Distributionally Robust Unit Commitment in Coordinated Electricity and District Heating Networks

A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: A case study of Mashhad

Wind power integration could be restricted by inflexible operation of combined heat and power (CHP) units due to the strong linkage between power generation and heating supply in winter. Utilization of the heat storage capacity of existing district heating network (DHN) is a cost-effective measure to enhance power system operational flexibility to accommodate large amounts of variable wind power. In this paper, transmission-constrained unit commitment (UC) with combined electricity and district heating networks (UC-CEHN) is formulated with a linear DHN model to coordinate short-term operation of electric power and district heating systems. The heat storage capacity of the DHN is modeled by capturing the quasi-dynamics of pipeline temperature. Both deterministic and robust models are developed to incorporate UC with the linear DHN model. Case studies are carried out for two test systems to show the potential benefits of the proposed method in terms of wind power integration and efficient operation.

Problem definition: In this paper, we study the fleet repositioning problem for a free-float vehicle sharing system, aiming to dynamically match the vehicle supply and travel demand at the lowest total cost of repositioning and lost sales. Academic/practical relevance: Besides the analytical results on the optimal repositioning policy, the proposed optimization framework is applicable to practical problems by its computational efficiency as well as the capability to handle temporally dependent demands. Methodology: We first formulate the problem as a stochastic dynamic program. To solve for a multiregion system, we deploy the distributionally robust optimization (DRO) approach that can incorporate demand temporal dependence, motivated by real data. We first propose a “myopic” two-stage DRO model that serves as both an illustration of the DRO framework and a benchmark for the later multistage model. We then develop a computationally efficient multistage DRO model with an enhanced linear decision rule (ELDR). Results: Under a two-region system, we find a simple reposition up-to and down-to policy to be optimal, when the demands are temporally independent. Such a structure is also preserved by our ELDR solution. We also provide new analytical insights by proving the optimality of ELDR in solving the single-period DRO problem. We then show that the numerical performance of the ELDR solution is close to the exact optimal solution from the dynamic program. Managerial implications: In a real-world case study of car2go, we quantify the “value of repositioning” and compare with several benchmarks to demonstrate that the ELDR solutions are computationally scalable and in general result in lower cost with less frequent repositioning. We also explore several managerial implications and extensions from the experiments.

Chapter 4 - Data-driven robust stochastic optimization for power systems operations

Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty

Quay cranes are among the most important resources in port terminals, and their efficient use is crucial for port terminals to remain competitive in the market. The problem of determining the sequence of tasks performed by the quay cranes that minimizes the turnaround time of a vessel is an NP‐hard problem known as the quay crane scheduling problem (QCSP).In this paper, we consider the unidirectional QCSP under uncertain processing times, where the cranes are only allowed to move in a specific direction. We start by presenting the first distributionally robust optimization (DRO) model for this problem and an exact decomposition algorithm to solve it. The proposed DRO model makes it possible to derive a stochastic programming model and a robust optimization model for the unidirectional QCSP by an appropriate choice of the risk‐averse parameter of the model. Through extensive numerical results, we compare these three approaches—stochastic programming, robust optimization, and DRO—to investigate whether significant differences among the solutions obtained exist. Finally, we propose a new method for helping practitioners to determine a representative set of different DRO solutions.

Commitment and dispatch of heat and power units via affinely adjustable robust optimization

Clean energy resources, like wind, have a stochastic nature, which involves uncertainties in the power system. Introducing energy storage systems (ESS) to the network can compensate for the uncertainty in wind plant output and allow the plant to participate in ancillary service markets. Advance in compressed air energy storage system (CAES) technologies and their fast response make them suitable for ancillary services. This paper investigates the participation of a combined energy system composed of wind plants and compressed air energy storage system (CAES) in the energy market from a private owner’s viewpoint, including trading in energy markets and bidding for frequency regulation and reserve capacity in ancillary service markets. Since this problem contains various uncertainties associated with market prices, wind generation levels, and regulation signals, distributionally robust optimization (DRO) is used to model the uncertainties and enhance the simultaneous participation of a combined wind-CAES system in day-ahead energy and ancillary service markets. This method combines the advantages of stochastic and robust optimization. In contrast to robust optimization (RO), the method consolidates specific statistical data to reduce conservative results. Simulation results demonstrate the proposed model’s effectiveness in handling uncertainties and provide a framework for investors in this area. In addition, case study analyses are applied to assess the model’s performance and validate the coordination of a wind plant and compressed air energy storage system in participating in a deregulated electricity market. Finally, DRO and RO are compared in modeling the uncertainties of the optimization problem. The optimal outputs demonstrate the effectiveness of DRO in terms of achieving higher realized profits with less conservative results.

Distributionally robust optimization (DRO) has been introduced for solving stochastic programs in which the distribution of the random variables is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that contains the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. As examples, we consider the shape-restricted confidence bands and the confidence bands constructed with a kernel density estimation technique. The former allows us to incorporate the prior knowledge of the shape of the underlying density function (e.g., unimodality and monotonicity), and the latter enables us to handle multidimensional cases. Furthermore, we establish the convergence of the optimal value of DRO to that of the underlying stochastic program as the sample size increases. The DRO with our ambiguity set involves functional decision variables and infinitely many constraints. To address this challenge, we apply duality theory to reformulate the DRO to a finite-dimensional stochastic program, which is amenable to a stochastic subgradient scheme as a solution method.

