A bi-level model of dynamic traffic signal control with continuum approximation

This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955; Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in Han et al. (2014), which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure

SOLVING TWO-LEVEL OPTIMIZATION PROBLEMS WITH APPLICATIONS TO ROBUST DESIGN AND ENERGY MARKETS

This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation

Dynamic optimization of energy systems with thermal energy storage

textThermal energy storage (TES), the storage of heat or cooling, is a cost-effective energy storage technology that can greatly enhance the performance of the energy systems with which it interacts. TES acts as a buffer between transient supply and demand of energy. In solar thermal systems, TES enables the power output of the plant to be effectively regulated, despite fluctuating solar irradiance. In district energy systems, TES can be used to shift loads, allowing the system to avoid or take advantage of peak energy prices. The benefit of TES, however, can be significantly enhanced by dynamically optimizing the complete energy system. The ability of TES to shift loads gives the system newfound degrees of freedom which can be exploited to yield optimal performance. In the hybrid solar thermal/fossil fuel system explored in this work, the use of TES enables the system to extract nearly 50% more solar energy when the system is optimized. This requires relaxing some constraints, such as fixed temperature and power control, and dynamically optimizing the over a one-day time horizon. In a district cooling system, TES can help equipment to run more efficiently, by shifting cooling loads, not only between chillers, but temporally, allowing the system to take advantage of the most efficient times for running this equipment. This work also highlights the use of TES in a district energy system, where heat, cooling and electrical power are generated from central locations. Shifting the cooling load frees up electrical generation capacity, which is used to sell power to the grid at peak prices. The combination of optimization, TES, and participation in the electricity market yields a 16% cost savings. The problems encountered in this work require modeling a diverse range of systems including the TES, the solar power plant, boilers, gas and steam turbines, heat recovery equipment, chillers, and pumps. These problems also require novel solution methods that are efficient and effective at obtaining workable solutions. A simultaneous solution method is used for optimizing the solar power plant, while a static/dynamic decoupling method is used for the district energy system.Chemical Engineerin

Modelling and analysing the impact of local flexibility on the business cases of electricity retailers

Demand side response are proposed to incentivise customers to shift their electricity usage from peak demand periods to off-peak demand periods and to curtail their electricity usage during peak demand periods, which show great potential to reduce the peak loads, electricity prices, customers’ bills and further stabilize the power systems. The investigation of this effect on the pricing strategies and the profits of electricity retailers has recently emerged as a highly interesting research area. However, the state-of-the-art, bi-level optimization modelling approach makes the unrealistic assumption that retailers treat wholesale market prices as exogenous, fixed parameters. On the other hand, distributed energy resources (DER) in electricity markets are proposed to bring the significant operating flexibility which can support system balancing and reduce demand peaks, thereby limiting the balancing costs of conventional generators and the investments costs of new generation and network assets. And, local energy markets (LEM) have recently attracted great interest as they enable effective coordination of small-scale DER at the customer side, and avoidance of distribution network reinforcements. However, the introduction of LEM has also significant implications on the strategic interactions between the customers and incumbent electricity retailers, which has not been explored. Furthermore, a specific demand response technology of electric vehicles (EV) exhibits the potential to support system balancing and limit demand peaks, thus improving significantly the cost-effectiveness of low-carbon electricity systems. And the effective pricing of EV charging by aggregators constitutes a key problem towards the realization of the significant EV flexibility potential in deregulated electricity systems and has been addressed by previous work through bi-level optimization formulations. However, the solution approach adopted in previous work cannot capture the discrete nature of the EV charging / discharging levels. Furthermore, aggregators suffering from communication and privacy limitations are hard to acquire the perfect knowledge of EV operating characteristics and traveling patterns. Given such a context, this thesis aims at addressing the above challenges and proposing strategic retail pricing-based energy response programs to study the interactions between the electricity retailer / aggregator and its served flexible customers / EV based on game theoretic modeling and learning based approaches. We conduct the research in three different application scenarios: 1) This thesis proposes a novel bi-level optimization problem which represents endogenously the wholesale market clearing process as an additional lower-level problem, thus capturing the realistic implications of a retailer’s pricing strategies and the resulting demand response on the wholesale market prices. This bi-level optimization problem is solved through converting it to a single-level Mathematical Programs with Equilibrium Constraints (MPEC). The scope of the examined case studies is threefold. First of all, they demonstrate the interactions between the retailer, the flexible consumers and the wholesale market and analyse the fundamental effects of the consumers’ time-shifting flexibility on the retailer’s revenue from the consumers, its cost in the wholesale market, and its overall profit. Furthermore, they analyse how these effects of demand flexibility depend on the retailer’s relative size in the market and the strictness of the regulatory framework. Finally, they highlight the added value of the proposed bi-level model by comparing its outcomes against the state-of-the-art bi-level modelling approach. 2) This thesis explores for the first time the interaction between electricity retailer and LEM by proposing a novel bi-level optimization problem, which captures the pricing decisions of a strategic retailer in the upper-level problem and the response of both independent customers and the LEM (both including flexible consumers, micro- generators and energy storages) in the lower-level problems. Since the lower-level problem representing the LEM is non-convex, a new analytical approach is employed for solving the developed bi-level optimization problem. The examined case studies demonstrate that the introduction of an LEM reduces the customers’ energy dependency on the retailer and limits the retailer’s strategic potential of exploiting the customers through large differentials between buy and sell prices. As a result, the profit of the retailer is significantly reduced while the customers, primarily the LEM participants and to a lower extent non-participating customer, achieve significant economic benefits. 3) This thesis proposes a reinforcement learning (RL) method that the EV aggregator gradually learns how to improve its pricing strategies by utilizing experiences acquired from its repeated interactions with the EV and the wholesale market. Although RL can tackle the challenge of imperfect information and MPEC reformulation, the state-of-the- art RL methods require discretization of state and / or action spaces and thus exhibit limitations in terms of solution optimality and computational requirements. This thesis proposes a novel deep reinforcement learning (DRL) method to solve the examined EV pricing problem, combining deep deterministic policy gradient (DDPG) principles with a prioritized experience replay (PER) strategy, and setting up the problem in multi-dimensional continuous state and action spaces. Case studies demonstrate that the proposed method outperforms state-of-the-art RL methods in terms of both solution optimality and computational requirements, and comprehensively analyze the economic impacts of smart-charging and vehicle-to-grid (V2G) flexibility on both aggregators and EV owners.Open Acces

