Applications of distributionally robust optimization in operation of power systems

Abstract

As power systems evolve, their components with uncertainties keep expanding. For example, renewable energy sources (RESs) including wind turbines and solar panels are growing to phase fossil fuels out. RESs are usually influenced by weather and therefore have unpredictable outputs. Besides, increasing penetration of electric vehicles (EVs) bring highly uncertain loads because they are influenced by human behaviours and the transportation system. With greater uncertainties, the operation of power systems becomes more challenging. As many operation tasks of power systems highly depend on optimization, it is crucial to have efficient approaches to solve optimization problems involving uncertainties. Stochastic optimization (SO) and robust optimization (RO) have been often used to solve optimization problems with uncertainties. SO assumes that the uncertainty follows a certain probabilistic distribution or uses discrete distributions based on selected scenarios to approximate the uncertainty distribution. As information about uncertainties is limited, there is ambiguity in the uncertainty distribution. SO ignores such ambiguity and thus often has inferior performance. Different from SO, RO constrains uncertainties within certain sets and focuses on the worst case in the considered sets. As the worst case rarely happens, over-conservative results are often obtained when RO is used to evaluate the economic efficiency of system operation. Without the drawbacks of SO and RO, a more recently developed approach named distributionally robust optimization (DRO) is becoming popular. In this thesis, potentials of DRO in economic dispatch are further exploited beyond the state-of-art literature and DRO is applied to facilitate balance responsible distribution systems (BRDSs) and balance responsible distribution companies (BR-DISCOs) to utilize the flexibility of electric vehicle aggregators (EVAs) for the first time. In works on economic dispatch using DRO, statistical moments are often adopted by DRO to depict the uncertainty distribution. As statistical moments are derived from limited samples of uncertainties, they may deviate from the actual moments. Instead of being ignored as in previous works, such deviations are considered by the DRO technique adopted here. Besides, because the output of RESs cannot be negative nor exceed the installed capacity, the uncertainty in RES outputs is always bounded. In this regard, ellipsoidal support sets are used to limit the range of possible uncertainty realizations under DRO. Moreover, as the proposed multi-period model is carried out as a rolling-plan, modelling of the first period is more important than that of the other periods. Therefore, a two-stage framework is employed here to model the first period without approximation. While for the other periods, segregated linear decision rule approximation is applied. With such structure, a proper trade-off between modelling accuracy and computational tractability is achieved by the proposed model. Furthermore, within the framework of DRO, RO is integrated to enhance the system security. A Constraint Generation algorithm is proposed to solve the proposed model. Through case studies, it is shown that the proposed model avoids over-conservative solutions and prevents inferior performance under limited uncertainty information through adopting more realistic DRO techniques. Also, the proposed model guarantees economical and secure long-term operation without causing excessive computational burden. Nowadays, many components of power systems are required to act as balance responsible parties (BRPs). BRPs should contribute to the energy balance of the entire system by maintaining their planned energy portfolio and will be penalized if they fail to do so, which means that BRPs should mitigate their forecast uncertainties. As BRPs, BRDSs with EVAs can utilize the flexibility of EVAs at the expense of disturbing their charging. Without influencing driving activities of EVs in the next day, a model is established here for BRDSs to delay uncertainties through the flexibility of EVAs and thus create opportunities for uncertainties from different times to offset each other. In the established model, linear decision rules approximation is used to reduce the computational complexity, based on which a scheme of uncertainty transferring is proposed to relieve disturbance to EVAs. DRO is used to evaluate the average performance of the operation plans. As the possibility that uncertainties offset each other depends on uncertainty correlations, temporal and spatial covariances of uncertainties are considered by DRO. Comprehensive case studies are carried out based on charging demands of EVAs simulated from real traffic data. The results show that the adopted DRO technique effectively avoids unnecessary costs. Also, the established model achieves the trade-off between cost savings brought by the flexibility of EVAs and the corresponding payments to EVAs. Overall, the operation costs of BRDSs are reduced with the established model. The model proposed for BRDSs is extended to be applicable for distribution companies serving as BRPs. Such distribution companies are here referred as BR-DISCOs. Different from BRDSs, BR-DISCOs would purchase energy to deliver to their customers and hence need to consider energy costs as well. Therefore, apart from mitigating uncertainties, the flexibility of EVAs can also be used by BR-DISCOs to shift EVA charging demands to hours with lower energy prices. Because using EVAs to mitigate uncertainties and shifting EVA charging demands would both disturb the charging of EVAs, their interactions need to be properly considered in the extended DRO model. Besides, it is assumed here that the disturbance to EVAs incurred from uncertainty mitigation needs not be fully recovered at the end of the day as long as the capability of EVAs in accepting disturbance is respected. As a result, the involved uncertainties appear to be eliminated as they will become deterministic information in the next day. Unrecovered disturbance to EVAs would influence their charging demands in the next day, but change on the average operation costs in the next day is little as the expectation of considered uncertainties is close to zero. Although recovering the disturbance to EVAs will cause BR-DISCO to deviate from its decided energy portfolio, it may be preferable in the earlier part of the day because otherwise payments to EVAs will keep increasing as time goes. Then, as in the previous model proposed for BRDSs, involved uncertainties can be regarded as being delayed through EVAs. Meanwhile, power losses in the charging and discharging of EVAs are used to reduce the scale of uncertainties and thus reduce penalties for energy deviations of BR-DISCO. Through case studies, the extended model is verified to be capable to coordinate the uses of EVAs in mitigating uncertainties and shifting their charging demands. It is found that the two forms of uncertainty mitigation, i.e., eliminating uncertainties and delaying uncertainties, could both reduce the operation costs of BR-DISCO and cooperatively achieve the minimum cost under the proposed model.