Optimization of battery management in telecommunications networks under energy market incentives

Abstract

Batteries are classically used as backup in case of power outages in telecommunications networks to keep the services always active. Recently, network operators use the batteries as a demand response lever, so as to reduce the energy costs and to generate revenues in the energy market. In this work, we study how the telecommunications operator can optimize the use of a battery over a given horizon to reduce energy costs and to perform load curtailments efficiently, as long as the safety usage rules are respected. First, we formulate the related optimization problem as a mixed integer program taking into account the telecommunication engineering and battery operation constraints, as well as the market rules. Second, we give two practical variants of the problem that can be solved to optimality in polynomial time thanks to a graph-oriented algorithm we propose. We also use this algorithm as a heuristic to solve efficiently the main problem. Finally, simulations based on real data from the French telecommunications operator Orange show the relevance of the model and of the graph-oriented algorithm: these prove to be computationally efficient in solving large scale instances, and significant savings and revenues can be generated through our optimized battery management policies.

Appendices

A: Linearizing (OBSC-MINLP)

For a product between a binary and a float variable $b_i$ and $f_j \in [0,F^{\max }]$ respectively, we can apply the McCormick strategy (see McCormick (1976)), which amounts to using a new variable $lin\_bf_i^j \in [0,F^{\max }]$ to replace this product $b_i f_j$, together with the following constraints:

$$\begin{aligned}&lin\_bf_i^j \le b_i{F}^{\max }.{} & {} \end{aligned}$$

(23)

$$\begin{aligned}&lin\_bf_i^j \le f_j.{} & {} \end{aligned}$$

(24)

$$\begin{aligned}&lin\_bf_i^j \ge f_j - (1-b_i){F}^{\max }. \end{aligned}$$

(25)

The non-linearities of this type in (3)-(19) are the products $x_ty_c$ (with $x_t \in [0, B^{\max }]$), $y_cp^{\max }_c$ (with $p^{\max }_c \in [0, P^{\max }]$) and $x_tz_t$ in (4), (14) and (12), respectively. Thanks to (5), we need to introduce only two new families of variables: $lin\_xy^c_t$ for all c in $\mathcal {C}$, t in $\mathcal {T}$ to linearize (4), and $lin\_ypmax_c$ for all c in $\mathcal {C}$ to linearize (14). Then, we can simply replace $x_tz_t$ by $\sum _{c \in \mathcal {C}_t} lin\_xy^c_t$ in (12).

Furthermore, to linearize $x=\min (a,b)$ for $a,b \in [M',M]$, we introduce a binary variable $y \in \{0,1\}$ such that, if $a > b$, then $y=1$, otherwise $y=0$. We can then rewrite x as follows:

$$\begin{aligned}&x \le a \text {, } x \le b.{} & {} \end{aligned}$$

(26)

$$\begin{aligned}&a-b \le (M-M')y \text {, } b-a \le (M-M')(1-y).{} & {} \end{aligned}$$

(27)

$$\begin{aligned}&x \ge a - (M-M')y \text {, } x \ge b - (M-M')(1-y).{} & {} \end{aligned}$$

(28)

In our case, we have two new families of binary variables: $lin\_side_t$ for all t in $\mathcal {T}$ to linearize (12), and19LLARSPS] for all c in $\mathcal {C}$ to linearize (15). In the case of (12), we have $u_t^B=(1-z_t)\min (a,b)$, where $a=B^{\max }{/\Delta }-x_t{/\Delta }$ and $b=\min (P_B,P^{\max }-W_t)$. In order to linearize this expression, we have to multiply all the terms a and b in (26) and (28) by $1-z_t$. Hence, we derive the following constraints, where $M'=0$ and $M=\max (P^{\max },B^{\max }{/\Delta })$:

$$\begin{aligned}&u_t^B \le (1-z_t)(B^{\max }{/\Delta }-x_t{/\Delta }) \text {, } u_t^B \le (1-z_t)\min (P_B,P^{\max }-W_t).{} & {} \end{aligned}$$

(29)

$$\begin{aligned}&(B^{\max }{/\Delta }-x_t{/\Delta })- \min (P_B,P^{\max }-W_t) \le Mlin\_side_t\text {, }{} & {} \nonumber \\ {}&\min (P_B,P^{\max }-W_t)- (B^{\max }{/\Delta }-x_t{/\Delta }) \le M(1-lin\_side_t).{} & {} \end{aligned}$$

