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We study uncertain linear complementarity problems (LCPs), i.e., problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ-robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon [16]. There, we studied Γ-robustified LCPs for l1- and box-uncertainty sets, whereas we now focus on ellipsoidal uncertainty set. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions.
Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty - especially using the concepts of robust optimization - is still in its infancy. In this paper, we extend the theory of strictly robust linear complementarity problems (LCPs) to Γ-robust settings, where existence of worst-case-hedged equilibria cannot be guaranteed. Thus, we study the minimization of the worst-case gap function of Γ-robust counterparts of LCPs. For box and l1-norm uncertainty sets we derive tractable convex counterparts for monotone LCPs and study their feasibility as well as the existence and uniqueness of solutions. To this end, we consider uncertainties in the vector and in the matrix defining the LCP. We additionally study so-called ρ-robust solutions, i.e., solutions of relaxed uncertain LCPs. Finally, we illustrate the Γ-robust concept applied to LCPs in the light of the above mentioned classical examples of bimatrix games and market equilibrium modeling.
We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Γ-robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Γ-robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.
Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost.
In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach.
Transport and trade of gas are decoupled after the liberalization of the European gas markets, which are now organized as so-called entry-exit systems. At the core of this market system are bookings and nominations, two special capacity-right contracts that grant traders access to the gas network. The latter is operated by a separate entity, known as the transmission system operator (TSO), who is in charge of the transport of gas from entry to exit nodes. In the mid to long term, traders sign a booking contract with the TSO to obtain injection and withdrawal capacities at entry and exit nodes, respectively. On a day-ahead basis, they then nominate within these booked capacities a balanced load flow of the planned amounts of gas to be injected into and withdrawn from the network the next day. The key property is that by signing a booking contract, the TSO is obliged to guarantee transportability for all balanced load flows in compliance with the booked capacities. To assess the feasibility of a booking, it is therefore necessary to check the feasibility of infinitely many nominations. As a result, deciding if a booking is feasible is a challenging mathematical problem, which we investigate in this dissertation.
Our results range from passive networks, consisting of pipes only, to active networks, containing controllable elements to influence gas flows. Since the study of the latter naturally leads to a bilevel framework, we first consider some more general properties of bilevel optimization. For the case of linear bilevel optimization, we consider the hardness of validating the correctness of big-Ms often used in solving these problems via a single-level reformulation. We also derive a family of valid inequalities to be used in a bilevel-tailored branch-and-cut algorithm as a big-M-free alternative.
We then turn to the study of feasible bookings. First, we present our results on passive networks, for which bilevel approaches are not required. A characterization of feasible bookings on passive networks is derived in terms of a finite set of nominations. While computing these nominations is a difficult task in general, we present polynomial complexity results for the special cases of tree-shaped or single-cycle passive networks. Finally, we consider networks with linearly modeled active elements. After obtaining a bilevel optimization model that allows us to determine the feasibility of a booking in this case, we derive various single-level reformulations to solve the problem. In addition, we obtain novel characterizations of feasible bookings on active networks, which generalize our characterization in the passive case. The performance of these various approaches is compared in a case study on two networks from the literature, one of which is a simplified version of the Greek gas network.
Value at risk approach to producer's best response in electricity market with uncertain demand
(2021)
We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided.
Bilevel problems are used to model the interaction between two decision makers in which the lower-level problem, the so-called follower's problem, appears as a constraint in the upper-level problem of the so-called leader. One issue in many practical situations is that the follower's problem is not explicitly known by the leader. For such bilevel problems with unknown lower-level model we propose the use of neural networks to learn the follower's optimal response for given decisions of the leader based on available historical data of pairs of leader and follower decisions. Integrating the resulting neural network in a single-level reformulation of the bilevel problem leads to a challenging model with a black-box constraint. We exploit Lipschitz optimization techniques from the literature to solve this reformulation and illustrate the applicability of the proposed method with some preliminary case studies using academic and linear bilevel instances.
This paper addresses unit commitment under uncertainty of load and power infeed from renewables in alternating current (AC) power systems. Beside traditional unit-commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming approach is used, which leads us to risk averse two-stage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented.
