Optimal stochastic bilevel scheduling of pumped hydro storage systems in a pay-as-bid energy market environment

Abstract This paper proposes a stochastic bilevel optimization approach to owners of pumped hydro storage systems (PHSSs) to participate in pay-as-bid power market and provide optimal bids and offers. The price offering of other generation units is modeled by stochastic programming. The upper-level of the proposed bilevel programming seeks maximization of the profit of the PHSS arbitrage, where the lower-level assures the optimal system dispatching (and market-clearing), and keeps the network security. The bilevel optimization is then transferred into a single-level equivalent via Karush-Kuhn-Tucker (KKT) complementarity conditions. The equations of the KKT conditions are linearly modeled using special ordered sets of type 1 (SOS1) variables. Furthermore, the bilinear objective function of the upper-level is approximated using McCormick envelopes relaxation method, in order to obtain the solutions as fast as possible and making the problem as mixed-integer linear programming. The proposed method is verified on the IEEE 24-bus reliability test system (RTS) considering different cases. The operation of a single PHSS is assessed in the network's normal and congested conditions. The results show that the PHSS achieve more revenue in a limited network as the offered prices go up to price cap in some periods. In the studied cases, the profit of energy arbitrage by the PHSS increases from $ 3302 in a normal network, to $ 8170 in a limited network. Moreover, the effect of wind generation uncertainty on the arbitrage problem is investigated using five sub-scenarios dedicated to wind generation. It is shown that the arbitrage profit is more sensitive on generation cost uncertainty rather than wind generation uncertainty. Furthermore, it is revealed that the expected profit in the presence of wind turbines is slightly lower than that without wind turbines. This is due to the fact that the wind generation power is always accepted and dispatched in the market and lowers the load demand and deteriorates the arbitrage opportunity. In this case the expected profit of the PHSS is decreased from $ 3302 to $ 3288. Furthermore, the effects of three PHSS units in the mentioned network is investigated. For a particular PHSS in the studied system, it is found that the increase of PHSS units in the network decreased the expected profit from $ 3302 to about $ 3176. Also, the location of PHSS units is deduced as an influential factor on profitability of merchant storage facilities.