Energy and Reserve Dispatch with Renewable Generation Using Data-Driven Distributionally Robust Optimization

With the increasing penetration of wind power, the uncertainty associated with it brings more challenges to the operation of the integrated energy system (IES), especially the power subsystem. However, the typical strategies to deal with wind power uncertainty have poor performance in balancing economy and robustness. Therefore, this paper proposes a distributionally robust joint chance-constrained dispatch (DR-JCCD) model to coordinate the economy and robustness of the IES with uncertain wind power. The optimization dispatch model is formulated as a two-stage problem to minimize both the day-ahead and the real-time operation costs. Moreover, the ambiguity set is generated using Wasserstein distance, and the joint chance constraints are used to ensure that the safety constraints (e.g., ramping limit and transmission limit) can be satisfied jointly under the worst-case probability distribution of wind power. The model is remodeled as a mixed-integer tractable programming issue, which can be solved efficiently by ready-made solvers using linear decision rules and linearization methods. Case studies on an electricity–gas–heat regional integrated system, which includes a modified IEEE 24-bus system, 20 natural gas-nodes, and 6 heat-node system, are investigated for verification. Numerical simulation results demonstrate that the proposed DR-JCCD approach effectively coordinates the economy and robustness of IES and can offer operators a reasonable energy management scheme with an acceptable risk level.

This paper focuses on economic dispatch (ED) in power systems with intermittent wind power, which is a very critical issue in future power systems. A stochastic ED problem is formed based on the recently proposed versatile probability distribution of wind power. The problem is then analyzed and proved to be strictly convex. Although such convex optimization is tractable in many cases, it may take a long time to solve due to its large scale. This paper proposes a dual decomposition method to decompose the large problem. Then, two methods are employed to solve the decomposed problem, namely, the subgradient method and a faster method, limited-memory BFGS with box constraints (L-BFGS-B, a quasi-Newton method). Case studies were conducted to verify the efficiency of the dual decomposition and L-BFGS-B method for solving the stochastic ED problem.

Aiming at the wind power prediction problem, a wind power probability prediction method based on the quantile regression of a dilated causal convolutional neural network is proposed. With the developed model, the Adam stochastic gradient descent technique is utilized to solve the cavity parameters of the causal convolutional neural network under different quantile conditions and obtain the probability density distribution of wind power at various times within the following 200 hours. The presented method can obtain more useful information than conventional point and interval predictions. Moreover, a prediction of the future complete probability distribution of wind power can be realized. According to the actual data forecast of wind power in the PJM network in the United States, the proposed probability density prediction approach can not only obtain more accurate point prediction results, it also obtains the complete probability density curve prediction results for wind power. Compared with two other quantile regression methods, the developed technique can achieve a higher accuracy and smaller prediction interval range under the same confidence level.

Wind power forecasting methods generally provide estimates of future wind power as point forecasts, but most of the decision making processes in electrical power systems management require more information than a single value. For this purpose, additional methods - complex or based on strong assumptions - have been developed for estimating so-called interval forecasts associated to point forecasts. The method proposed by the authors is based on the use of discrete time Markov chain models of a proper order, developed starting from wind power time series analysis. It allows to directly obtain in an easy way an estimate of the wind power distributions on a very short-term horizon, without requiring restrictive assumptions on wind power probability distribution. With reference to an application, results obtained via a First and Second Order Markov Chain Model, respectively, are compared to those of Persistent Model evaluating the related prediction errors.

To settle the large-scale integration of wind power, it is necessary to enhance the wind power hosting capacity of distribution network. To this end, this paper proposes a stochastic mixed-integer linear programming (MILP) model to enhance the hosting capacity of distribution network with active network management (ANM) strategies. The considered ANM strategies include power factor control, reactive power compensation and network reconfiguration. The coordination of these strategies enables the distribution network to accommodate more wind power without violating its constraints. The proposed model relaxes the original nonlinear power flow equations into linear format, so that a near optimal solution can be obtained with moderate computational burden. The wind power and load demand are inherently variable due to the influence of various factors. Moreover, the output powers of multiple wind turbines (WTs) located at adjacent sites of a wind farm are spatially correlated. To represent the uncertain loads, the roulette wheel mechanism is used to generate the load samples. In addition, a combined method of inverse transformation and Nataf transformation is established to deal with the uncertain and correlated wind power. Then, scenario combination and reduction are conducted to generate representative scenarios. Finally, the feasibility and accuracy of the proposed method are verified via numerical tests on the modified IEEE 33-bus test system.

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

The wind model based on Markov process was proposed to determine the best installed capacity of wind farm. Firstly, piecewise probability distribution model based on Markov process was proposed to establish the scheduling multiple state probability distribution of wind power. Secondly, the probability production simulation method using the sequential convolution and non-sequential convolution was proposed to calculate the evaluation index which includes annual utilization hours of wind farm, wind curtailment, cost of traditional units and benefit of wind farm. Then the best installed capacity of wind farm could be calculated according to the above evaluation index with the aim to minimize the cost of the system. A case study was employed to illustrate the effectiveness of the proposed approach.

We consider distributionally robust two-stage stochastic linear optimization problems with higher-order (say \(p\ge 3\) and even possibly irrational) moment constraints in their ambiguity sets. We suggest to solve the dual form of the problem by a semi-infinite programming approach, which deals with a much simpler reformulation than the conic optimization approach. Some preliminary numerical results are reported.