In this paper we analyze peak-load pricing in the presence of network constraints. In our setup, firms facing fluctuating demand decide on the size and location of production facilities. They make production decisions constrained by the invested capacities, taking into account that market prices reflect scarce transmission capacities. We state general conditions for existence and uniqueness of the market equilibrium and provide a characterization of equilibrium investment and production. The presented analysis covers the cases of perfect competition and monopoly - the case of strategic firms is approximated by a conjectural variations approach. Our result is a prerequisite for analyzing regulatory policy options with computational multilevel equilibrium models, since uniqueness of the equilibrium at lower levels is of key importance when solving these models. Thus, our paper contributes to an evolving strand of literature that analyzes regulatory policy based on computational multilevel equilibrium models and aims at taking into account individual objectives of various agents, among them not only generators and customers but also, e.g., the regulator deciding on network expansion.
We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.
We consider uniqueness and multiplicity of market equilibria in a short-run setup where traded quantities of electricity are transported through a capacitated network in which power flows have to satisfy the classical lossless DC approximation. The firms face fluctuating demand and decide on their production, which is constrained by given capacities. Today, uniqueness of such market outcomes are especially important in more complicated multilevel models for measuring market (in)efficiency. Thus, our findings are important prerequisites for such studies. We show that market equilibria are unique on tree networks under mild assumptions and we also present a priori conditions under which equilibria are unique on cycle networks. On general networks, uniqueness fails to hold and we present simple examples for which multiple equilibria exist. However, we prove a posteriori criteria for the uniqueness of a given solution and characterize situations in which multiple solutions exist.
We consider the numerical approximation of linear damped wave systems
by Galerkin approximations in space and appropriate time-stepping schemes. Based on
a dissipation estimate for a modified energy, we prove exponential decay of the physical
energy on the continuous level provided that the damping is effective everywhere in the
domain. The methods of proof allow us to analyze also a class of Galerkin approximations
based on a mixed variational formulation of the problem. Uniform exponential stabil-
ity can be guaranteed for these approximations under a general compatibility condition
on the discretization spaces. As a particular example, we discuss the discretization by
mixed finite element methods for which we obtain convergence and uniform error esti-
mates under minimal regularity assumptions. We also prove unconditional and uniform
exponential stability for the time discretization by certain one-step methods. The valid-
ity of the theoretical results as well as the necessity of some of the conditions required
for our analysis are demonstrated in numerical tests
Ongoing policy discussions on the reconfiguration of bidding zones in European electricity markets induce uncertainty about the future market design. This paper deals with the question of how this uncertainty affects market participants and their long-run investment decisions in generation and transmission capacity. Generalizing the literature on pro-active network expansion planning, we propose a stochastic multilevel model which incorporates generation capacity investment, network expansion, and market operation, taking into account uncertainty about the future bidding zone configuration. Using a stylized two-node network, we disentangle different effects that uncertainty has on market outcomes. If there is a possibility that future bidding zone configurations provide improved regional price signals, welfare gains materialize even if the change does not actually take place. As a consequence, welfare gains of an actual change of the bidding zone configuration are substantially lower due to those anticipatory effects. Additionally, we show substantial distributional effects in terms of both expected gains and risks, between producers and consumers and between different generation technologies.
In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) action is made, often but not always understood as some "compensation".
In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis.
We consider risk neutral and risk averse two-stage stochastic semidefinite programs with continuous and mixed-integer recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems.
Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed.
With power demand (load) and in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.
We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case.
The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for
systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework.
We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying dynamics. Our strategy combines the construction of suboptimal quasi-turnpike trajectories via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by the large-layer regime of residual neural networks, commonly used in deep learning applications. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes.
One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4, 7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1–3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system.
The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a pde model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state.
This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research.
We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks.
In this paper, optimal control problems subject to a nonlinear scalar conservation law are
studied. Such optimal control problems are challenging both at the continuous and at the discrete
level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore
discretization of the underlying optimal control problem should be designed with care. Here the
discretize-then-optimize approach is employed where first the full discretization of the objective
function as well as the underlying PDE is considered. Then, the derivative of the reduced objective
is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta
(TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods,
also called strong stability preserving (SSP), are originally designed to preserve total variation of
the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to
ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is
too strong. Also approximation properties that the discrete adjoint inherits from the discretization
of the state equation are studied. Moreover order conditions are derived. In addition, optimal
choices with respect to CFL constant are discussed and numerical experiments are presented.
The dynamical, boundary optimal control problems
on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential
Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for
optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation
on the graph. The boundary control problem by Neumann
conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented.
In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.
Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems
(2020)
In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process.
We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems.