(30)

$$\begin{aligned}&u_t^B \ge (1-z_t)(B^{\max }{/\Delta }-x_t{/\Delta }) - M(1-z_t)lin\_side_t \text {, }{} & {} \nonumber \\&u_t^B \ge (1-z_t)\min (P_B,P^{\max }-W_t) - M(1-z_t)(1-lin\_side_t).{} & {} \end{aligned}$$

(31)

Note that, since $u_t^B \in [0,P^B]$, Constraints (31) can be replaced by:

$$\begin{aligned}&u_t^B \ge (1-z_t)(B^{\max }{/\Delta }-x_t{/\Delta }) - Mlin\_side_t \text {, } u_t^B \ge (1-z_t)\nonumber \\&\quad \min (P_B,P^{\max }-W_t) - M(1-lin\_side_t). \end{aligned}$$

(32)

Indeed, when $z_t=0$, (31) and (32) are equivalent, and, when $z_t=1$, (29) together with (32) and $u_t^B \in [0,P^B]$ ensure that $u_t^B=0$.

In the case of (15), we can rewrite such constraints as $p^{\max }_c=-x=-\min (0,-\frac{\sum _{t'=f_c-1}^{l_c}W_{t'} + x_{f_c}{/\Delta }-x_{f_c-1}{/\Delta }}{l_c - f_c +2}+ P_{TSO})$, and linearize them by considering the terms $a=0$ and $b=-\frac{\sum _{t'=f_c-1}^{l_c}W_{t'} + x_{f_c}{/\Delta }-x_{f_c-1}{/\Delta }}{l_c - f_c +2}+ P_{TSO}$, with $b \in [-P^ {\max },P^ {\max }]$.

The complete linear version of (OBSC-MINLP), referred to as (OBSC-MILP), can then be written as follows:

$$\begin{aligned} \min \sum _{t \in \mathcal {T}}E_t(u_t^B + u_t^D) - \left\{ \begin{array}{lr} \sum \limits _{ t \in \mathcal {T}} R_{t}(W_t - u^D_{t}), &{} \quad \quad \quad (3)\\ \sum \limits _{c \in \mathcal {C}} (R_{f_c}{/\Delta })(lin\_xy_{f_c}^c-lin\_xy_{l_c+1}^c). &{} \end{array}\right. \end{aligned}$$

(33)

$$\begin{aligned}&p^{\max }_c \ge \frac{\sum _{t'=f_c-1}^{l_c}W_{t'} + x_{f_c}{/\Delta }-x_{f_c-1}{/\Delta }}{l_c - f_c +2}- P_{TSO}\quad \forall c \in \mathcal {C}. \end{aligned}$$

(41)

$$\begin{aligned}&\frac{\sum _{t'=f_c-1}^{l_c}W_{t'} + x_{f_c}{/\Delta }-x_{f_c-1}{/\Delta }}{l_c - f_c +2}- P_{TSO} \le 2P^{\max } lin\_sidepmax_c \quad \forall c \in \mathcal {C}. \langle \rangle \end{aligned}$$

(42)

$$\begin{aligned}&-\frac{\sum _{t'=f_c-1}^{l_c}W_{t'} + x_{f_c}{/\Delta }-x_{f_c-1}{/\Delta }}{l_c - f_c +2}+ P_{TSO} \le 2P^{\max } (1-lin\_sidepmax_c) \quad \forall c \in \mathcal {C}. \end{aligned}$$

(43)

$$\begin{aligned}&p^{\max }_c {\le } \frac{\sum _{t'=f_c-1}^{l_c}W_{t'} {+} x_{f_c}{/\Delta }{-}x_{f_c-1}{/\Delta }}{l_c {-} f_c +2}{-} P_{TSO} {+} 2P^{\max }(1-lin\_sidepmax_c)\quad \forall c \in \mathcal {C}. \end{aligned}$$

(44)

$$\begin{aligned}&p^{\max }_c \le 2P^{\max }lin\_sidepmax_c\quad \forall c \in \mathcal {C}. \end{aligned}$$

