Solution sensitivity-based scenario reduction for stochastic unit commitment

Abstract

A two-stage stochastic program is formulated for day-ahead commitment of thermal generating units to minimize total expected cost considering uncertainties in the day-ahead load and the availability of variable generation resources. Commitments of thermal units in the stochastic reliability unit commitment are viewed as first-stage decisions, and dispatch is relegated to the second stage. It is challenging to solve such a stochastic program if many scenarios are incorporated. A heuristic scenario reduction method termed forward selection in recourse clusters (FSRC), which selects scenarios based on their cost and reliability impacts, is presented to alleviate the computational burden. In instances down-sampled from data for an Independent System Operator in the US, FSRC results in more reliable commitment schedules having similar costs, compared to those from a scenario reduction method based on probability metrics. Moreover, in a rolling horizon study, FSRC preserves solution quality even if the reduction is substantial.

Appendix A: Concrete stochastic reliability unit commitment model

1.1 Appendix A.1: Notation

Sets and indices

$\mathcal {B}$ :

Set of buses

$\mathcal {G}$ :

Set of thermal units

$\mathcal {L}$ :

Set of transmission lines, modeled as directed arcs

$\mathcal {L}_{I}(b)$ :

Set of transmission lines to bus $b$

$\mathcal {L}_{O}(b)$ :

Set of transmission lines from bus $b$

$\mathcal {K}_g$ :

Set of time intervals of stairwise start-up cost function of thermal unit $g$

$\mathcal {S}$ :

Set of scenarios

$\mathcal {T}$ :

Set of time periods

${BF}_{\ell }, {BT}_{\ell }$ :

Buses located at the two ends of transmission line $\ell$, representing the bus injecting power to and absorbing power from line $\ell$, respectively

Parameters

$\xi _s$ :

Probability of scenario $s$

$\varDelta _{g}^{ru}$ :

Ramping-up limit of unit $g$ (MW/h)

$\varDelta _{g}^{rd}$ :

Ramping-down limit of unit $g$ (MW/h)

$\varDelta _{g}^{su}$ :

Start-up ramping limit of unit $g$ (MW/h)

$\varDelta _{g}^{sd}$ :

Shut-down ramping limit of unit $g$ (MW/h)

$\bar{P}_g$ :

Capacity of unit $g$ (MW)

$\underline{P}_g$ :

Minimum power output of unit $g$ (MW)

$B_{\ell }$ :

Negative susceptance of line $\ell \in \mathcal {L}$

$d_{bts}$ :

Net load at bus $b$ in period $t$ in scenario $s$ (MWh)

$F_{\ell }$ :

Maximum capacity of transmission line $\ell$ (MW)

$H_g^+, H_g^-$ :

Time of unit $g$ has been on, or off at the beginning of scheduling

$\varrho _{gk}$ :

Startup cost of thermal unit $g$ for time intervals $q \in \mathcal {K}_g$ ($)

$R_{t}$ :

Reserve requirement in period $t$ (MWh)

$T_g^U, T_g^D$ :

Minimum up and down times of unit $g$

$\varGamma _{\alpha }^{+},\varGamma _{\alpha }^{-}$ :

Penalties on load imbalance ($/MWh)

$\varGamma _{\beta }^{+},\varGamma _{\beta }^{-}$ :

Penalties on reserve requirement imbalance ($/MWh)

$J$ :

Number of blocks of the piecewise linear generation cost function of unit $g$

$\delta _{jgts}$ :

Energy generated in block $j$ of the piecewise linear generation cost function of unit $g$ in period $t$ in scenario $s$ (MWh)

$\lambda _{jg}$ :

Slope of block $j$ of the piecewise linear generation cost function of unit $g$ ($/MWh)

$\gamma _{jg}$ :

Upper limit of block $j$ of the piecewise linear generation cost function (MWh)

$a_g$ :

No-load cost of unit $g$ ($)