In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study the effect of a delay in the implementation of L 2-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as solutions of problems of exact optimal control, that is if they are implemented without delay, they steer the system to a position of rest in a given finite time T. We show that arbitrarily small delays δ > 0 can have a destabilizing effect in the sense that we can find initial states such that if the optimal control u is implemented in the form yx(t, 1) = u(t − δ) for t > δ, the energy of the system state at the terminal time T is almost twice as big as the initial energy. We also show that for more regular initial states, the effect of a delay in the implementation of the optimal control is bounded above in the sense that for initial positions with derivatives of BV-regularity and initial velocities with BV-regularity, the terminal energy is bounded above by the delay δ multiplied with a factor that depends on the BV-norm of the initial data. We show that for more general hyperbolic optimal exact control problems the situation is similar. For systems that have arbitrarily large eigenvalues, we can find terminal times T and arbitrarily small time delays δ, such that at the time T + δ, in the optimal control loop with delay the norm of the state is twice as large as the corresponding norm for the initial state. Moreover, if the initial state satisfies an additional regularity condition, there is an upper bound for the effect of time delay of the order of the delay with a constant that depends on the initial state only.
One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P=NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem.
In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension.
Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates.
We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].
Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.
In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to an optimal control problem. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation.
In order to carry out our study, we use the setting of a finite-dimensional linear-quadratic optimal control problem, for which the turnpike property is well understood. We prove that, when the time horizon T tends to infinity, the value function converges to a travelling-front like solution of the form W(x) + c T + λ. In addition, we provide a control interpretation of each of these three terms in the spirit of the turnpike theory. Finally, we compare this asymptotic decomposition with the existing results on long-time behavior for Hamilton-Jacobi equations. We stress that in our case, the Hamiltonian is not coercive in the momentum variable, a case rarely considered in the classical literature about Hamilton-Jacobi equations.
In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls.
We prove that, when the time horizon TTT tends to infinity, the value function asymptotically behaves as W(x)+c T+λW(x) + c\, T + \lambda W(x)+cT+λ, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property.
As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we provide a new turnpike result for the linear-quadratic optimal control problem with constrained control. As a main feature, our turnpike result applies to the case when the steady optimum may saturate the control constraints. This prevented us from proving the turnpike property with an exponential rate, which is well-known to hold for the unconstrained case.
We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function [katex]u_T[/katex] and a time horizon [katex]T > 0[/katex], we aim to construct all the initial conditions for which the viscosity solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of [katex]u_T[/katex] , is the one obtained by reversing the direction of time in the equation, considering [katex]u_T[/katex] as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when [katex]u_T[/katex] is reachable, we construct the set of all initial conditions for which the solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. Note that in general, such initial conditions are not unique. Finally, for the case when the target [katex]u_T[/katex] is not necessarily reachable, we study the projection of [katex]u_T[/katex] on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of [katex]u_T[/katex] , i.e. the smallest reachable target bounded from below by [katex]u_T[/katex] .
Pricing of access to energy networks is an important issue in liberalized energy sectors because of the natural monopoly character of the underlying transport infrastructures. We introduce a general pricing framework for potential-based energy flows in arbitrarily structured transport networks. In different specifications of our general pricing model we discuss first- and second-best pricing results and compare different pricing outcomes of potential-free and potential-based energy flow models. Our results show that considering nonlinear laws of physics leads to significantly different pricing results on networks and that these differences can only be seen in sufficiently complex, e.g., cyclic, networks as they can be found in real-world situations.
Many long-term investment planning models for liberalized electricity markets either optimize for the entire electricity system or focus on confined jurisdictions, abstracting from adjacent markets. In this paper, we provide models for analyzing the impact of the interdependencies between a core electricity market and its neighboring markets on key long-run decisions. This we do both for zonal and nodal pricing schemes. The identification of welfare optimal investments in transmission lines and renewable capacity within a core electricity market requires a spatially restricted objective function, which also accounts for benefits from cross-border electricity trading. This leads to mixed-integer nonlinear multilevel optimization problems with bilinear nonconvexities for which we adapt a Benders-like decomposition approach from the literature. In a case study, we use a stylized six-node network to disentangle different effects of optimal regional (as compared to supra-regional) investment planning. Regional planning alters investment in transmission and renewable capacity in the core region, which affects private investment in generation capacity also in adjacent regions and increases welfare in the core region at the cost of system welfare. Depending on the congestion-pricing scheme, the regulator of the core region follows different strategies to increase welfare causing distributional effects among stakeholders.
