accommodating uncertain loads and renewable generation by accounting for both the frequency regulation effect and optimal participation mechanism of secondary regulation reserves for conventional units in response to uncertainties in the robust optimization counterpart of security constrained economic dispatch.The FARED is formulated as a robust optimization problem.In this formulation the allowable frequency deviation and the possible load or renewable generation curtailments are expressed in terms of variable uncertainty sets.The variables in the formulation are described as interval variables and treated in affine form.In order to improve the computational tractability,the dominant constraints which canbe the candidates of tight transmission constraints are determined by complementarity constraints.Then the robust optimization problem is simplified to a bilinear programming problem based on duality theory.Finally,the effectiveness and efficiency of the proposed method are illustrated based on several study cases.

A VSC-based Model for Power Flow Assessment of Multi-terminal VSC-HVDC Transmission Systems

This paper puts forward a new practical voltage source converter(VSC)based AC-DC converter model suitable for conducting power flow assessment of multi-terminal VSCbased high-voltage direct current(VSC-MTDC)systems.The model uses an advanced method to handle the operational limits and control modes of VSCs into the power flow formulation.The new model is incorporated into a unified framework encompassing AC and DC power grids and is solved by using the Newton-Raphson method to enable quadratically convergent iterative solutions.The use of complementarity constraints,together with the Fischer-Burmeister function,is proposed to enable the seamless incorporation of operational control modes of VSC and automatic enforcement of any converter’s operational limits that become violated during the iterative solution process.Thus,a dedicated process for checking limits is no longer required.Furthermore,all existing relationships between the VSC control laws and their operational limits are considered directly during the solution of the power flow problem.The applicability of the new model is demonstrated with numerical examples using various multi-terminal AC-DC transmission networks,one of which is a utility-sized power system.