Coordinated Voltage Control of Weak Sub-transmission Networks Considering Wind Power Variability

This paper presents an adjustable interval optimization model for the energy and reserve clearance while treating the wind power variation and uncertainty. Instead of conventional predicted intervals (PIs), adjustable intervals (AIs) as subsets of PIs are proposed as more judicious choices for economically covering wind uncertainties. Ramp-capability reserve is modeled to ensure the existence of sufficient ramp capability needed to follow up wind power variation within AIs. However, realization of wind power beyond AIs, which is no longer compensable by ramp-capability reserve, would necessarily lead to either load shedding or wind spillage. In order to optimally determine AIs, apart from the energy and reserve procurement costs, the costs associated with load shedding and wind spillage are incorporated in the objective function. The confidence level theory is employed as well for tuning the robustness and conservatism of the final solution. Performance of the proposed method is examined on several case studies on an 8-bus and modified IEEE 118-bus test systems. Sensitivity analyses are conducted to specify the impacts of important parameters on the obtained solution. The results confirm the applicability and effectiveness of the proposed methodology.

Renewable energy sources (RES) and low-carbon technology users play a vital role in modern power systems. However, RES generation is easily affected by the environment. Meanwhile, the load, such as electric vehicles (EVs) and prosumers, accounts for most low-carbon technology users. Their power is usually superimposed on peak loads without dispatching, which also exacerbates the instability of the power system. Current optimal dispatching mechanisms mainly rely on centralized organizations, while their dispatching process is not open and transparent. In this article, we propose a blockchain-based trustworthy dispatching approach for the distribution network in high renewable energy penetrated power systems. We first develop an optimal dispatching model considering EVs’ charging behavior and the prosumers’ economic benefits. With the model, prosumers can be dispatched to balance power and consume renewable energy, reducing the impact of disorderly charging on the grid and the abandonment of RES generation. An orderly charging iteration optimization (OCIO) algorithm is proposed to implement orderly EV charging while considering the charging cost and the period. We also propose a modified particle swarm optimization (mPSO) algorithm to publish dispatching tasks based on real-time power balance. Furthermore, blockchain is applied as an open and transparent ledger to record each entity’s power generation and consumption information, ensuring that the dispatching process is trustworthy. Finally, the effectiveness of the dispatching approach is verified in the modified IEEE 33-bus test system and Ethereum-based smart contracts.

Modern power systems which integrate renewable energy sources (RESs), such as wind or solar energy sources and plug-in electric vehicles (PEVs), need to carry out the uncertainty due to the access of demanded or injected power. Therefore, it is necessary to consider uncertainty costs in optimal power flow problems. To develop the uncertainty cost functions, log-normal probability distribution function (PDF) for solar irradiance, normal PDF for unloading and loading behavior of PEVs, and Rayleigh PDF for wind speed are considered. In this work, a dynamic optimal power flow (DOPF) model is formulated by considering the uncertainties of RESs and PEVs. A newly hybrid metaheuristic algorithm named cross entropy (CE) covariance matrix adaption evolutionary strategy (CMAES) is proposed to evaluate the DOPF problem in a modified IEEE 118-bus system. The application of adaptive step length in the CMAES method improves the local exploitation capability, while the CE method is used in the initial stage for global exploration due to its fast convergence. Simulation results indicate that the proposed technique can solve the DOPF problems with RESs and PEVs effectively and can provide high-quality solutions compared to different techniques. The conventional statistical method called ANOVA test is performed for comparative analysis of different techniques. The results of this test show the validation of CE-CMAES compared to different optimization techniques. Therefore, the proposed method could be easily and efficiently applied by power system operators in real-time optimal power flow operations.

This study describes a computationally efficient model for the optimal sizing and siting of Electrical Energy Storage Devices (EESDs) in Smart Grids (SG), accounting for the presence of time-varying electricity tariffs due to Demand Response Program (DRP) participation. The joint planning and operation problem for optimal siting and sizing of the EESD is proposed in a two-stage optimization problem. In this regard, the long-term decision variables deal were the size and location of the EESDs and have been considered at the master level while the operating point of the generation units and EESDs is determined by the slave stage of the model utilizing a standard mixed-integer linear programming model. To examine the effectiveness of the model in the slave sub-problem, the operation model is solved for different working days of different seasons. Binary Particle Swarm Optimization (BPSO) and Binary Genetic Algorithm (BGA) have been used at the master level to propose different scenarios for investment in the planning stage. The slave problem optimizes the model in terms of the short-term horizon (day-ahead). Additionally, the slave problem determines the optimal schedule for an SG considering the presence of EESD (with sizes and locations provided by the upper level). The electricity price fluctuates throughout the day, according to a Time-of-Use (ToU) DRP pricing scheme. Moreover, the impacts of DRPs have been addressed in the slave stage. The proposed model is examined on a modified IEEE 24-Bus test system.

When conducting the wind power (WP) planning, it is very important for electric power companies to evaluate the penetration limit of the grid-accommodable WP. This paper proposes a probabilistic method for determining grid-accommodable WP capacity based on the multiscenario analysis. Typical power system operation scenarios are generated from the combination of different WP scenarios and demand scenarios. A power system operation simulation model is proposed and implemented to the generated scenarios. The operation results are further used as the basis of the proposed probabilistic method. The validity and effectiveness of the new method are demonstrated in two cases, i.e., the IEEE 39-bus test system and a real large power system in China, respectively.

This paper formalizes the two-stage recourse model and focuses on a stochastic generalization of the ordinary set cover problem to illustrate our technique for solving two-stage stochastic linear programs. The integrality constraints can be relaxed to yield a linear program. We give a compact formulation where only have the stage-1 variables and explain the basic idea by considering the two-stage stochastic set cover problem (SSC) and its formulation given by (SSC-P2). Furthermore, theorem forms for tacking various two-stage stochastic optimization problem be presented