In this paper we propose an equilibrium model in order to analyze the impact of electricity market design on generation and transmission expansion in liberalized electricity markets. In a multi-level structure, our framework takes into account that generation investment and operation is decided by private investors, while network expansion and redispatch is decided by a regulated transmission system operator — as well as the different objectives of firms (profit maximization) and the regulator (welfare maximization). In order to illustrate the possibilities to quantify long term economic effects with our framework, we calibrate our model for the German electricity market. We consider various moderate adjustments of the market design: (i) the division of the market area into two price zones, (ii) the efficient curtailment of renewable production and (iii) a cost-benefit-driven balance between network expansion and network management measures. We then analyze the impact of these market designs on generation and transmission investment in case those design elements are anticipated upon network development planning. The resulting investment and production decisions are compared to a benchmark that reflects the current German electricity market design and to an overall optimal first-best benchmark. Our results reveal that price zones do have a significant impact on locational choice of generators and result in a reduced need for network expansion, but lead to only moderate annual welfare gains of approximately 0.9% of annual total system costs. Anticipation of optimal curtailment of renewables and a cost-benefit-driven use of redispatch operations upon network expansion planning, however, implies a welfare gain of over 4.9% of annual total system costs per year as compared to the existing market design, which equals 85% of the maximum possible welfare gain of the first-best benchmark.
The economics of global green ammonia trade – "Shipping Australian wind and sunshine to Germany"
(2023)
This paper contributes to understanding the transformation of global energy trade to green energy carriers, focusing on green ammonia as the foreseeable first green hydrogen carrier. We provide a comprehensive overview of today's ammonia trade and assess scaling options for the trade of green ammonia. To that aim, we develop an optimization model for the integrated assessment of the green ammonia value chain that covers all steps from green ammonia production in an exporting country, up to delivery to a harbor in an importing country. The model endogenously chooses among different technology options and determines cost minimal operation. In a case study, we apply the model to the large-scale import of ammonia from Australia to Germany in a scenario for 2030. The results show that green ammonia can reach cost parity with gray ammonia even for moderate gas prices (but not necessarily with blue ammonia) if CO2 prices are high enough. We also provide a sensitivity analysis with respect to the interest rate and other key technical and economic parameters and show that cracking ammonia to provide pure hydrogen comes at a 45 % cost markup per MWh at the destination.
The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities
(2018)
It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function---in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the ``cost of not knowing enough'' is rather high by comparing our approach with the open-source global solver SCIP. This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.
Liberalized gas markets in Europe are organized as entry-exit regimes so that gas trade and transport are decoupled. The decoupling is achieved via the announcement of technical capacities by the transmission system operator (TSO) at all entry and exit points of the network. These capacities can be booked by gas suppliers and customers in long-term contracts. Only traders who have booked capacities up-front can "nominate" quantities for injection or withdrawal of gas via a day-ahead market. To ensure feasibility of the nominations for the physical network, the TSO must only announce technical capacities for which all possibly nominated quantities are transportable. In this paper, we use a four-level model of the entry-exit gas market to analyze possible welfare losses associated with the decoupling of gas trade and transport. In addition to the multilevel structure, the model contains robust aspects to cover the conservative nature of the European entry-exit system. We provide several reformulations to obtain a single-level mixed-integer quadratic problem. The overall model of the considered market regime is extremely challenging and we thus have to make the main assumption that gas flows are modeled as potential-based linear flows. Using the derived single-level reformulation of the problem, we show that the feasibility requirements for technical capacities imply significant welfare losses due to unused network capacity. Furthermore, we find that the specific structure of the network has a considerable influence on the optimal choice of technical capacities. Our results thus show that trade and transport are not decoupled in the long term. As a further source of welfare losses and discrimination against individual actors, we identify the minimum prices for booking capacity at the individual nodes.
Dynamical systems with sub-processes evolving on many different time scales are ubiquitous in applications. Their efficient solution is greatly enhanced by automatic time step variation. This paper is concerned with the theory, construction and application of IMEX-Peer methods that are super-convergent for variable step sizes and A-stable in the implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods which keep their higher order for variable step sizes and exhibit favourable linear stability properties, we derive necessary and sufficient conditions on the nodes and coefficient matrices and apply an extrapolation approach based on already computed stage values. New super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are given as result of additional order conditions which maintain the super-convergence property independent of step size changes. Numerical experiments and a comparison to other super-convergent IMEX-Peer methods show the potential of the new methods when applied with local error control.
Super-convergence and post-processing for mixed finite element approximations of the wave equation
(2017)
We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes.
