Ramping constraint formulations under consideration of reserve activation in Unit Commitment Problems

Abstract

Recently, the European Commission passed the Guideline on Electricity Balancing to standardize future reserve products in European electricity markets. Strict regulatory requirements are imposed on ramping behavior that must be adhered to, which brings ramping constraints into renewed focus. In a literature review, we find that prevalent ramping constraint formulations cannot guarantee that those regulatory requirements are fully satisfied. Since reserve activation in relation to ramping constraints is not discussed in literature, we aim to fill this research gap with a focus on the impact of reserve activation on ramping feasibility, availability of reserve, and thereby induced imbalances. We argue for the use of a simple, yet consequential and more robust, extended version of ramping constraints to account for intertemporal dependencies of reserve in general and to satisfy regulatory requirements in particular. Prevalent formulations significantly overestimate available reserve compared to the robust formulation. Worst-case bounds on the overestimation factor are provided for common operating modes. Computational experiments on standard benchmark sets confirm that the prevalent formulation overestimates reserve in realistic scenarios. Reserve activations are simulated from German data in a comprehensive ex-post analysis to identify reserve overestimation as a hitherto neglected source of imbalances and to quantify its extent.

Appendices

Derivation of bounds on reserve overestimation

Let $\Delta_{t}=y_{t}-y_{t-1}$ denote the change in power output. For prevalent ramping constraints (13)–(14),

$$\begin{aligned}r_{t}^{+} & \leq\overline{V}-\Delta_{t}\,,\end{aligned}$$

(24)

$$\begin{aligned}r_{t}^{-} & \leq\underline{V}+\Delta_{t}\,,\end{aligned}$$

(25)

and for robust ramping constraints (16)–(17),

$$\begin{aligned}r_{t}^{+}+r_{t-1}^{-} & \leq\overline{V}-\Delta_{t}\,,\end{aligned}$$

(26)

$$\begin{aligned}r_{t-1}^{+}+r_{t}^{-} & \leq\underline{V}+\Delta_{t}\,.\end{aligned}$$

(27)

1.1 Case A—constant power

With constant power output $\Delta_{t}=0$, $r_{t}^{+}=r_{t-1}^{+}$, and $r_{t}^{-}=r_{t-1}^{-}$, the following inequalities hold. For prevalent ramping constraints (24)–(25),

$$\begin{aligned}r_{t}^{+} & \leq\overline{V}\,,\end{aligned}$$

(28)

$$\begin{aligned}r_{t}^{-} & \leq\underline{V}\,,\end{aligned}$$

(29)

which gives an upper bound on total reserve of

$$r_{t}^{+}+r_{t}^{-}\leq\overline{V}+\underline{V}.$$

(30)

For robust ramping constraints (26)–(27),

$$\begin{aligned}r_{t}^{+}+r_{t}^{-} & \leq\overline{V}\,,\end{aligned}$$

(31)

$$\begin{aligned}r_{t}^{+}+r_{t}^{-} & \leq\underline{V}\,,\end{aligned}$$

(32)

s.t.

$$r_{t}^{+}+r_{t}^{-}\leq\min\left\{\overline{V},\underline{V}\right\}.$$

(33)

To determine the factor of reserve overestimation, compare the bound on total reserve of prevalent ramping constraints (30) to that of robust ramping constraints (33). In the worst-case of constant power output, prevalent ramping constraints overestimate total available reserve by a factor of

$$\frac{\overline{V}+\underline{V}}{\min\left\{\overline{V},\underline{V}\right\}}\geq 2,$$

(34)

compared to robust ramping constraints. For symmetric ramping rate limits $V=\overline{V}=\underline{V}$, the bound on the overestimation factor decreases to $2V/V=2$.

1.2 Case B—ramping up or down

Apply the same logic as before, assume the plant ramps down with the maximum ramp rate $\Delta_{t}=-\underline{V}$, and again $r_{t}^{+}=r_{t-1}^{+}$ as well as $r_{t}^{-}=r_{t-1}^{-}$. Then, the prevalent formulation yields

$$\begin{aligned}r_{t}^{+} & \leq\overline{V}+\underline{V}\,,\end{aligned}$$

(35)

$$\begin{aligned}r_{t}^{-} & \leq 0\,,\end{aligned}$$

(36)

and the robust formulation

$$\begin{aligned}r_{t}^{+}+r_{t}^{-} & \leq\overline{V}+\underline{V}\,,\end{aligned}$$

(37)

$$\begin{aligned}r_{t}^{+}+r_{t}^{-} & \leq 0\,.\end{aligned}$$

(38)

With (35)–(36), only positive reserve can be provided, whereas with (37)–(38), neither positive nor negative reserve can be provided. Therefore, in the ramp-down case, overestimation of positive reserve is theoretically unbounded for the prevalent formulation. Equally, overestimation of negative reserve is unbounded in the ramp-up case. It is, of course, bounded by minimum and maximum power output levels that will be reached eventually.