$c_{gt}^{u}(\cdot )$ :

Commitment cost function of unit $g$ in period $t$ ($)

$c_{gts}^{p}(\cdot )$ :

Generation cost function of unit $g$ in period $t$ in scenario $s$ ($)

Decision variables

$v_{gt} \in \{0,1\}$ :

First-stage decision, binary variable, equal to $1$ if unit $g$ is on in period $t$, and 0 otherwise

$p_{gts} \ge 0$ :

Generation level of unit $g$ in period $t$ in scenario $s$ for $g \in \mathcal {G}$ (MW)

$\bar{p}_{gts} \ge 0$ :

Maximum available power generation for unit $g$ in period $t$ in scenario $s$ for $g \in \mathcal {G}$ (MW)

$\alpha _{bts}^{+},\,\alpha _{bts}^{-} \ge 0$ :

Auxiliary variables, representing shortage and excess in load supply at bus $b$ in period $t$ in scenario $s$, respectively (MWh)

$\beta _{ts}^{+}$, $\beta _{ts}^{-} \ge 0$ :

Ancillary variables, representing shortage and excess in reserve requirement in period $t$ in scenario $s$ (MWh)

$\theta _{bts}$ :

Phase angle at bus $b$ in period $t$ in scenario $s$ (radians)

$\omega _{\ell ts}$ :

Line power of transmission line $\ell$ in period $t$ in scenarios $s$, unrestricted in sign because power can flow in either directions on a line

1.2 Appendix A.2: Mathematical model

The full concrete formulation of SRUC extends the deterministic UC model in Carrión and Arroyo (2006) to a two-stage stochastic program. The commitments of thermal units are considered as first-stage decisions. Second-stage decision variables include generation level of each unit, and corresponding maximum available generation level. The following presents a two-stage stochastic program by viewing each hour as a period.

1.2.1 Appendix A.2.1: Objective function

$$\begin{aligned} \min \sum _{t\in \mathcal {T}}\sum _{g \in \mathcal {G}}c_{gt}^u(v_{gt}) + \sum _{s\in \mathcal {S}}\xi _s\zeta _s \end{aligned}$$

(31)

The objective function (31) consists of two parts: the cost related to commitments of units, like startup, shutdown and no-load costs; and the cost related to generation and penalties on load and reserve requirement imbalances, upon realization of a scenario in the second stage, as shown in (32).

$$\begin{aligned} \zeta _s = \sum _{t \in \mathcal {T}}\sum _{g \in \mathcal {G}}c_{gts}^p(p_{gts}) + \sum _{t \in \mathcal {T}}\sum _{b \in \mathcal {B}}\left( \varGamma _{\alpha }^{+}\alpha _{bts}^{+} + \varGamma _{\alpha }^{-}\alpha _{bts}^{-}\right) + \sum _{t \in \mathcal {T}} \left( \varGamma _{\beta }^{+}\beta _{ts}^{+} + \varGamma _{\beta }^{-}\beta _{ts}^{-}\right) \end{aligned}$$

(32)

The goal of SRUC is to minimize total commitment cost, expected generation cost and expected penalties on imbalances in generation and reserve. The following section presents operational constraints.

1.2.2 Appendix A.2.2: Constraints

Energy balance at each bus:

$$\begin{aligned}&\sum _{g \in \mathcal {G}(b)}p_{gts} + \sum _{\ell \in \mathcal {L}_I(b)}\omega _{\ell ts} - \sum _{\ell \in \mathcal {L}_O(b)}\omega _{\ell ts} \nonumber \\&\qquad + \alpha _{bts}^+ - \alpha _{bts}^- = d_{bts}, \forall b \in \mathcal {B}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S} \end{aligned}$$

(33)

Formula (33) describes the relationship between load demand and generated and net transmitted energy for each bus.