(45)

$$\begin{aligned}&lin\_xy^c_t \le y_c{B}^{\max } \quad \forall c \in \mathcal {C}, \forall t \in \mathcal {T}.\end{aligned}$$

(46)

$$\begin{aligned}&lin\_xy^c_t \le x_t \quad \forall c \in \mathcal {C}, \forall t \in \mathcal {T}. \end{aligned}$$

(47)

$$\begin{aligned}&lin\_xy^c_t \ge x_t - (1-y_c){B}^{\max } \quad \forall c \in \mathcal {C}, \forall t \in \mathcal {T}. \end{aligned}$$

(48)

$$\begin{aligned}&lin\_ypmax_c \le y_c{P}^{\max } \quad \forall c \in \mathcal {C}. \end{aligned}$$

(49)

$$\begin{aligned}&lin\_ypmax_c \le p_c^{\max } \quad \forall c \in \mathcal {C}. \end{aligned}$$

(50)

$$\begin{aligned}&lin\_ypmax_c \ge p_c^{\max } - (1-y_c){P}^{\max } \quad \forall c \in \mathcal {C}. \end{aligned}$$

(51)

$$\begin{aligned}&u_t^D \in [0,W_t],~u_t^B \in [0,P_B],~x_t \in [{B}^{\min },{B}^{\max }],~z_t \in \{0,1\},lin\_side_t \in \{0,1\} \quad \hspace{-0.5cm}\quad \forall \, t \in \mathcal {T}. \end{aligned}$$

(52)

$$\begin{aligned}&p^{\max }_c \in \mathbb {R}^+,~y_{c} \in \{0,1\}, lin\_ypmax_c \in [0,P^{\max }],lin\_sidepmax_c \in \{0,1\} \quad \forall \, c \in \mathcal {C}. \end{aligned}$$

(53)

$$\begin{aligned}&lin\_xy^c_t \in [0,{B}^{\max }] \quad \forall t \in \mathcal {T},\forall c \in \mathcal {C}. \end{aligned}$$

(54)

B: Proof of Theorem 1

In this section, we provide the proof of Theorem 1. Beforehand, for the sake of a better understanding, we provide a reformulation of the objective function (3) and (4), that we now recall. In order to do this, and for the sake of simplicity, we define the set $\mathcal {F}$ of all the feasible solutions, i.e., $\mathcal {F}=\{ (y,x,z,p^{\max },u^D,u^B) \in \{0,1\}^{\mathcal {C}}\!\times \!\mathbb {R}^{+\mathcal {T}}\!\times \!\{0,1\}^{\mathcal {T}}\!\times \!\mathbb {R}^{+\mathcal {C}}\!\times \!\mathbb {R}^{+\mathcal {T}}\!\times \!\mathbb {R}^{+\mathcal {T}}\,|\, (5)-(19)\text { are satisfied} \}$. Then, we have:

$$\begin{aligned} \min _{(y,x,z,p^{\max },u^D,u^B) \in \mathcal {F}}\sum \limits _{t \in \mathcal {T}}E_t(u_t^B + u_t^D) - {\left\{ \begin{array}{ll} \sum \limits _{ t \in \mathcal {T}} R_{t}(W_t - u^D_{t}),\hspace{2.00cm} \text {if OTR}, &{} \\ \sum \limits _{c \in \mathcal {C}} R_{f_c}y_c(x_{f_c}{/\Delta }-x_{l_c+1}{/\Delta }),\hspace{.72cm} \text {if FTR}. &{}\\ \end{array}\right. } \end{aligned}$$

From Constraints (10), we have that $u_t^B + u_t^D = W_t + x_{t+1}/\Delta - x_t/\Delta$ for each t.

Let us consider in what follows the following cases related to the three possible battery states at each time period t:

• Battery in discharge ($x_t-x_{t+1}>0$): from Constraints (6), we get $z_t=1$, and then, from Constraints (12), we derive $u^B_t=0$. Hence, we get $u_t^B + u_t^D = u_t^D = W_t - (x_t/\Delta - x_{t+1}/\Delta )$.

• Battery in charge ($x_t-x_{t+1}<0$): from Constraints (7), we have that $z_t=0$, and, from Constraints (13), $u^D_t \ge W_t$. Since $u^D_t\le W_t$, we get $u^D_t = W_t$, and thus $u_t^B + u_t^D = u_t^B + W_t$.

• Battery in rest mode ($x_t-x_{t+1}=0$): from Constraints (8), (13) and (12), we have that $z_t=0$, that $u^D_t=W_t$, and that $u^B_t=0$. In this case, 19we get $u_t^B + u_t^D = W_t$.

Given that we are always in one of the three above cases for a given t, but not in two at the same time, and that the power demand $W_t$ is present in the sum $u_t^B + u_t^D$ in all these cases, we can then group the power demand terms over t as $\sum _{t \in \mathcal {T}}W_t$. Additionally, for any curtailment $c \in \mathcal {C}$ performed (i.e., such that $y_c=1$), the term $x_{t+1}/\Delta - x_{t}/\Delta = -(x_{t}/\Delta - x_{t+1}/\Delta )$ is present only during the battery discharge periods of c (i.e., for $t \in \{f_c,\dots ,l_c\}$), and the term $u^B_t$ is present only during the battery recharge periods of c (i.e., for $t \in \{l_c+1,\dots ,t^B_c\}$). Hence, we obtain:

$$\begin{aligned}&\sum _{t \in \mathcal {T}} (u_t^B + u_t^D) = \sum _{t \in \mathcal {T}:x_t-x_{t+1}>0} (u_t^B + u_t^D) + \sum _{t \in \mathcal {T}:x_t-x_{t+1}<0} (u_t^B + u_t^D) + \sum _{t \in \mathcal {T}:x_t-x_{t+1}=0} (u_t^B + u_t^D)\\&\quad = \sum _{t \in \mathcal {T}:x_t-x_{t+1}>0} (W_t - (x_t/\Delta -x_{t+1}/\Delta )) + \sum _{t \in \mathcal {T}:x_t-x_{t+1}<0} (W_t + u_t^B) + \sum _{t \in \mathcal {T}:x_t-x_{t+1}=0} W_t\\&\quad = \sum _{t \in \mathcal {T}}W_t +\sum _{c \in \mathcal {C}}y_c \Bigg (- \sum _{t=f_c}^{l_c}(x_t/\Delta -x_{t+1}/\Delta ) + \sum _{t=l_c+1}^{t^B_c}u^B_t\Bigg ). \end{aligned}$$

For similar reasons (since, for each $t \in \mathcal {T}$, we can as well rewrite $E_t(u_t^B + u_t^D)$ as $E_t(W_t - (x_t/\Delta -x_{t+1}/\Delta ))$, $E_t (W_t+u_t^B)$, or $E_t W_t$, with respect to the three above cases), we obtain:

$$\begin{aligned} \sum _{t \in \mathcal {T}} E_t(u_t^B + u_t^D) = \sum _{t \in \mathcal {T}}E_t W_t +\sum _{c \in \mathcal {C}}y_c \Bigg (- \sum _{t=f_c}^{l_c} E_t (x_t/\Delta -x_{t+1}/\Delta ) + \sum _{t=l_c+1}^{t^B_c} E_t u^B_t\Bigg ). \end{aligned}$$

By replacing $\sum _{t\in \mathcal {T}}E_t(u^B_t+u^D_t)$ by this new expression in the objective function, we get:

$$\begin{aligned}&\min _{(y,x,z,p^{\max },u^D,u^B) \in \mathcal {F}} \underbrace{\sum _{t \in \mathcal {T}}E_tW_t}_\text {T1} - \underbrace{\sum _{c \in \mathcal {C}}y_c \Bigg ( \sum _{t=f_c}^{l_c}E_t(x_t/\Delta -x_{t+1}/\Delta ) - \sum _{t=l_c+1}^{t^B_c}E_tu^B_t\Bigg )}_\text {T2} \quad \nonumber \\&\hspace{5cm} - \underbrace{{\left\{ \begin{array}{ll} \sum _{ t \in \mathcal {T}} R_{t}(W_t - u^D_{t}), \hspace{1.53cm}\text {if OTR,} &{} \\ \sum _{c \in \mathcal {C}} R_{f_c}y_c(x_{f_c}{/\Delta }-x_{l_c+1}{/\Delta }), \hspace{.25cm}\text {if FTR.}&{} \end{array}\right. }}_\text {T3} \quad \end{aligned}$$

(55)

Since $W_t-u^D_t = x_t/\Delta -x_{t+1}/\Delta$ for any battery discharge period t of any curtailment performed (and $W_t-u^D_t = 0$ otherwise), we have $\sum _{ t \in \mathcal {T}} R_{t}(W_t - u^D_{t}) = \sum _{c \in \mathcal {C}}y_c \sum _{t=f_c}^{l_c} R_t(x_t/\Delta -x_{t+1}/\Delta )$, and, since $x_{f_c}-x_{l_c+1}=\sum _{t = f_c}^{l_c}(x_{t}-x_{t+1})$, we have $\sum _{c \in \mathcal {C}} R_{f_c}y_c(x_{f_c}{/\Delta }-x_{l_c+1}{/\Delta }) = \sum _{c \in \mathcal {C}} y_c \sum _{t=f_c}^{l_c} R_{f_c}(x_t/\Delta -x_{t+1}/\Delta )$. Hence, we can merge the terms T2 and T3 in Eq.  (55):

$$\begin{aligned}&\min _{(y,x,z,p^{\max },u^D,u^B) \in \mathcal {F}} \underbrace{\sum _{t \in \mathcal {T}}E_tW_t}_\text {T1} \nonumber \\&- \sum _{c \in \mathcal {C}}y_c \Bigg (\underbrace{ \sum _{t=f_c}^{l_c}{\left\{ \begin{array}{ll} (E_t+R_t)(x_t/\Delta -x_{t+1}/\Delta ) \\ (E_t+R_{f_c})(x_t/\Delta -x_{t+1}/\Delta ) \end{array}\right. }}_{f_1(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B)} - \underbrace{\sum _{t=l_c+1}^{t^B_c}E_tu^B_t}_{f_2(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B)}\Bigg ). \quad \end{aligned}$$

(56)

Note that, since the standard cost term T1 is a constant, minimizing this objective function can be seen as maximizing the objective function $\sum _{c \in \mathcal {C}}y_c(f_1(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B)-f_2(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B))$ over $\mathcal {F}$.

Moreover, we introduce a new variable $d_c \ge 0$ for each $c \in \mathcal {C}$, which represents the overall battery discharge level during curtailment c, and which is not needed in our MIP formulation, but that will be useful to rewrite the objective function. By definition, the value of $d_c$ for each curtailment c is $x_{f_c}-x_{l_c+1}$ if c is performed (i.e., if $y_c=1$), and 0 otherwise.

This yields the following new constraints:

$$\begin{aligned}{} & {} d_c = y_c(x_{f_c}-x_{l_c+1}){} & {} \forall c \in \mathcal {C}.{} & {} \end{aligned}$$

(57)

Note that, once the values of the variables $y_c$ are known, so are the ones of the variables $z_t$ (and vice-versa), thanks to Constraints (5). Similarly, once the values of the variables $d_c$ are known, so are the ones of the variables $p_c^{\max }$, thanks to Constraints (15) and the fact that, from Property 1 and Lemma 1, the value of $p_c^{\max }$ for each $c \in \mathcal {C}$ can be computed from the values of the variables $d_c$ only. Furthermore, we define a new set $\mathcal {F'}$ as follows: $\mathcal {F'} = \{(d_c)_{c \in \mathcal {C}} \text { and } ((y_c)_{c \in \mathcal {C}},(x_t)_{t \in \mathcal {T}},(z_t)_{t \in \mathcal {T}},(p_c^{\max })_{c \in \mathcal {C}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}}) \in \mathcal {F} \text { satisfying (57)}\}$.

For each $(\overline{y},\overline{d})=((\overline{y}_c)_{c \in \mathcal {C}}, (\overline{d}_c)_{c \in \mathcal {C}})$, let $\mathcal {F}^*(\overline{y},\overline{d}) = \{((x_t)_{t \in \mathcal {T}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}}) \text { such that }$ $(\overline{d}_c)_{c \in \mathcal {C}},$ $(\overline{y}_c)_{c \in \mathcal {C}},(x_t)_{t \in \mathcal {T}},(z_t)_{t \in \mathcal {T}},(p_c^{\max })_{c \in \mathcal {C}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}} \in \mathcal {F}'\}$. We also define the set $\mathcal {F}^{*} = \{((\overline{y}_c)_{c \in \mathcal {C}}, (\overline{d}_c)_{c \in \mathcal {C}}) \text { such that } \mathcal {F}^*(\overline{y},\overline{d}) \ne \emptyset \}$. Then, we have:

$$\begin{aligned}&\max _{((y_c)_{c \in \mathcal {C}},(x_t)_{t \in \mathcal {T}},(z_t)_{t \in \mathcal {T}},(p_c^{\max })_{c \in \mathcal {C}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}}) \in \mathcal {F}} \sum _{c \in \mathcal {C}}y_c \\ {}&\Bigg ( f_1(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B) - f_2(y_c,x_t,z_t,p_c^{\max },u_t^D,u_t^B)\Bigg )\\ =&\max _{((y_c)_{c \in \mathcal {C}},(x_t)_{t \in \mathcal {T}},(z_t)_{t \in \mathcal {T}},(p_c^{\max })_{c \in \mathcal {C}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}},(d_c)_{c \in \mathcal {C}}) \in \mathcal {F'}} \sum _{c \in \mathcal {C}}y_c \\ {}&\Bigg ( \sum _{t=f_c}^{l_c}{\left\{ \begin{array}{ll} (E_t+R_t)(x_t/\Delta -x_{t+1}/\Delta ) \\ (E_t+R_{f_c})(x_t/\Delta -x_{t+1}/\Delta ) \end{array}\right. } - \sum _{t=l_c+1}^{t^B_c}E_tu^B_t\Bigg )\\ =&\max _{(\overline{y},\overline{d})\in \mathcal {F}^{*}} \Bigg (\max _{((x_t)_{t \in \mathcal {T}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}}) \in \mathcal {F}^*(\overline{y},\overline{d})} \sum _{c \in \mathcal {C}}\overline{y}_c \\ {}&\Bigg ( \sum _{t=f_c}^{l_c}{\left\{ \begin{array}{ll} (E_t+R_t)(x_t/\Delta -x_{t+1}/\Delta ) \\ (E_t+R_{f_c})(x_t/\Delta -x_{t+1}/\Delta ) \end{array}\right. } - \sum _{t=l_c+1}^{t^B_c}E_tu^B_t\Bigg )\Bigg ) \\ =&\max _{(\overline{y},\overline{d})\in \mathcal {F}^{*}} \Bigg (\sum _{c \in \mathcal {C}}\overline{y}_c \max _{((x_t)_{t \in \mathcal {T}},(u_t^D)_{t \in \mathcal {T}},(u_t^B)_{t \in \mathcal {T}}) \in \mathcal {F}_c^*(\overline{y},\overline{d})} \\ {}&\Bigg ( \underbrace{ \sum _{t=f_c}^{l_c}{\left\{ \begin{array}{ll} (E_t+R_t)(x_t/\Delta -x_{t+1}/\Delta ) \\ (E_t+R_{f_c})(x_t/\Delta -x_{t+1}/\Delta ) \end{array}\right. }}_{f_c^S(x_t,u_t^D,u_t^B)} - \underbrace{\sum _{t=l_c+1}^{t^B_c}E_tu^B_t}_{f_c^B(x_t,u_t^D,u_t^B)}\Bigg )\Bigg ). \end{aligned}$$

where $\mathcal {F}_c^*(\overline{y},\overline{d})$ is the restriction of the set $\mathcal {F}^*(\overline{y},\overline{d})$ to the variables $x_t$, $u^D_t$ and $u^B_t$ for all $t \in \mathcal {T}$ such that $c \in \mathcal {C}_t$. Note that the last equality comes from the fact that, once the values of all the variables $y_c$ are known, it can be checked that the only constraints in (5)-(19) and (57) linking the variables associated with time periods of different curtailments that are performed are Constraints (14) and (15). Hence, once the values of all the variables $y_c$ and of all the variables $d_c$ are known, there is no remaining links between the variables associated with time periods of different curtailments that are performed.