1.3 Case C—oscillating power output

For example, consider alternating ramp-up and ramp-down phases with $\Delta_{t-1}=-\underline{V}$, $\Delta_{t}=\overline{V}$, and $\Delta_{t+1}=\Delta_{t-1}=-\underline{V}$. For prevalent ramping constraints, for $t$

$$\begin{aligned}r_{t}^{+}\leq 0, & & r_{t}^{-}\leq\underline{V}+\overline{V}\,,\end{aligned}$$

(39)

and in $t-1$

$$\begin{aligned}r_{t-1}^{+}\leq\overline{V}+\underline{V}, & & r_{t-1}^{-}\leq 0\,.\end{aligned}$$

(40)

For robust ramping constraints, for $t$

$$\begin{aligned}r_{t}^{+}+r_{t-1}^{-}\leq 0, & & r_{t-1}^{+}+r_{t}^{-}\leq\underline{V}+\overline{V}\,,\end{aligned}$$

(41)

and for $t+1$ by substituting $\Delta_{t+1}=\Delta_{t-1}$, $r_{t+1}^{+}=r_{t-1}^{+}$, and $r_{t+1}^{-}=r_{t-1}^{-}$,

$$\begin{aligned}r_{t}^{-}+r_{t-1}^{+}\leq\overline{V}+\underline{V}, & & r_{t}^{+}+r_{t-1}^{-}\leq 0\,.\end{aligned}$$

(42)

Note that constraints (41) and (42) are identical.

From prevalent ramping constraints (39)–(40) follows

$$r_{t-1}^{+}+r_{t}^{-}\leq 2\left(\overline{V}+\underline{V}\right),$$

(43)

whereas for robust ramping constraints (41)–(42),

$$r_{t-1}^{+}+r_{t}^{-}\leq\overline{V}+\underline{V}.$$

(44)

This time, even for the asymmetrical case $\overline{V}\neq\underline{V}$, overestimation is bounded by a factor of two. Interestingly, with robust constraints (41)–(42), the same amount of reserve can be allocated as with prevalent constraints (39)–(40) but only for a single period, e.g. $r_{t}^{-}=\overline{V}+\underline{V}$. For the robust formulation, this comes at the expense of positive reserve in adjoining periods, s.t. $r_{t-1}^{+}=r_{t+1}^{+}=0$, whereas with the prevalent formulation positive reserve in adjoining periods is unaffected, s.t. $r_{t-1}^{+},r_{t+1}^{+}\leq\overline{V}+\underline{V}$.

Mathematical model for the computational study

The model with robust ramping constraints is given as the maximization of

$$\begin{aligned} \sum\limits_{t\in\mathcal{T}}\left(P_{t}^{\mathrm{x}}x_{t}+\sum\limits_{i\in\mathcal{I}}\right.&\left(P_{t}^{+}r_{it}^{+}+P_{t}^{-}r_{it}^{-}-C_{i}^{\mathrm{u}}u_{it}-C_{i}^{\mathrm{v}}v_{it}\right.\\ &\left.\left.-\sum\limits_{j\in\mathcal{J}_{i}}C_{ij}^{\mathrm{v}}y_{ijt}-C_{i}^{\mathrm{w}}w_{it}\right)\right)\,, \end{aligned}$$

(45)

subject to

$$\begin{aligned}y_{it} & =\underline{P}_{i}v_{it}+\sum\limits_{j\in\mathcal{J}_{i}}y_{ijt}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(46)

$$\begin{aligned}\underline{P}_{i}v_{it} & \leq y_{it}-r_{it}^{-}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(47)

$$\begin{aligned}y_{it}+r_{it}^{+} & \leq\overline{P}_{i}v_{it}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(48)

$$\begin{aligned}y_{ijt} & \leq\overline{P}_{ij}v_{it}, & & \forall i\in\mathcal{I},\forall j\in\mathcal{J}_{i},\forall t\in\mathcal{T}\,,\end{aligned}$$

