Approximately Adaptive Distributionally Robust Optimization for Energy and Reserve Dispatch

Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: An exact and physically-bounded formulation

In this paper, we consider dispatchability as the set of all admissible nodal wind power injections that will not cause infeasibility in real-time dispatch (RTD). Our work reveals that the dispatchability of the affine policy based RTD (AF-RTD) is a polytope whose coefficients are linear functions of the generation schedule and the gain matrix of affine policy. Two mathematical formulations of the dispatchability maximized energy and reserve dispatch (DM-ERD) are proposed. The first one maximizes the distance from the forecast to the boundaries of the dispatchability polytope subject to the available production cost or reserve cost. Provided the forecast value and variance of wind power, the generalized Gauss inequality (GGI) is adopted to evaluate the probability of infeasible RTD without the exact probability distribution of wind power. Combining the first formulation and the GGI approach, the second one minimizes the total cost subject to a desired reliability level through dispatchability maximization. Efficient convex optimization based algorithms are developed to solve these two models. Different from the conventional robust optimization method, our model does not rely on the specific uncertainty set of wind generation and directly optimizes the uncertainty accommodation capability of the power system. The proposed method is also compared with the affine policy based robust energy and reserve dispatch (AR-ERD). Case studies on the PJM 5-bus system illustrate the proposed concept and method. Experiments on the IEEE 118-bus system demonstrate the applicability of our method on moderate sized systems and its scalability to large dimensional uncertainty.

This paper proposes a two-stage distributionally robust optimization model for the joint energy and reserve dispatch (D-RERD for short) of bulk power systems with significant renewable energy penetration. Distinguished from the prevalent uncertainty set-based and worst-case scenario oriented robust optimization methodology, we assume that the output of volatile renewable generation follows some ambiguous distribution with known expectations and variances, the probability distribution function (pdf) is restricted in a functional uncertainty set. D-RERD aims at minimizing the total expected production cost in the worst renewable power distribution. In this way, D-RERD inherits the advantages from both stochastic optimization and robust optimization: statistical characteristic is taken into account in a data-driven manner without requiring the exact pdf of uncertain factors. We present a convex optimization-based algorithm to solve the D-RERD, which involves solving semidefinite programming (SDP), convex quadratic programming (CQP), and linear programming (LP). The performance of the proposed approach is compared with the emerging adaptive robust optimization (ARO)-based model on the IEEE 118-bus system. Their respective features are discussed in case studies.

This paper proposes a distance-based distributionally robust energy and reserve (DB-DRER) dispatch model via Kullback–Leibler (KL) divergence, considering the volatile of renewable energy generation. Firstly, a two-stage optimization model is formulated to minimize the expected total cost of energy and reserve (ER) dispatch. Then, KL divergence is adopted to establish the ambiguity set. Distinguished from conventional robust optimization methodology, the volatile output of renewable power generation is assumed to follow the unknown probability distribution that is restricted in the ambiguity set. DB-DRER aims at minimizing the expected total cost in the worst-case probability distributions of renewables. Combining with the designed empirical distribution function, the proposed DB-DRER model can be reformulated into a mixed integer nonlinear programming (MINLP) problem. Furthermore, using the generalized Benders decomposition, a decomposition method is proposed and sample average approximation (SAA) method is applied to solve this problem. Finally, simulation result of the proposed method is compared with those of stochastic optimization and conventional robust optimization methods on the 6-bus system and IEEE 118-bus system, which demonstrates the effectiveness and advantages of the method proposed.