Optimization Approaches for Electricity Generation Expansion Planning Under Uncertainty

In this dissertation, we study the long-term electricity infrastructure investment planning problems in the electrical power system. These long-term capacity expansion planning problems aim at making the most effective and efficient investment decisions on both thermal and wind power generation units. One of our research focuses are uncertainty modeling in these long-term decision-making problems in power systems, because power systems\u27 infrastructures require a large amount of investments, and need to stay in operation for a long time and accommodate many different scenarios in the future. The uncertainties we are addressing in this dissertation mainly include demands, electricity prices, investment and maintenance costs of power generation units. To address these future uncertainties in the decision-making process, this dissertation adopts two different optimization approaches: decision-dependent stochastic programming and adaptive robust optimization. In the decision-dependent stochastic programming approach, we consider the electricity prices and generation units\u27 investment and maintenance costs being endogenous uncertainties, and then design probability distribution functions of decision variables and input parameters based on well-established econometric theories, such as the discrete-choice theory and the economy-of-scale mechanism. In the adaptive robust optimization approach, we focus on finding the multistage adaptive robust solutions using affine policies while considering uncertain intervals of future demands. This dissertation mainly includes three research projects. The study of each project consists of two main parts, the formulation of its mathematical model and the development of solution algorithms for the model. This first problem concerns a large-scale investment problem on both thermal and wind power generation from an integrated angle without modeling all operational details. In this problem, we take a multistage decision-dependent stochastic programming approach while assuming uncertain electricity prices. We use a quasi-exact solution approach to solve this multistage stochastic nonlinear program. Numerical results show both computational efficient of the solutions approach and benefits of using our decision-dependent model over traditional stochastic programming models. The second problem concerns the long-term investment planning with detailed models of real-time operations. We also take a multistage decision-dependent stochastic programming approach to address endogenous uncertainties such as generation units\u27 investment and maintenance costs. However, the detailed modeling of operations makes the problem a bilevel optimization problem. We then transform it to a Mathematic Program with Equilibrium Constraints (MPEC) problem. We design an efficient algorithm based on Dantzig-Wolfe decomposition to solve this multistage stochastic MPEC problem. The last problem concerns a multistage adaptive investment planning problem while considering uncertain future demand at various locations. To solve this multi-level optimization problem, we take advantage of affine policies to transform it to a single-level optimization problem. Our numerical examples show the benefits of using this multistage adaptive robust planning model over both traditional stochastic programming and single-level robust optimization approaches. Based on numerical studies in the three projects, we conclude that our approaches provide effective and efficient modeling and computational tools for advanced power systems\u27 expansion planning

Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems