Uses modified Hotelling model Must regulate: 1. Supply of energy over yearly cycle - low demand in summer - high demand in winter 2. Current supply - most flexible balancing power Assume we have a reservoir: - Want to find the time pattern of water use in the reservoir - Water used for power today could be used tomorrow - Aim to maximize social surplus - Assumed that hydro generation has zero marginal cost ### Cost Structure - Not considering new investments, want to find optimal management for existing capacities ### Variables - $R_t\quad$ amount in reservoir - $w_t\quad$ inflow of water - $r_t\quad$ outflow of water Use a transformation coefficient to translate water flow to electricity - $e_t^H \leq \frac{1}{a}r_t$ Water management based on filling and emptying reservoir: $R_t \leq R_{t-1} + w_t -e^H_t$ - conservation of water in reservoir ### Basic Hydro Model Assumptions: - unlimited transferability b/w time periods - infinite reservoir - no emptying of reservoir until last period - start with water "endowment" - amount of water to use over time horizon *T* - no discounting (short period) - Electricity production during each time period is equal to the endowment $\underbrace{\sum_{t=1}^T e_t^H}_{\substack{\text{Total} \\ \text{production}}}=\underbrace{W}_{\substack{\text{Water} \\ \text{enodowment}}}$ - demand function is marginal utility - marginal willingness to pay --> demand Optimization problem: - Maximize sum of utilities from electricity consumption for the given time period, without using more electricity than the water endowment provides $\text{max}\quad\sum_{t=1}^TU_t(e_t^H)$ $\text{subject to}\quad\sum_{t=1}^Te_t^H\leq W, e_T^H \geq 0, t= 1,\dots,T$ #### Results - Solution: how to allocate water over time to maximize surplus - Shadow price - Change in sum of utilities due to a change in water endowment - 0 if the reservoir is not emptied - marginal utility is constant, equal to shadow price #### Bathtub Diagram Looks at two periods - "x" axis is total energy for both periods - assume all water is used - left y-axis price for period 1 - right y-axis price for period 2 - demand curves - left to right for period 1 - right to left for period 2