(49)

$$\begin{aligned}\left(y_{it}+r_{it}^{+}\right)-\left(y_{it-1}-r_{it-1}^{-}\right) \leq\overline{V}_{i}v_{it-1}+\overline{U}_{i}u_{i}\,,\\ \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(50)

$$\begin{aligned}\left(y_{it-1}+r_{it-1}^{+}\right)-\left(y_{it}-r_{it}^{-}\right) \leq\underline{V}_{i}v_{it}+\underline{W}_{i}w_{it}\,,\\ \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(51)

$$\begin{aligned}y_{i0} & =Y_{i0}, & & \forall i\in\mathcal{I}\,,\end{aligned}$$

(52)

$$\begin{aligned}r_{i0}^{+} & =R_{i0}^{+}, & & \forall i\in\mathcal{I}\,,\end{aligned}$$

(53)

$$\begin{aligned}r_{i0}^{-} & =R_{i0}^{-}, & & \forall i\in\mathcal{I}\,,\end{aligned}$$

(54)

$$\begin{aligned}v_{it}-v_{it-1} & =u_{it}-w_{it}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(55)

$$\begin{aligned}v_{i0} & =V_{i0}, & & \forall i\in\mathcal{I}\,,\end{aligned}$$

(56)

$$\begin{aligned}u_{it}+w_{it} & \leq 1, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(57)

$$\begin{aligned}\sum\limits_{i\in\mathcal{I}}y_{it} & =x_{t}+X_{t}, & & \forall t\in\mathcal{T}\,,\end{aligned}$$

(58)

$$\begin{aligned}u_{it},v_{it},w_{it} & \in\mathbb{B}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(59)

$$\begin{aligned}x_{t} & \in\mathbb{R}_{\geq 0}, & & \forall t\in\mathcal{T}\,,\end{aligned}$$

(60)

$$\begin{aligned}r_{it}^{+},r_{it}^{-},y_{it} & \in\mathbb{R}_{\geq 0}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(61)

$$\begin{aligned}y_{ijt} & \in\mathbb{R}_{\geq 0}, & & \forall i\in\mathcal{I},\forall j\in\mathcal{J}_{i},\forall t\in\mathcal{T}\,.\end{aligned}$$

(62)

For the prevalent formulation, robust ramping constraints (50)–(51) are replaced by

$$\begin{aligned}\left(y_{it}+r_{it}^{+}\right)-y_{it-1} & \leq\overline{V}_{i}v_{it-1}+\overline{U}_{i}u_{i}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,,\end{aligned}$$

(63)

$$\begin{aligned}y_{it-1}-\left(y_{it}-r_{it}^{-}\right) & \leq\underline{V}_{i}v_{it}+\underline{W}_{i}w_{it}, & & \forall i\in\mathcal{I},\forall t\in\mathcal{T}\,.\end{aligned}$$

(64)

For fixed reserve requirements, we additionally introduce

$$\begin{aligned}\sum\limits_{i\in\mathcal{I}}r_{it}^{+} & =R_{t}^{+} & & \forall t\in\mathcal{T}\,,\end{aligned}$$

(65)

$$\begin{aligned}\sum\limits_{i\in\mathcal{I}}r_{it}^{-} & =R_{t}^{-} & & \forall t\in\mathcal{T}\,,\end{aligned}$$

(66)

where $R_{t}^{+}$ and $R_{t}^{-}$ are the reserve requirements given in the IEEE benchmark dataset (Knueven et al. 2020; Krall et al. 2012; Barrows et al. 2019).

List of symbols

Indices

$i$ :

Plant id

$j$ :

Output level

$t$ :

Time period

Parameters

$C_{i}^{\mathrm{u}}$ :

Start-up costs of plant $i$

$C_{i}^{\mathrm{v}}$ :

Operating costs at the minimum power output level of plant $i$

$C_{ij}^{\mathrm{v}}$ :

Marginal costs of power output of plant $i$ at output level $j$

$C_{i}^{\mathrm{w}}$ :

Shut-down costs of plant $i$

$\overline{P}_{i}$ :

Maximum power output of plant $i$

$\underline{P}_{i}$ :

Minimum power output of plant $i$

$\overline{P}_{ij}$ :

Maximum power output of plant $i$ at output level $j$

$P_{t}^{\mathrm{x}}$ :