Probability functions figure prominently in optimization problems
of engineering. They may be nonsmooth even if all input data are smooth.
This fact motivates the consideration of subdifferentials for such typically just
continuous functions. The aim of this paper is to provide subdifferential formulae
of such functions in the case of Gaussian distributions for possibly
infinite-dimensional decision variables and nonsmooth (locally Lipschitzian)
input data. These formulae are based on the spheric-radial decomposition of
Gaussian random vectors on the one hand and on a cone of directions of
moderate growth on the other. By successively adding additional hypotheses,
conditions are satisfied under which the probability function is locally Lipschitzian
or even differentiable.
We present a new balancing-based structure-preserving model reduc-
tion technique for linear port-Hamiltonian descriptor systems. The pro-
posed method relies on a modification of a set of two dual generalized
algebraic Riccati equations that arise in the context of linear quadratic
Gaussian balanced truncation for differential algebraic systems. We de-
rive an a priori error bound with respect to a right coprime factorization
of the underlying transfer function thereby allowing for an estimate with
respect to the gap metric. We further theoretically and numerically ana-
lyze the influence of the Hamiltonian and a change thereof, respectively.
With regard to this change of the Hamiltonian, we provide a novel proce-
dure that is based on a recently introduced Kalman–Yakubovich–Popov
inequality for descriptor systems. Numerical examples demonstrate how
the quality of reduced-order models can significantly be improved by first
computing an extremal solution to this inequality.
We examine interpolatory model reduction methods that are well-suited for treating large scale port-Hamiltonian differential-algebraic systems in a way that is able to preserve and indeed, take advantage of the underlying structural features of the system. We introduce approaches that incorporate regularization together with prudent selection of interpolation data. We focus on linear time-invariant systems and present a systematic treatment of a variety of model classes that include combinations of index-1 and index-2 systems, describing in particular how constraints may be represented in the transfer function and then preserved with interpolatory methods. We propose an algorithm to generate effective interpolation data and illustrate its effectiveness via two numerical examples.
We extend the modeling framework of port-Hamiltonian descriptor systems to include under- and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Numerical examples are provided.
In this work we analyze the structural properties of the set of feasible bookings in the European entry-exit gas market system. We present formal definitions of feasible bookings and then analyze properties that are important if one wants to optimize over them. Thus, we study whether the sets of feasible nominations and bookings are bounded, convex, connected, conic, and star-shaped. The results depend on the specific model of gas flow in a network. Here, we discuss a simple linear flow model with arc capacities as well as nonlinear and mixed-integer nonlinear models of passive and active networks, respectively. It turns out that the set of feasible bookings has some unintuitive properties. For instance, we show that the set is nonconvex even though only a simple linear flow model is used.
In this work we study polyhedra in the context of network flow problems, where the flow value on each arc lies in one of several predefined intervals. This is motivated by nonlinear
problems on transportation networks, where nonlinearities are handled by piecewise linear approximation or relaxation - a common and established approach in many applications.
Several methods for modeling piecewise linear functions are known which provide a complete description for a single network arc. However, in general this property is lost when considering multiple arcs. We show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities. For the case of paths of degree-two-nodes we give a complete description of the polyhedron projected to the integer variables. Our model is based on - but not limited to - the multiple choice method; we also show how to transfer our results to a formulation based on the incremental method. Computational results show that a state-of-the-art MIP-solver greatly benefits from using our cutting planes for random and realistic network topologies.
This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classical welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash–Cournot models. Thus, an analogue of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.
In this paper we propose a bi-level equilibrium model that allows to analyze the impact of different regulatory frameworks on storage and network investment in distribution networks. In our model, a regulated distribution system operator decides on network investment and operation while he anticipates the decisions of private agents on storage investment and operation. Since, especially in distribution networks, voltage stability and network losses have a decisive influence on network expansion and operation, we use a linearized AC power flow formulation to adequately account for these aspects. As adjustments of the current regulatory framework, we consider curtailment of renewable production, the introduction of a network fee based on the maximum renewable feed-in, and a subsidy scheme for storage investment. The performance of the different alternative frameworks is compared to the performance under rules that are commonly applied in various countries today, as well as to a system-optimal (first-best) benchmark. To illustrate the economic effects, we calibrate our model with data from the field project Smart Grid Solar. Our results reveal that curtailment and a redesign of network fees both have the potential to significantly reduce total system costs. On the contrary, investment subsidization of storage capacity has only a limited impact as long as the distribution system operator is not allowed to intervene in storage operation.