Reserve requirements:

$$\begin{aligned} \sum _{g \in \mathcal {G}}\bar{p}_{gts} + \beta _{ts}^+ - \beta _{ts}^- = \sum _{b\in \mathcal {B}}d_{bts} + R_{t} , \forall t \in \mathcal {T}, \forall s \in \mathcal {S} \end{aligned}$$

(34)

Reserve requirements (34) maintain reliability if contingencies that are not modeled in scenarios occur; e.g., outages of generators or transmission lines.

Line Power:

$$\begin{aligned} \omega _{\ell ts}&= B_{\ell }(\theta _{BF_{\ell }ts} - \theta _{BT_{\ell }ts}), \forall t \in \mathcal {T}, \forall \ell \in \mathcal {L}, \forall s \in \mathcal {S}\end{aligned}$$

(35)

$$\begin{aligned} - F_{\ell }&\le \omega _{\ell ts} \le F_{\ell },\forall t \in \mathcal {T}, \forall \ell \in \mathcal {L}, \forall s \in \mathcal {S} \end{aligned}$$

(36)

Power flow in each transmission line is formulated and restricted in (35) and (36). Formulation (35) is a linear, lossless DC approximation of the relationship between phase angles and power flow on a transmission line.

Generation limits:

$$\begin{aligned} \underline{P}_{g}v_{gt} \le p_{gts} \le \bar{p}_{gts} \le \bar{P}_g v_{gt}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S} \end{aligned}$$

(37)

Available power generation level of a thermal unit depends on its operational status, as in (37). The difference between maximum available generation level $\bar{p}_{gts}$ and actual generation level $p_{gts}$ indicates the contribution of unit $g$ to the reserve requirement in period $t$ in scenario $s$. In addition, the maximum available generation level of a thermal unit in a period is coupled by possible generation levels in preceding and succeeding periods.

Ramp rate limits:

$$\begin{aligned} \mathbf{} \bar{p}_{gts}&\le p_{g,t-1,s} + \varDelta _g^{ru}v_{g,t-1} + \varDelta _g^{su}(v_{gt} - v_{g,t-1}) \nonumber \\&\quad + \bar{P}_g(1-v_{gt}), \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(38)

$$\begin{aligned} \bar{p}_{gts}&\le \bar{P}_gv_{g,t+1} \nonumber \\&\quad + \varDelta _{g}^{sd}(v_{gt} - v_{g,t+1}), \forall g \in \mathcal {G}, \forall t = 1, \cdots , |\mathcal {T}|-1\end{aligned}$$

(39)

$$\begin{aligned} p_{g,t-1,s} - p_{gts}&\le \varDelta _g^{rd}v_{gt} + \varDelta _g^{sd}(v_{g,t-1}- v_{gt}) \nonumber \\&\quad + \bar{P}_g(1-v_{g,t-1}), \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S} \end{aligned}$$

(40)

Formulas (38)–(40) represent maximum available changes in generation levels of each unit between two consecutive periods.

Minimum up-time constraints:

$$\begin{aligned}&\displaystyle \sum _{t=1}^{H_g^+}(1-v_{gt}) = 0, \forall g \in \mathcal {G}\end{aligned}$$

(41)

$$\begin{aligned}&\displaystyle \sum _{i=t}^{t+T_g^U-1}v_{gi} \ge (v_{gt} - v_{g,t-1}), \forall g \in \mathcal {G}, \forall t = H_g^{+}+1, \ldots , |\mathcal {T}| - T_g^U+1\end{aligned}$$

(42)

$$\begin{aligned}&\displaystyle \sum _{i=t}^{|\mathcal {T}|}(v_{gi}- v_{gt} + v_{g,t-1})\ge 0, \forall g \in \mathcal {G}, \forall t = |\mathcal {T}|-T_g^U+2, \ldots , |\mathcal {T}| \end{aligned}$$

(43)

Minimum down-time constraints:

$$\begin{aligned}&\displaystyle \sum _{t=1}^{H_g^-}v_{gt} = 1, \forall g \in \mathcal {G} \end{aligned}$$

