Decomposition algorithm for large-scale two-stage unit-commitment

Abstract

Everyday, electricity generation companies submit a generation schedule to the grid operator for the coming day; computing an optimal schedule is called the unit-commitment problem. Generation companies can also occasionally submit changes to the schedule, that can be seen as intra-daily incomplete recourse actions. In this paper, we propose a two-stage formulation of unit-commitment, wherein both the first and second stage problems are full unit-commitment problems. We present a primal-dual decomposition approach to tackle large-scale instances of these two-stage problems. The algorithm makes extensive use of warm-started bundle algorithms, and requires no specific knowledge of the underlying technical constraints. We provide an analysis of the theoretical properties of the algorithm, as well as computational experiments showing the interest of the approach for real-life large-scale unit-commitment instances.

Appendices

Appendix 1: Description of the unit-commitment model

This appendix provides more information about the models of subproblems used in the numerical experiments of Sect. 5.

1.1 Hydro valley subproblems

The hydro valley subproblems deal with optimizing the power production of turbines and pumps for a given price signal. The turbines and pumps connect various reservoirs together. For a given topology, one readily establishes the flow equations that deal with updating the reservoirs levels through time. These reservoir levels have to remain between a lower and upper bound for all T time steps. Turbines and pumps are moreover subject to natural bounds on production levels. The most challenging feature to take into account is the turbining efficiency function that associates with each turbined quantity $(\mathrm{m}^3 / \mathrm{h})$ and water head (reservoir level in uphill reservoir, in $\mathrm{m}^3$) the amount of produced power (MW). This function can be highly non-linear, non-concave and may even contain forbidden zones of production, see e.g., Finardi and Silva (2006), Borghetti et al. (2013).

A common assumption in the French system (see, e.g., Merlin et al. 1981) is that the water-head effect for large reservoirs can be neglected as the volumetric difference that can be achieved during the T time steps is quite small. For smaller reservoirs the effect caused on the amount of produced power would be quite small. Moreover following the set of assumptions made in Merlin et al. (1981), the power efficiency function becomes concave and is approximated with an a priori piecewise linearization. This makes the hydro valley subproblem a linear program. More details can be found in van Ackooij et al. (2014).

1.2 Thermal subproblems

As the thermal subproblems are concerned, we set up a usual model, similar to the one of Frangioni et al. (2011), that incorporates, minimum generation levels, minimal up and down times, start up costs, fixed generation costs and ramping rates. We provide here a short description for convenience. To simplify notation, we do not include a reference to the specific unit the problem belongs to,

The decision variables are $p \in \mathbb {R}_+^T$ providing the amount of generated power in MW, $u \in \left\{ 0,1\right\} ^T$ the on/off status of the unit for each time step and $z \in \left\{ 0,1\right\} ^T$ an auxiliary variable indicating an effective start of the unit. Problem data describing cost are $c \in \mathbb {R}_+^T$ in €/MWh, a proportional cost of production, $c_f \in \mathbb {R}_+^T$ in € / h, a fixed production cost and $c_s \in \mathbb {R}_+^T$ in €, a start up cost. Bounds on production levels expressed in MW, when producing are given by $p_{\min } \in \mathbb {R}^T_+$ and $p_{\max } \in \mathbb {R}^T_+$. Ramping rate related data is $g_+, g_- > 0$ expressed in MW / h and correspond to the ramping up gradient and ramping down gradient respectively. The numbers $s_+, s_- > 0$ express similar quantities but for starting and stopping ramping rates. Finally $\tau _+, \tau _-$ expressed in a number of time steps correspond to the minimum up and down times respectively. We make the assumption that when a unit is online for exactly $\tau _+$ time steps the minimum up constraint is satisfied [while Frangioni et al. (2011) assume this for $\tau _+ + 1$ time steps]. The optimization problem can then be stated as follows, where $\lambda \in \mathbb {R}^T$ is a Lagrangian price multiplier (€/MWh):

$$\begin{aligned} \min \nolimits _{p,u,z \in \mathbb {R}^T_+ \times \left\{ 0,1\right\} ^{2T}}&(c - \lambda )^{\mathsf {T}}p \Delta t + c_f^{\mathsf {T}}u \Delta t + c_s^{\mathsf {T}}z \\ \hbox {s.t.} \qquad&p_{\min }(t)u(t) \le p(t) \le p_{\max }(t)u(t), \;\hbox {for}\; t=1,\ldots ,T \\&p(t) \le p(t-1) + u(t-1)g_+ \Delta t + (1 - u(t-1))s_+ \Delta t, \;\hbox {for}\; t=1,\ldots ,T \\&p(t-1) \le p(t) + u(t)g_- \Delta t + (1 - u(t))s_- \Delta t, \;\hbox {for}\; t=1,\ldots ,T \\&u(t) \ge u(r) - u(r-1), \;\hbox {for}\; t = t_0 + 1, \ldots , T, r = t -\tau _+ + 1, \ldots , t-1 \\&u(t) \le 1 - u(r-1) + u(r), \;\hbox {for}\; t = t_0 + 1, \ldots , T, r = t - \tau _- + 1, \ldots , t-1 \\&u(t) - u(t-1) \le z(t), \;\hbox {for}\; t=1,\ldots ,T. \end{aligned}$$

Here $\Delta t$ corresponds to the size of each time step expressed in hours, p(0) to the initial power output and $t_0$ is defined according the amount of time $\tau _0$ (in time steps) the unit has spend producing or is offline. More specifically,

$$\begin{aligned} t_0 = \left\{ \begin{array}{cc} \max \left\{ 0, \tau _+ - \tau _0\right\} &{}\qquad \hbox {if } p(0) > 0 \\ \max \left\{ 0, \tau _- - \tau _0\right\} &{}\quad \hbox {otherwise} \\ \end{array} \right. \end{aligned}$$

Obviously $u(0) = 1$ in the first case and $u(0) = 0$ in the second.