Implementing integrated electric-heat systems (IEHSs) with coupled power distribution networks and district heating networks is an essential means to solve current energy problems. However, prosumers with multiple energy forms coupled and renewable energy sources with natural uncertainties pose challenges to the operation of IEHSs. This paper proposes a joint energy and reserve dispatch model for IEHSs based on transactive energy, which is a coordinated combination of a bi-level Stackelberg game and two-stage robust optimization. The bi-level Stackelberg game is used to realize the equilibrium of interests among three transacting parties, namely, integrated energy service provider (IESP), multi-carrier prosumer (MCP), and load aggregator (LA). The two-stage robust optimization is employed to ensure the reliability of the system operation under renewable energy uncertainty. In the upper level of the Stackelberg game, the IESP perform pricing and reserve dispatch, while the MCP and LA maximize their benefits via energy management in the lower level. Linearization techniques are utilized to approximate the bi-level Stackelberg game model into a single-level mixed-integer linear programming problem. The converted single-level game model is subsequently regarded as the first stage, while the real-time feasibility check is regarded as the second stage to form a two-stage robust optimization model, which is solved by a modified C&CG algorithm. Case studies demonstrate that the proposed joint energy and reserve dispatch method effectively achieves economic and reliable operation.

With resorting to the tool of distributionally robust optimization, this paper proposes a risk-based distributionally robust approach to address the wind power uncertainty in the energy and reserve dispatch. The proposed model minimizes the total cost including dispatch cost and risk cost. The dispatch cost refers to the cost of scheduling energy and spinning reserve while the risk cost is the expected cost of load shedding and wind spillage. Unlike the previous approach which predefined distribution of random variables, the proposed approach takes the ambiguity of distributions into account. It extracts probabilistic information from historical data of random variables, and constructs ambiguity set to contain possible distributions. Then the worst-case distribution over ambiguity set is used to evaluate risks to hedge against the ambiguity. In this paper, a novel metric—Wasserstein-moment metric (WM-metric) is introduced to construct ambiguity set. Compared with Wasserstein-metric and Moment-metric, WM-metric considers more probabilistic information and thus can further mitigate the conservativeness of ambiguity set. The performance of the proposed approach is tested by a 6-bus system for illustrative purpose.

Global warming and environmental pollution concerns have promoted dramatic integrations of renewable energy sources all over the world. Associated with benefits of environmental conservation, essentially uncertain and variable characteristics of such energy resources significantly challenge the operation of power systems. In order to implement reliable and economical operations, a robust energy and reserve dispatch (RERD) model is proposed in this paper, in which the operating decisions are divided into pre-dispatch and re-dispatch. A robust feasibility constraint set is imposed on pre-dispatch variables, such that operation constraints can be recovered by adjusting re-dispatch after wind generation realizes. The model is extended to more general dispatch decision making problems involving uncertainties in the framework of adjustable robust optimization. By revealing the convexity of the robust feasibility constraint set, a comprehensive mixed integer linear programming based oracle is presented to verify the robust feasibility of pre-dispatch decisions. A cutting plane algorithm is established to solve associated optimization problems. The proposed model and method are applied to a five-bus system as well as a realistic provincial power grid in China. Numeric experiments demonstrate that the proposed methodology is effective and efficient.

A Lagrangian relaxation solution method is applied by the authors to the problem of joint energy and reserve dispatch in a pool-oriented electricity market.

Continuous power supply is crucial for any developing economy and its cost efficiency and reliability is highly dependent on operating reserve. Globally, the power systems are adopting a market based structure to enhance performance. In conventional electricity markets reserve gets a second place as its dispatch is performed after energy allocation but in the competitive market these two commodities are dispatched simultaneously to maximize efficiency and reliability. This study proposes a technique based on interior point algorithm for optimizing the dynamic combined energy and reserve dispatch problem with practical complex equality and inequality constraints such as power balance, generation and reserve capacity limits, ramp-up/down limits, and reserve-energy coupling constraints. To simulate practical outage conditions generator contingencies and their effect on operating cost and reserve dispatch is observed. The proposed algorithm is simulated on MATLAB platform and tested for three different cases of the IEEE 57 bus system with 17 generating units. It is found that the proposed method converges to optimal solution for all tested cases and produces feasible solutions where all complex constraints are completely satisfied.

