Graphical Models and Belief Propagation-hierarchy for Optimal Physics-Constrained Network Flows

In this manuscript we review new ideas and first results on application of the Graphical Models approach, originated from Statistical Physics, Information Theory, Computer Science and Machine Learning, to optimization problems of network flow type with additional constraints related to the physics of the flow. We illustrate the general concepts on a number of enabling examples from power system and natural gas transmission (continental scale) and distribution (district scale) systems. 1.1 Introductory remarks In this chapter we discuss optimization problems which appears naturally in the classical settings describing flows over networks constrained by the physical nature Michael Chertkov Theoretical Division, T-4 & CNLS, Los Alamos National Laboratory Los Alamos, NM 87545, USA and Energy System Center, Skoltech, Moscow, 143026, Russia,  chertkov@lanl. gov Sidhant Misra Theoretical Division, T-5, Los Alamos National Laboratory Los Alamos, NM 87545, USA, e-mail: sidhant@lanl.gov Marc Vuffray Theoretical Division, T-4, Los Alamos National Laboratory Los Alamos, NM 87545, USA, e-mail: sidhant@lanl.gov Krishnamurthy Dvijotham Pacific Northwest National Laboratory, PO Box 999, Richland, WA 99352, USA e-mail: krishnamurthy.dvijotham@pnnl.gov Pascal Van Hentenryck University of Michigan, Department of Industrial & Operations Engineering Ann Arbor, MI 48109, USA,  pvanhent@umich.edu 1 ar X iv :1 70 2. 01 89 0v 1 [ cs .S Y ] 7 F eb 2 01 7 2 M. Chertkov, S. Misra, M. Vuffray, D. Krishnamurthy, and P. Van Hentenryck of the flows which appear in the context of electric power systems, see e.g. [27, 44], and natural gas application, see e.g. [13] and references there in. Other examples of physical flows where similar optimization problem arise include pipe-flow systems, such as district heating [75, 1] and water [54], as well as traffic systems [40]. We aim to show that the network flow optimization problem can be stated naturally in terms of the so-called Graphical Models (GM). In general, GMs for optimization and inference are wide spread in statistical disciplines such as Applied Probability, Machine Learning and Artificial Intelligence [53, 29, 16, 12, 32, 50], Information Theory [55] and Statistical Physics [47]. Main benefit of adopting GM methodology to the physics-constrained network flows is in modularity and flexibility of the approach – any new constraints, set of new variables, and any modification of the optimization objective can be incorporated in the GM formulation with an ease. Besides, if all (or at least majority of) constraints and modifications are factorized, i.e. can be stated in terms of a small subset of variables, underlying GM optimization or GM statistical inference problems can be solved exactly or approximately with the help of an emerging set of techniques, algorithms and computational approaches coined collectively Belief Propagation (BP), see e.g. an important original paper [74] and recent reviews [47, 55, 66]. It is also important to emphasize that an additional benefit of the GM formulation is in its principal readiness for generalization. Even though we limit our discussion to application of the GM and BP framework to deterministic optimizations, many probabilistic and/or mixed generalizations (largely not discussed in this paper) fit very naturally in this universal framework as well. We will focus on optimization problems associated with Physics-Constrained Newtork Flow (PCNF) problems. Structure of the networks will obviously be inherited in the GM formulation, however indirectly through graphand variabletransformations and modifications. Specifically, next Section 1.2 is devoted solely to stating a number of exemplary energy system formulations in GM terms. Thus, in Section 1.2.1 and Section 1.2.2 we consider dissipation optimal and respectively general physics-constrained network flow problems. In particular, Section 1.2.2 includes discussion of power flow problems in both power-voltage, Section 1.2.2.1, and current-voltage, Section 1.2.2.2, formats, as well as discussion of the gas flow formulation in Section 1.2.2.3 and general k-component physics-constrained network flow problem in Section 1.2.2.4. Section 1.2.3 describes problems of the next level of complexity – these including optimization over resources. In particular, general optimal physics-controlled network flow problem is discussed in Section 1.2.3.1 and more specific cases of optimal flows, involving optimal power flow (in both power-flow and current-voltage formulations) and gas flows are discussed in Sections 1.2.3.2,1.2.3.3,1.2.2.3, respectively. Section 1.2.4 introduces a number of feasibility problems, all stated as special kinds of optimizations. Here we discuss the so-called instanton, Section 1.2.4.1, containment Section 1.2.4.2, and state estimation, Section 1.2.4.3, formulation. The long introductory section concludes with a discussion in Section 1.2.5 of an exemplary (and even more) complex optimization involving split of resources between participants/aggregators. 1 Graphical Models for Optimal Flows 3 In Section 1.3 we describe how any of the aforementioned PCNF and optimal PCNF problems can be re-stated in the universal Graphical Model format. Then, in Section 1.4, we take advantage of the factorized form of the PCNF GM and illustrate how BP methodology can be used to solve the optimization problems exactly and/or approximately. Specifically, in Section 1.4.1 we restate the optimization (Maximum Likelihood) GM problem as a Linear Programming (LP) in the space of beliefs (proxies for probabilities). The resulting LP is generally difficult as working with all variables in a combination. We take advantage of the GM factorization and introduce in Section 1.4.2 the so-called Linear Programming Belief Propagation (LP-BP) relaxation, providing a provable lower bound for the optimal. Finally, in Section 1.4.3 we construct a tractable relaxation of LP-BP based on an interval partitioning of the underlying space. Section 1.5 discuss hierarchies which allow to generalize, and thus improve LPBP. The so-called LP-BP hierarchies, related to earlier papers on the subject [65, 30, 63] are discussed in Section 1.5.1. Then, relation between the LP-BP hierarchies and classic LP-based Sherali-Adams [59] and Semi-Definite-Programming based Lasserre hierarchies [41, 36, 52, 37] are discussed in Section 1.5.2. Section 1.6 discuss the special case of a GM defined over a tree (graph without loops). In this case LP-BP is exac, equivalent to the so-called Dynamic Programming approach, and as such it provides a distributed alternative to the global optimization through a sequence of graph-element-local optimizations. However, even in the tree case the exact LP-BP and/or DP are not tractable for GM stated in terms of physical variables, such as flows, voltages and/or pressures, drawn from a continuous set. Following, [18] we discuss here how the problem can be resolved with a proper interval-partitioning (discretization). We conclude the manuscript presenting summary and discussing path forward in Section 1.7. 1.2 Problems of Interest: Formulations In this Section we formulate a number of physics-constrained network flow problems which we will then attempt to analyze and solve with the help of Graphical Model (GM)/Belief Propagation (BP) approaches/techniques in the following Sections. 1.2.1 Dissipation-Optimal Network Flow We start introducing/discussing Network Flows constrained by a minimum dissipation principle, i.e. one which can be expressed as an unconstrained optimization/minimization of an energy function (potential). 4 M. Chertkov, S. Misra, M. Vuffray, D. Krishnamurthy, and P. Van Hentenryck Consider a static flow of a commodity over an undirected graph, G = (V ,E ) described through the following network flow equations i ∈ V : qi = ∑ j:(i, j)∈E φi j, (1.1) where qi stand for injection, qi > 0, or consumption, qi < 0, of the flow at the node i and φi j =−φ ji stands for the value of the flow through the directed edge (i, j) – in the direction from i to j 1. We consider a balanced network, ∑i∈V qi = 0. We constraint the flow requiring that the minimum dissipation principle is obeyed min φ ∑ {i, j}∈E Ei j(φi j) ∣∣∣∣ Eq. (1.1) , (1.2) where φ . = (φi j = −φ ji|{i, j} ∈ E ), and Ei j(x) are local (energy) functions of their arguments for all {i, j} ∈ E . The local energy functions Ei j(x) are required to be convex at least on a restricted domain. We call the sum of local energy functions E(φ) = ∑{i, j}∈E Ei j(φi j) the global energy function or simply the energy function. Versions of this problem appear in the context of the feasibility analysis of the dissipative network flows, that is flows whose redistribution over the network is constrained by potentials, e.g. voltages or pressures in the context of resistive electric networks and gas flow networks, respectively [22, 48, 64]. Note, that the formulation (1.2) can also be supplemented by additional flow or potential constraints. Requiring Karush-Kuhn-Tucker (KKT) stationary point conditions on the optimization problem stated in Eq. (1.2) leads to the following set of equations ∀{i, j} ∈ E : E ′ i j(φi j) = λi−λ j, (1.3) where λi is a Lagrangian multiplier corresponding to the i’s equation (1.1). The problem becomes fully defined by the pair of Eqs. (1.1,1.3), which can also be restated solely in terms of the λ -variables i ∈ V : qi = ∑ j:{i, j}∈E ( E ′ i j )−1 (λi−λ j). (1.4) 1.2.2 General Physics-Constrained Network Flows We call “unconstrained” a network flow for which only conservation of flow(s), described by Eq. (1.1), is enforced. Contrariwise we call “Physics-constrained” a 1 In the following we will use notation {i, j} for the undirected graph and (i, j) for the respective directed graph. When the meaning is clear we slightly abuse notations denoting by E both the set of undirected and directed edges. 1 Graphical Models for Optimal Flows 5 network flow t