Wholesale price in period $t$

$P_{t}^{-}$ :

Negative reserve price in period $t$

$P_{t}^{+}$ :

Positive reserve price in period $t$

$R_{i0}^{-}$ :

Initial negative reserve of plant $i$ in period 0

$R_{t}^{-}$ :

Exogenous negative reserve requirement in period $t$

$R_{i0}^{+}$ :

Initial positive reserve of plant $i$ in period 0

$R_{t}^{+}$ :

Exogenous positive reserve requirement in period $t$

$\overline{U}_{i}$ :

Start-up rate of plant $i$

$\overline{V}_{i}$ :

Maximum ramp-up rate of plant $i$

$\underline{V}_{i}$ :

Maximum ramp-down rate of plant $i$

$V_{i}$ :

Initial commitment status of plant $i$

$\underline{W}_{i}$ :

Shut-down rate of plant $i$

$X_{t}$ :

Power delivery obligation in period $t$

$Y_{i0}$ :

Initial power output of plant $i$ in period 0

Sets

$\mathcal{I}$ :

Set of power plants

$\mathcal{J}_{i}$ :

Set of output levels of plant $i$

$\mathcal{T}$ :

Set of time periods

Variables

$r_{it}^{-}$ :

Negative reserve of plant $i$ in period $t$

$r_{it}^{+}$ :

Positive reserve of plant $i$ in period $t$

$u_{it}$ :

Start-up status of plant $i$ in period $t$

$v_{it}$ :

Commitment status of plant $i$ in period $t$

$w_{it}$ :

Shut-down status of plant $i$ in period $t$

$x_{t}$ :

Wholesale market bid size in period $t$

$y_{it}$ :

Power output of plant $i$ in period $t$

List of abbreviations

aFRR:

Automatic Frequency Restoration Reserves

BSP:

Balancing Service Provider

CPP:

Conventional Power Plant

mFRR:

Manual Frequency Restoration Reserves

TSO:

Transmission System Operator

UCP:

Unit Commitment Problem

Supplementary data

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Fig. 12

Average potential ramping violations per period resulting from the prevalent formulation, according to cross activation (18)–(19) and deactivation (20)–(21)

Fig. 13

Average potential ramping violations per period resulting from the prevalent formulation for given reserve requirements, according to cross activation (18)–(19) and deactivation (20)–(21)

Fig. 14

Difference in the number of plants that provide reserve relative to the number of plants per instance between solutions of the prevalent and robust formulation. Positive values mean that the prevalent formulation uses more plants to satisfy the given reserve requirements compared to the robust formulation, and vice versa for negative values. By fixing the reserve requirements, two instances in the ferc region and one in the rts_gmlc region become infeasible for both formulations and are excluded from this analysis. In some cases, unexpectedly, the prevalent formulation requires more power plants to provide the same amount of reserve. This can be attributed to complex cost structures involving negative costs, equal marginal costs, and high market price volatility

Fig. 15

Kernel density estimate of four hour interval manual frequency restoration reserves (mFRR) activation data of Germany in 2019. Full data in dark, whereas red shows activations of the 90th percentile of reserve activation probabilities. a Negative mFRR, b Positive mFRR

Fig. 16

Ex-post analysis of total absolute ramping violations resulting from the prevalent formulation over the planning horizon of 48 periods. Each column uses a different combination of activation rate and activation ratio for the simulation of plant specific reserve activations. The first two columns show realistic activation scenarios simulated from historical German data. The third column shows an arbitrarily chosen high risk activation scenario with an activation rate of 0.5 and an activation ratio of 0.8

Fig. 17

Ex-post analysis of total absolute ramping violations resulting from the prevalent formulation for fixed reserve requirements over the planning horizon of 48 periods. Each column uses a different combination of activation rate and activation ratio for the simulation of plant specific reserve activations. The first two columns show realistic activation scenarios simulated from historical German data. The third column shows an arbitrarily chosen high risk activation scenario with an activation rate of 0.5 and an activation ratio of 0.8

Fig. 18

Ex-post analysis of absolute number of violations per plant resulting from the prevalent formulation over the planning horizon of 48 periods. Each column uses a different combination of activation rate and activation ratio for the simulation of plant specific reserve activations. The first two columns show realistic activation scenarios simulated from historical German data. The third column shows an arbitrarily chosen high risk activation scenario with an activation rate of 0.5 and an activation ratio of 0.8