The integration of distributed energy resources into a virtual power plant (VPP) can realise the scale merit. This study presents a novel approach for determining the optimal offering strategy of a VPP participating in the day‐ahead (DA), the spinning reserve (SR), and the real‐time (RT) markets. To hedge against multi‐stage uncertainties, the authors propose a robust mixed integer linear programming optimisation model that comprises four levels: (i) the optimal DA energy and reserve dispatch; (ii) the worst‐case realisation of uncertainties involved in DA market energy prices, SR market capacity prices, stochastic power production, and called balancing power; (iii) the optimal RT energy re‐dispatch; and (iv) the worst‐case realisation of uncertain RT market energy prices. Moreover, a tractable solution method based on strong duality theory and the column‐and‐constraint generation algorithm to solve the proposed four‐level formulation was developed. Finally, numerical results for a realistic case study demonstrate the efficiency and applicability of the proposed approach. The commercial benefits of this strategy are also evaluated.

Performance comparison of enhanced PSO and DE variants for dynamic energy/reserve scheduling in multi-zone electricity market

A robust optimization approach to energy and reserve dispatch in electricity markets

This paper proposes an approach for considering the impact of demand side management (DSM) in energy and reserve service scheduling problem considering index of customers' reliability in electricity markets. The market clearance is performed in the presence of demand side response and both energy and reserve are dispatched optimally while accounting for different load levels, previous operating points, unit ramping rates and possible contingencies that could occur in the system and the associated loss of loads. Alongside it accounts for the constraints including voltage and transmission limits in the system's steady state and probable outages. On the contrary of the existing research, this paper first maximizes the benefit of the customers with optimizing the load using demand side program and then the overall costs of energy, spinning reserve and interruption are minimized with this load. This paper studies the impact of DSM in two stages. In the first stage, optimal energy and reserve dispatch are performed with and without considering DSM. In the second stage two different types of DSM programs are compared and the difference in the optimum operating point of the market whilst considering each program is investigated. The effectiveness of the proposed method is examined by application to the IEEE 24 bus test system.

Traditional robust optimization methods use box uncertainty sets or gamma uncertainty sets to describe wind power uncertainty. However, these uncertainty sets fail to utilize wind forecast error probability information and assume that the wind forecast error is symmetrical and independent. This assumption is not reasonable and makes the optimization results conservative. To avoid such conservative results from traditional robust optimization methods, in this paper a novel data driven optimization method based on the nonparametric Dirichlet process Gaussian mixture model (DPGMM) was proposed to solve energy and reserve dispatch problems. First, we combined the DPGMM and variation inference algorithm to extract the GMM parameter information embedded within historical data. Based on the parameter information, a data driven polyhedral uncertainty set was proposed. After constructing the uncertainty set, we solved the robust energy and reserve problem. Finally, a column and constraint generation method was employed to solve the proposed data driven optimization method. We used real historical wind power forecast error data to test the performance of the proposed uncertainty set. The simulation results indicated that the proposed uncertainty set had a smaller volume than other data driven uncertainty sets with the same predefined coverage rate. Furthermore, the simulation was carried on PJM 5-bus and IEEE-118 bus systems to test the data driven optimization method. The simulation results demonstrated that the proposed optimization method was less conservative than traditional data driven robust optimization methods and distributionally robust optimization methods.

With the increasing penetration of renewable generation such as wind power in modern power systems, there are many new challenges arising in power system operation with respect to reliability and economy. In this work, we study a two-stage data-driven distributionally robust (DR) energy and reserve dispatch problem with uncertain wind power. Different from the general moment-based ambiguity set, we design a new distance-based ambiguity set to describe the uncertain probability distribution of wind power, which can be constructed in a data-driven manner from historical data. Base on this new ambiguity set, the second-stage worst-case expectation of the problem is reformulated to a combination of conditional value-at-risk (CVaR) and an expected cost with respect to a reference distribution. Thus, the proposed two-stage DR model becomes a two-stage stochastic optimization problem which can be readily solved. Case studies are carried out to verify the effectiveness of the proposed method based on the IEEE 6-bus test system and modified IEEE 118-bus test system. Simulation results show the value of data in controlling the conservatism of the problem, and the DR problem converges to the stochastic problem with fixed distribution as the data size goes to infinity.