We now return to the proof of Theorem 1, that we first recall:

Theorem 1 Let $\mathcal {F}_{opt}$ be the set of vectors $((y_c)_{c \in \mathcal {C}},(d_c)_{c \in \mathcal {C}})$ such that:

• for each $c \in \mathcal {C}$, $y_c \in \{0,1\}$ and $d_c \ge 0$,

• for each $c\in \mathcal {C}$ and $c'\in \mathcal {C}{\setminus }\{c\}$, if $y_c=y_{c'}=1$, then $\{f_c,\dots ,t^B_c\} \cap \{f_{c'},\dots ,t^B_{c'}\} = \emptyset$, where, for each $c\in \mathcal {C}$, $t^B_c > l_c$ is the integer such that $d_c \in ]{\Delta }\sum _{t=l_c+1}^{t^B_c-1} \min (P_B,P^{\max }-W_{t}),{\Delta }\sum _{t=l_c+1}^{t^B_c} \min (P_B,P^{\max }-W_{t})]$,

• for each $c\in \mathcal {C}$, if $y_c = 1$, then we must have $d_c \in [\sum _{t=f_c}^{l_c}{\Delta }\max (\min (W_t,D^{\min }),W_t-p_c^{\max }), \min (\sum _{t=f_c}^{l_c}{\Delta }\min (W_t,D^{\max }), B^{\max }-B^{\min })]$, where the value of $p_c^{\max }$ is computed using Property 1 and Lemma 1.

In any feasible solution to an instance of OBSC, we have $((y_c)_{c \in \mathcal {C}},(d_c)_{c \in \mathcal {C}}) \in \mathcal {F}_{opt}$. Moreover, for any $((y_c)_{c \in \mathcal {C}},(d_c)_{c \in \mathcal {C}}) \in \mathcal {F}_{opt}$, one can obtain a feasible solution of value:

$$\begin{aligned} \sum _{t \in \mathcal {T}}E_tW_t - \sum _{c \in \mathcal {C}}y_c f^G_c(d_c,d_{c^-}), \end{aligned}$$

where $f^G_c(d_c,d_{c^-})$ is a function that can be computed in linear time, and where, for each curtailment c such that $y_c=1$, $c^-$ is the only curtailment such that $y_{c^-}=1$ and there exists no curtailment $c'$ such that $y_{c'}=1$ and $l_{c^-}< f_{c'} \le l_{c'} < f_{c}$. In other words, for each curtailment c, $c^-$ is the curtailment that immediately precedes c.

In particular, the optimal value of OBSC can be rewritten as follows:

$$\begin{aligned} \sum _{t \in \mathcal {T}}E_tW_t - \max _{((y_c)_{c \in \mathcal {C}},(d_c)_{c \in \mathcal {C}}) \in \mathcal {F}_{opt}}\sum _{c \in \mathcal {C}}y_c f^G_c(d_c,d_{c^-}). \end{aligned}$$

Furthermore, in the case where the value of $p^{\max }_c$ depends only on the value of $y_c$ for each c, $f^G_c(d_c,d_{c^-})$ is a continuous piecewise linear function of $d_c$ having O(T) segments.

Proof

Our goal is to show that, once the values $\overline{y}_c$ of the variables $y_c$ are known, the value

$\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B) - f_c^B(x_t,u_t^D,u_t^B)\big )$ for each c can be written as a function $f_c^G$ of the values $\overline{d}_{c'}$ of the variables $d_{c'}$ for all $c' \in \mathcal {C}$. More precisely, for each curtailment c with $\overline{y}_{c}=1$, $\overline{d}_{c^-}$ and $\overline{d}_{c}$ are the only ones needed to express $f_{c}^G$, where $c^-$ is the curtailment with $\overline{y}_{c^-}=1$ that directly precedes c in the solution. Moreover, for each c, whenever the computation of $p_c^{\max }$ only depends on the values $\overline{y}_{c'}$ (see also Sect. 5.2), and not on the values $\overline{d}_{c'}$, for all $c' \in \mathcal {C}$, $f_c^G$ is a continuous piecewise linear function of $\overline{d}_c$ only.

The first part of the proof will show that, once the values $\overline{y}_c$ of all the variables $y_c$ are known, $f_c^B(x_t,u_t^D,u_t^B)$ for each c is in fact in itself a function of $\overline{d}_c$ (and of $\overline{d}_c$ only). Hence, once the value $\overline{d}_c$ is fixed, so is the value of $f_c^B(x_t,u_t^D,u_t^B)$. Intuitively, $f_c^B(x_t,u_t^D,u_t^B)$ is in fact equal to the battery recharging cost (in monetary units) of curtailment c.

