A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Lasserre, Jean-Bernard

doi:10.1137/090759240

Cited by 32 publications

(54 citation statements)

References 27 publications

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“…This imposes the moments of ϕ on µ such that the optimal value of this SDP relaxation approximates E ϕ (ρ(Q heat )) from below [19].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Using the polynomial representation outlined above, we cast the selection of an optimal set-point as a polynomial two-stage stochastic program with recourse [18]. We develop an approximation of polynomial two-stage problems based on semi-definite relaxations inspired by [19]. Using duality arguments, we derive a Generalized Moment Problem (GMP) that is equivalent to the two-stage problem and provide a sparse hierarchy of semidefinite programming (SDP) relaxations, which returns an estimate of the optimal setpoint and an expected cost estimate.…”

Section: Summary Of Contributionsmentioning

confidence: 99%

“…For each x 1 and y, let x * 2 (x 1 , y) be the optimal second stage solution. Under Assumption 1, the optimal map x * 2 (x 1 , y) is measurable [19,Theorem 2.2]. Thus, we can compute the expected value E ϕ (v * (x 1 , y)).…”

Section: Two-stage Stochastic Program With Recoursementioning

confidence: 99%

“…The infinite-dimensional GMP (4) can be approximated by an SDP relaxation involving a finite number of moments of µ [17]. GMPs with constraints on certain marginal distributions have been approximated using SDP relaxations in the literature for other purposes [19,23], and the derivation in this section leading to (9) and (11) is closely related to these. There is a trade-off between the accuracy of the approximation and the computational cost involved, and this is controlled by the choice of relaxation degree k ∈ N. The lowest admissible degree is determined by the degrees of the polynomial functions defining the problem.…”

Section: Tractable Relaxation Of Gmpmentioning

confidence: 99%

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A two-stage polynomial approach to stochastic optimization of district heating networks

Hohmann

Warrington

Lygeros

2019

Sustainable Energy, Grids and Networks

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In this paper, we use stochastic polynomial optimization to derive high-performance operating strategies for heating networks with uncertain or variable demand. The heat flow in district heating networks can be regulated by varying the supply temperature, the mass flow rate, or both simultaneously, leading to different operating strategies. The task of choosing the set-points within each strategy that minimize the network losses for a range of demand conditions can be cast as a two-stage stochastic optimization problem with polynomial objective and polynomial constraints. We derive a generalized moment problem (GMP) equivalent to such a two-stage stochastic optimization problem, and describe a hierarchy of moment relaxations approximating the optimal solution of the GMP. Under various network design parameters, we use the method to compute (approximately) optimal strategies when one or both of the mass flow rate and supply temperature for a benchmark heat network. We report that the performance of an optimally-parameterized fixed-temperature variable-mass-flow strategy can approach that of a fully variable strategy. systems tend to have variable mass flow control [2]. Each strategy is subject to a trade-off in terms of losses; higher supply and return temperatures lead to increased heat losses, whereas higher mass flow rates increase the hydraulic losses in the pipes. This trade-off has been studied in different contexts. A comparison of strategies for primary networks 1 can be found in [3]. Hydraulic control strategies for the primary and secondary network were optimized in [4]. New mass flow regulation schemes using pumps were compared to the traditional strategy of controlling the consumer side heat flow using valves in [5]. The performance of district heating networks with multiple sources was studied in [6].Significant effort has also been invested in simplified models of these systems. The steady-state thermal losses of a network can be modelled as an exponential temperature drop along a pipe segment [6,7,8,9].By replacing the exponential by its first order Taylor approximation, the authors of [7] and [9] obtain a polynomial representation of the pipe output temperature. The hydraulic losses, namely the pressure drop along pipes and substations, are mass flow rate dependent and can be characterised implicitly by the nonlinear Colebrook-White equation [6]. To simplify this representation, it is often assumed that a pipe segment has a constant coefficient of resistance [7,9,10], making the absolute pressure losses quadratic in the mass flow rate. Thus, both types of network loss can be modelled using polynomial functions. Two-stage stochastic programsIf the operating strategy keeps a control variable fixed (either temperature or mass flow rate), it is desirable that the fixed choice leads to acceptable performance over a range of demand conditions. Here we define this performance as the expected operational cost incurred by hydraulic and thermal losses with respect to a probability distribution of the heat de...

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“…This imposes the moments of ϕ on µ such that the optimal value of this SDP relaxation approximates E ϕ (ρ(Q heat )) from below [19].…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Summary Of Contributionsmentioning

confidence: 99%

Section: Two-stage Stochastic Program With Recoursementioning

confidence: 99%

Section: Tractable Relaxation Of Gmpmentioning

confidence: 99%

See 2 more Smart Citations

A two-stage polynomial approach to stochastic optimization of district heating networks

Hohmann

Warrington

Lygeros

2019

Sustainable Energy, Grids and Networks

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show abstract

“…Rather than solving this two-stage problem, we approximate the linearization point using SDP relaxations of a GMP inspired by [13]. Specifically, we use the first moment, an output of the SDP relaxation, as a linearization point, and show that it minimizes the model discrepancy defined above.…”

Section: Introductionmentioning

confidence: 99%

Optimal Linearizations of Power Systems With Uncertain Supply and Demand

Hohmann

Warrington

Lygeros

2019

IEEE Trans. Power Syst.

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Linearized models of power systems are often desirable to formulate tractable control and optimization problems that still reflect real-world physics adequately under various operating conditions. In this paper, we propose an approach that can make use of known data concerning the distribution of demand, and/or intermittent supply, to minimize expected model inconsistency with respect to the original non-linear model. The optimal linearization is obtained by approximating a generalized moment problem with a hierarchy of sparse semi-definite relaxations. The output is a linearization point that minimizes the expected absolute constraint violation with respect to the uncertain supply and demand. Numerical results for different power systems networks demonstrate the accuracy and scalability of our linearization method.

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A “Joint+Marginal” Approach in Optimization

Lasserre

2011

International Series in Operations Research &Amp; Management Science

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A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Cited by 32 publications

References 27 publications

A two-stage polynomial approach to stochastic optimization of district heating networks

A two-stage polynomial approach to stochastic optimization of district heating networks

Optimal Linearizations of Power Systems With Uncertain Supply and Demand

A “Joint+Marginal” Approach in Optimization

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