Appendix 2: Lagrangian heuristics

As in decomposition approaches for deterministic unit-commitment, heuristics play an important role in our algorithm—more precisely in Step 2. In our numerical experiments, we use three heuristics inspired from Borghetti et al. (2003) and Takriti and Birge (2000). This section describes them briefly.

1.1 Three Lagrangian heuristics

The three heuristic use information returned by the bundle algorithm maximizing (11) used in Step 1. More precisely, denoting by p is the number of iterations of this algorithm and $x^j$ a primal iterate obtained at iteration $j\in \{1,\ldots ,p\}$, the heuristics use the following quantities:

• 1. the dual simplicial multipliers $\alpha$ of the quadratic program solved at the last iteration of the bundle method;

• 2. the so-called pseudo schedule $\hat{x}$ is defined as $\sum _{j=1}^p \alpha _j x^j$, see Daniildis and Lemaréchal (2005), Dubost et al. (2005);

• 3. the pseudo costs $(\hat{c}_1,\ldots ,\hat{c}_m)$ defined as $\hat{c}_i = \sum _{j=1}^p \alpha _j c^j_i$ where $c^j_i$ the pure production cost of subproblem i at iteration j;

• 4. the pseudo commitment decisions defined as $\hat{u}^j_i = \sum _{j=1}^p \alpha _j u^j_i$, where $u^j_i \in \left\{ 0,1\right\} ^T$ are the commitment decisions of each thermal plant for each iteration $j=1,\ldots ,p$.

Another common ingredient is the resolution of an economic dispatch problem: for a fixed set of commitment decisions, we let a continuous optimization problem adjust production levels in order to generate a solution in $X^2$.

We begin by remarking that the pseudo-schedule is a technically feasible solution as hydro valleys are concerned (since these sub-problems have convex feasible sets, see the previous section). Also the pseudo-schedule is directly related to offer-demand equilibrium constraints through bundle stopping criteria. We therefore keep the pseudo-schedule as hydro-valleys are concerned and remove their generation from the load D in order to form $\tilde{D}$. The production parc is such that the obtained net load is always strictly positive. The heuristics are therefore mostly concerned with thermal plants. For convenience of notation we will still use m to index the number of thermal plants.

1.2 Commitment based heuristic

This heuristic is inspired from Borghetti et al. (2003). We begin with an initial guess for the commitment decisions called $\tilde{u}$, for instance one of the commitment decisions encountered during the optimization of problem (11). We now build a priority list in two different ways. The first is time-independent and related to sorting the pseudo costs divided by total generated pseudo power in increasing order. A lower value indicates a unit with higher priority (best cost to power ratio). The second is a time-dependent priority list in which we divide the pseudo-commitment decisions by the above pseudo cost over pseudo power ratio. A higher value indicates a unit more likely to be started.

Starting from our initial commitment guess $\tilde{u}$ we first begin by computing the generation envelope, i.e., the minimum and maximum power the plants can generate over the whole time horizon at these commitment decisions. We now move from the first time step to the last one, if $\tilde{D}$ is in the generation envelope, nothing more needs to be done. If generation is insufficient, we check if we can start the highest priority unit (if not done so already), we continue in this manner until generation covers load. If generation is in excess, we try to decommit the lowest priority unit (if not already off) and continue in this manner until the minimum generation is below load. The hence generated commitment decision is post-processed with an economic dispatch in order to finely adjust generation to actual load. We also post-process any of the generated commitment decisions $u^j$, $j=1,\ldots ,p$ in order to retain the best one.

1.3 Recombining heuristic

This method inspired from Takriti and Birge (2000) recombines the earlier obtained primal iterates in order to find a feasible solution. For additional flexibility we add a slack variable, and therefore solve the following mixed-integer problem:

$$\begin{aligned} \min \nolimits _{z \in \left\{ 0,1\right\} ^{mp},s \in \mathbb {R}^T_+}&\sum _{j=1}^p \sum _{i=1}^m c^j_i z^j_i + \sum _{t=1}^T c^{\mathtt{imb}}_t s_t \\ \hbox {s.t.} \qquad&\sum _{j=1}^p z^j_i = 1, i=1,\ldots ,m \quad \text {and}\quad \tilde{D} - \sum _{i=1}^m A^i \sum _{j=1}^p z^j_i x^j_i - s \le s^u, \end{aligned}$$

where $c^{\mathtt{imb}}$ is a large imbalance related cost. The resulting optimal solution builds an initial commitment decision $\tilde{u}$, which is post-processed as explained previously. In order to keep the mixed-integer program small, we use the dual variables $\alpha$ and insert only elements of iteration j if $\alpha _j$ is sufficiently large, (e.g., $\alpha _j > 10^{-3}$).