Mixed-integer nonlinear programs featuring “on/off” constraints

Hijazi, Hassan; Bonami, Pierre; Cornuéjols, Gérard; Ouorou, Adam

doi:10.1007/s10589-011-9424-0

Cited by 61 publications

(67 citation statements)

References 17 publications

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“…In order to provide global optimality bounds, it is necessary to consider convex relaxations of the nonlinear constraints in (1). While possible relaxation approaches have been outlined in [19,22], the study of tailored convex relaxations for the problem of optimal placement and operation of control valves in water distribution networks is outside the scope of this manuscript and will be investigated in future work.…”

Section: Case Studymentioning

confidence: 99%

Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks

2016

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In this paper, we investigate the application of penalty and relaxation methods to the problem of optimal placement and operation of control valves in water supply networks, where the minimization of average zone pressure is the objective. The optimization framework considers both the location and settings of control valves as decision variables. Hydraulic conservation laws are enforced as nonlinear constraints and binary variables are used to model the placement of control valves, resulting in a mixed-integer nonlinear program. We review and discuss theoretical and algorithmic properties of two solution approaches. These include penalty and relaxation methods that solve a sequence of nonlinear programs whose stationary points converge to a stationary point of the original mixed-integer program. We implement and evaluate the algorithms using a benchmarking water supply network. In addition, the performance of different update strategies for the penalty and relaxation parameters are investigated under multiple initial conditions. Practical recommendations on the numerical implementation are provided.

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Section: Case Studymentioning

confidence: 99%

Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks

2016

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“…This result can be extended to the general case (l 0 ‰ u 0 ) when functions g are monotonic [21]. Specifically, in the linear case where gpxq " α T x´β, the convex hull is given by…”

Section: Relaxations For Reconfiguration Problemsmentioning

confidence: 92%

“…Let us emphasise that this constraint is not sufficient for defining the convex hull as shown in [21], therefore, one can strengthen the relaxation by introducing the remaining nondominated constraints. In what follows, we use the results in (7)-(10) to formulate on/off constraints in the reconfiguration framework.…”

Section: Relaxations For Reconfiguration Problemsmentioning

confidence: 99%

Optimal AC Distribution Systems Reconfiguration

Hijazi

Thiébaux

2014

2014 Power Systems Computation Conference

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Abstract-Convex power flow relaxations have become popular to alleviate difficulties with embedding the non-convex AC power flow equations into optimisation models, and to provide guarantees on the quality of feasible solutions generated by heuristic approaches. However, their use has almost universally been limited to purely continuous problems. This paper extends the reach of relaxations to reconfiguration problems with binary decision variables, such as minimal power loss, load balancing and power supply restoration. This is achieved by extending the relaxations of AC power flows to bear on the on/off nature of constraints featured in reconfiguration problems. This leads to an approach producing AC feasible solutions with provable optimality gaps, and to global optimal solutions in some cases. In terms of run time, the new models are competitive with stateof-the-art approximations which lack formal guarantees.Keywords-Distribution Systems Reconfiguration, Power Supply Restoration, AC Power Flow, DC Power Flow, On/Off Constraints, Global Optimisation, Convex Relaxations, MINLP I. INTRODUCTION The steady-state alternating current (AC) power flow equations are at the core of virtually every computational problem in the field of power systems. Unfortunately, these equations form a system of non-convex constraints, raising significant challenges in any optimisation framework. When embedding these constraints into optimisation models, global non-linear programming (NLP) solvers do not scale and may not converge, even if all decision variables are continuous. These issues are exacerbated in the case of reconfiguration and power supply restoration problems, where discrete decisions are made that affect the grid topology. On the one hand, discrete variables increase the computational difficulty of the problem. On the other hand, topology changes lead to departure from the normal operating conditions that are used to hot-start non-linear solvers. Yet utilities are facing an increasing need for efficient reconfiguration methods, in order to optimise their operations to better handle the intermittent nature of distributed generation, and to quickly resupply customers in fault situations.There is a rich set of optimisation approaches in the power systems literature that tries to circumvent these problems. Black-box heuristic methods, which push power flow calculations outside the optimisation solver, have become very popular, owing to their broad applicability [1]- [3]. Yet, these approaches fail to exploit the problem structure and lack formal guarantees on the quality of the solutions returned. Another popular approach is to approximate the power flow equations, usually linearly [4]-[6], or with more accurate convex models [7]. This enables the use of a new generation of mathematical programming solvers including mixed-integer linear (MIP),

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“…This result can be extended to the general case (l 0 -u 0 ) when functions g are monotonic [22]. Specifically, in the linear case where gðxÞ ¼ a T x À b, the convex hull is given by…”

Section: On/off Constraintsmentioning

confidence: 93%