Stochastic hydro-thermal unit commitment via multi-level scenario trees and bundle regularization

Abstract

For an electric power mix subject to uncertainty, the stochastic unit-commitment problem finds short-term optimal generation schedules that satisfy several system-wide constraints. In regulated electricity markets, this very practical and important problem is used by the system operator to decide when each unit is to be started or stopped, and to define how to generate enough energy to meet the load. For hydro-dominated systems, an accurate description of the hydro-production function involves non-convex relations. This feature, combined with the fine time discretization needed to represent uncertainty of renewable generation, yields a large-scale mathematical optimization model that is nonlinear and has mixed-integer variables. To make the problem tractable, a novel solution strategy, based on multi-horizon scenario trees, is proposed. The approach deals in a first level with the integer decision variables representing whether units are on or off. Once units are committed, the expected operational cost is minimized by solving a continuous second-level problem, which is separable by scenarios. The coordination between the two decision levels is done by means of a bundle-like variant of Benders decomposition that proves very efficient for the considered setting. To assess the quality of the optimal commitment on out-of-sample scenarios, a new simulation technique, based on certain sustainable pseudo-distance is proposed. For the numerical experiments, a mix of hydro, thermal, and wind power plants extracted from the Brazilian power system is considered. The results confirm the interest of the approach, particularly regarding a more efficient management of hydro-plants, because non-convex operational regions are considered by the model.

Appendices

Appendices

Details for the stochastic unit commitment solver and the test system are given below, respectively in Appendices 1 and 2.

Appendix 1: Algorithm with the full SUC solver

In Algorithm 4 below, the probability of operational scenario j, conditioned to the strategic scenario i is denoted by ${\mathfrak {p}}^{ij}.$

Appendix 2: System details

Below we provide the details about the system considered in the experiments presented in Sect. 5. As shown in Fig. 4, the system has three hydro-plants, each of which having a water reservoir. Table 5 presents the initial, minimum, and maximum volume of the reservoirs of the hydro-plants, and the maximum volume of water spilled per hour. All volumes are in $\hbox {hm}^3$. The hydro-plants numbered 1, 2, and 3 have 3, 5, and 4 units, respectively. Units belonging to the same hydro-plant are identical. Table 6 shows the characteristics of the units of each hydro-plant. The second column shows the maximum number of times each unit can be switched on within the time horizon; the third and fourth columns present the minimum and maximum generation of each unit (in MW); and the fifth and sixth columns show the minimum and maximum outflow of each unit (in $\hbox {hm}^3$). Hydro-plant number 1 is upstream to hydro-plant number number 2 and has 1-h water travel time.

Table 5 Characteristics of each hydro-plant

Table 6 Characteristics of each individual unit of each hydro-plant

The system has seven thermal units. The details about each thermal unit are displayed in Tables 7 and 8. The first column of each table shows the unit numbers. In Table 7, the second column shows the buses in which the units are located (see Fig. 4); the third column presents the start-up costs; the fourth and fifth columns show the minimum and maximum power generation (in MW) of each unit when they are on; the sixth and seventh columns exhibit the ramp-up and -down rates (in MW). In Table 8, the second and third columns show the minimum time (in hours) the units must be on (after they are switched on) and off (after they are switched off), respectively; the fourth column indicates whether the units are on or off at the beginning; while the fifth column informs how long (in hours) they are in that state; and, finally, the last column shows the generation (in MW) of the units at the beginning.

Table 7 Characteristics of each thermal unit

Table 8 Characteristics of each thermal unit (continuation)

As shown in Fig. 4, there are four load demands, located at buses 1, 2, 3, and 4. We have considered the same demand for each scenario. Table 9 presents the demand for each time step at each bus.

Table 9 Demand (MW) at each bus

The water inflow at time t in plant h is given by

$$\begin{aligned} \xi ^{\text {inflow}}_{th} = \xi ^{\text {inflow}}_{t-1,h} \max (0, \zeta + \phi ) \end{aligned}$$

where the random variable $\zeta$ has a normal distribution with mean 1 and standard deviation 0.025, $\phi$ has a uniform distribution on the interval $[-0.1,0.1]$, and $\xi ^{\text {inflow}}_{0,1} = 151$, $\xi ^{\text {inflow}}_{0,2} = 229$, and $\xi ^{\text {inflow}}_{0,3} = 260$. The wind power produced at time t in bus b is given by

$$\begin{aligned} \xi ^{\text {wind}}_{tb} = \left\{ \begin{array}{ll} \min (0.1 \vartheta , \xi ^{\text {wind}}_{t-1,b} + 0.05 \vartheta \beta ), &{} \text {if this quantity is non-negative},\\ \xi ^{\text {wind}}_{t-1,b} (1 + \varrho ), &{} \text {otherwise}, \end{array} \right. \end{aligned}$$

where $\vartheta$ is the average of the total demand, $\xi ^{\text {wind}}_{0,b} = 0.05 \vartheta$, $\beta$ has a uniform distribution on the interval $[-1,1]$, and $\varrho$ has a uniform distribution on the interval [1, 2].