A novel two stage robust optimization model is presented in this research paper for scheduling of multi-energy Combined Cooling, Heating, and Power (CCHP) microgrid. This model will effectively address uncertainties in renewable energy source (RES) generation. The approach presented employs a twostage framework that integrates Maximum Entropy Ellipsoid (MEE) uncertainty modelling, the Big-M method, and the Column and Constraint Generation (C&CG) algorithm. The MEE set look after realistic uncertainties by accounting for correlations between RES sources, like wind turbine (WT) or photovoltaic (PV). These RES are often influenced by shared environmental factors. The Big-M method guarantees efficient handling of conditional constraints within the optimization, especially critical in scenarios. C&CG method is used to solve the Two Stage Robust Optimization model, which utilizes worst-case scenarios under MEE balancing both robust feasibility and computational efficiency. Through this robust optimization approach, the study aims to minimize operational costs, reduce CO2 emissions, and improve the overall reliability of energy supply. Special attention is given to managing demand-side responses and incorporating real-time energy consumption patterns, ultimately leading to a more sustainable and cost-effective energy solution for CCHP microgrids in the face of volatile renewable energy conditions.

Massive adoptions of combined heat and power (CHP) units necessitate the coordinated operation of the power system and district heating system (DHS). Exploiting the reconfigurable property of district heating networks (DHNs) provides a potentially cost-effective solution to enhance the flexibility of the power system by readjusting the configuration for heat supply, which has not been addressed in the literature. To address this issue, a unit commitment considering combined electricity and reconfigurable heating network (UC-CERHN) is proposed in this paper to coordinate the day-ahead scheduling of power system and DHS. The DHS is formulated as a nonlinear and mixed-integer model considering the reconfigurable DHN. To make the commitment problem tractable, an auxiliary heat quantity variable is introduced, and the DHS model is reformatted to a linear energy flow model by approximation of heat loss, where the computational burdens are significantly reduced. Extensive case studies are presented to validate the effectiveness and accuracy of the approximated model and illustrate the potential benefits of the proposed method with respect to congestion management and wind power accommodation.

Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power.

Coordinated energy scheduling and management strategies in the energy hub plan are essential to achieve optimal economic performance. In this paper, the scheduling and management framework of an energy hub (EH) is presented with the aim of energy profit maximization in partnership with electricity, natural gas, and district heating networks (EGHNs) considering the coordinated multi-energy management based on the day-ahead market. The optimum capacity of EH equipment, including photovoltaic and wind renewable energy sources, a combined heat and power system (CHP), a boiler, energy storage, and electric vehicles is determined in the day-ahead market using the improved Fick’s law algorithm (IFLA), considering the energy profit maximization and also satisfying the linear network and hub constraints. The conventional FLA is inspired by the concept of Fick’s diffusion law, and, in this study, its performance against premature convergence is improved by using Rosenbrock’s direct rotational method. The performance of the IFLA when applied to EH coordinated scheduling and management problems with the aim of profit maximization is compared with the conventional FLA, particle swarm optimization (PSO), and manta ray foraging optimization (MRFO) methods. The results show that the proposed scheduling and multi-energy management framework achieves more energy profit in the day-ahead electricity, gas, and heating markets by satisfying the operation and EH constraints compared to other methods. Furthermore, according to the findings, the increased (decreased) demand and the forced outage rate caused a decrease (increase) in the EH profit. The results show the effectiveness of the proposed framework to obtain the EH maximum energy profit in the day-ahead market.

We present a new methodology to support an energy trading company (ETC) to devise contracting strategies under an optimal risk-averse renewable portfolio. The uncertainty in the generation of renewable energy sources is accounted for by exogenously simulated scenarios, as is customary in stochastic programming. However, we recognize that spot prices largely depend on unpredictable market conditions, making it difficult to capture its underlying stochastic process, which challenges the use of fundamental approaches for forecasting. Under such framework, industry practices make use of stress tests to validate portfolios. We then adapt the robust optimization approach to perform an endogenous stress test for the spot prices as a function of the buy-and-sell portfolio of contracts and renewable energy generation scenarios. The optimal contracting strategy is built through a bilevel optimization model that uses a hybrid approach, mixing stochastic and robust optimization. The proposed model is flexible to represent the traditional stochastic programming approach and to express the ETC's uncertainty aversion in the case where the price distribution cannot be precisely estimated. The effectiveness of the model is illustrated with examples from the Brazilian market, where the proposed approach is contrasted to its stochastic counterpart and both are benchmarked against observed market variables.

This paper presents a new approach for the contingency-constrained single-bus unit commitment problem. The proposed model explicitly incorporates an n - K security criterion by which power balance is guaranteed under any contingency state comprising the simultaneous loss of up to K generation units. Instead of considering all possible contingency states, which would render the problem intractable, a novel method based on robust optimization is proposed. Using the notion of umbrella contingency, the robust counterpart of the original problem is formulated. The resulting model is a particular instance of bilevel programming which is solved by its transformation to an equivalent single-level mixed-integer programming problem. Unlike previously reported contingency-dependent approaches, the robust model does not depend on the size of the set of credible contingencies, thus providing a computationally efficient framework. Simulation results back up these conclusions.