(44)

$$\begin{aligned}&\displaystyle \sum _{i=t}^{t+T_g^D-1}(1-v_{gt}) \ge (v_{g,t-1} - v_{gt}), \forall g \in \mathcal {G}, \forall t = H_g^{-}+1, \ldots , |\mathcal {T}|-T_g^D+1\end{aligned}$$

(45)

$$\begin{aligned}&\displaystyle \sum _{i=t}^{|\mathcal {T}|}(1-v_{gi}-v_{g,t-1}+v_{gt}) \ge 0, \forall g \in \mathcal {G}, \forall t = |\mathcal {T}|-T_g^{D}+2, \ldots , |\mathcal {T}| \end{aligned}$$

(46)

Thermal units cannot be shut down (started up) immediately after being started up (shut down), because these operations can only be performed under gradual change of temperature, which translates to time periods. Formulas (41)–(46) use binary variables to describe these restrictions on thermal units.

Commitment cost:

$$\begin{aligned} c_{gt}^u(v_{gt}) \ge \varrho _{gk}\left( v_{gt} - \sum _{i=1}^{\min (t-1,k)}v_{g,t-i} \right) + a_gv_{gt}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall k \in \mathcal {K}_g \end{aligned}$$

(47)

The start-up cost function is monotonically increasing along the time of a thermal unit has been off. Stairwise start-up cost function is adopted and formulated in the first term of (47), where $\mathcal {K}_g = \{k_1, \ldots , k_{\mathcal {N}_g}\}$, and $\varrho _{gk_i} \le \varrho _{gk_{i+1}}$, $i = 1, \ldots , \mathcal {N}_g - 1$. Notice that, $\varrho _{gk}$ includes shutdown cost in the case study. The second term is no-load cost of unit $g$ which will occur once a unit is committed.

Generation cost function:

$$\begin{aligned}&c_{gts}^p(p_{gts}) = \sum _{j=1}^J\lambda _{jg}\delta _{jgts}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(48)

$$\begin{aligned}&p_{gts} = \underline{P}_{g} + \sum _{j = 1}^{J}\delta _{jgts}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(49)

$$\begin{aligned}&\delta _{1gts} \le \gamma _{1j} - \underline{P}_{g}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(50)

$$\begin{aligned}&\delta _{jgts} \le \gamma _{jg} - \gamma _{j-1,g}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}, \forall j = 2, \ldots , J-1\end{aligned}$$

(51)

$$\begin{aligned}&\delta _{Jgts} \le \bar{P}_{g} - \gamma _{J-1, g}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(52)

$$\begin{aligned}&\delta _{jgts} \ge 0, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}, \forall j = 1, \ldots , J \end{aligned}$$

(53)

Equations (48)–(53) compute piecewise-linear generation costs.

A mixed integer linear program (MILP) extensive form of SRUC has been formulated in (31)–(56) , with boundary defined for each decision variable, as follows.

1.2.3 Appendix A.2.3: Bounds

$$\begin{aligned}&0 \le p_{gts} \le \bar{P_g}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(54)

$$\begin{aligned}&0 \le \bar{p}_{gts} \le \bar{P_g}, \forall g \in \mathcal {G}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S}\end{aligned}$$

(55)

$$\begin{aligned}&-\pi \le \theta _{bts} \le \pi , \forall b \in \mathcal {B}, \forall t \in \mathcal {T}, \forall s \in \mathcal {S} \end{aligned}$$

(56)

This model can be easily extended to the situation in which variable generation, such as wind, is viewed as dispatchable resource. Set $\mathcal {G}$ will include variable energy generators, and $d_{bts}$ will represent load rather than net load. In formula (37), we can set $v_{gt}$ to 1 for any variable energy generator $g$, and allow $\bar{P}_{g}$ to vary by scenario. Then other constraints will be suitable for variable energy generators.