So, we begin by fixing some $c\in \mathcal {C}$. At each recharging time period $t \in \{l_c+1, \dots , t_c^B\}$, no curtailment is performed and thus $z_t=0$. In addition, Constraints (5), (13) and (14) impose that $u^D_t=W_t$. Hence, summing Eq.s  (10) from $l_c+1$ to $t-1$ yields:

$$\begin{aligned}&x_t = x_{l_c+1} + {\Delta } \sum _{l_c+1 \le t' < t}u^B_{t'}.{} & {} \end{aligned}$$

(58)

Note that, because of Constraints (9), the curtailment c can start only if the battery is fully charged, i.e., if $x_{f_c} = B^{\max }$. Hence, from Constraints (57), we have $\overline{d}_c = B^{\max }-x_{l_c+1}$. Using this relation and replacing $x_t$ in (12) by its expression provided by (58), we have that:

$$\begin{aligned}&u^B_t=\min (\overline{d}_c{/\Delta }-\sum _{l_c+1 \le t' < t}u^B_{t'},P_B,P^{\max }-W_t).{} & {} \end{aligned}$$

(59)

This yields:

$$\begin{aligned} f^B_c(x_t,u_t^D,u_t^B)=\sum _{t=l_c+1}^{t_c^B}E_t \min (\overline{d}_c{/\Delta }-\sum _{l_c+1 \le t' < t}u^B_{t'},P_B,P^{\max }-W_t). \end{aligned}$$

(60)

By definition of $t_c^B$ (see also the proof of Property 1), the value of $u^B_t$ for each recharging time period $l_c+1 \le t<t_c^B$ is $\min (P_B, P^{\max }-W_t)$, and, for all $t > t_c^B$, no power bought for recharging the battery is related to c. The value of $u^B_t$ depends on the value of $\overline{d}_c$ only at the last recharging time period $t_c^B$, as shown in Property 1. Thus, we can extend the sum from $t_c^B$ to T, and $f^B_c$ can be rewritten as follows:

$$\begin{aligned}&f^B_c(x_t,u_t^D,u_t^B)=\sum _{t=l_c+1}^{T}E_t \Big [ \min \big (P_B,P^{\max }-W_t, \overline{d}_c{/\Delta } \nonumber \\&\quad - \min (\overline{d}_c{/\Delta }, \sum _{t'=l_c+1}^{t-1}\min (P_B,P^{\max }-W_{t'}))\big )\Big ]. \end{aligned}$$

(61)

Note that, at each time period t after the complete battery recharge (i.e., when $\sum _{t'=l_c+1}^{t-1} \min (P_B,P^{\max }-W_{t'}) \ge \overline{d}_c{/\Delta }$, meaning that $\min (\overline{d}_c{/\Delta }, \sum _{t'=l_c+1}^{t-1} \min (P_B,P^{\max }-W_{t'}))=\overline{d}_c{/\Delta }$), we have $\min (P_B,P^{\max } -W_{t}, \overline{d}_c{/\Delta }-\overline{d}_c{/\Delta })=0$, and thus no power is bought for the recharge related to c, as requested. Let us fix a time period $\overline{t} \in \{l_c+1,\dots ,T\}$, and assume that the value of $\overline{d}_c \in [{\Delta }\sum _{t=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_t), {\Delta }\sum _{t=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_t)]$. Then, the sum in (61) for t from $l_c+1$ to T can be decomposed into three parts:

• for t from $l_c+1$ to $\overline{t}-1$,

• for $t=\overline{t}$,

• and for t from $\overline{t}+1$ to T.

In the first case, for any recharging time period t such that $l_c+1 \le t < \overline{t}$, we have $\overline{d}_c \ge {\Delta } \sum _{t'=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_{t'}) \ge {\Delta } \sum _{t'=l_c+1}^{t}\min (P_B,P^{\max }-W_{t'})$, which implies, on the one hand, that $\min (\overline{d}_c{/\Delta }, \sum _{t'=l_c+1}^{t-1}\min (P_B,P^{\max }-W_{t'}))=\sum _{t'=l_c+1}^{t-1}\min (P_B,P^{\max }-W_{t'})$, and, on the other hand, that $\overline{d}_c{/\Delta }-\sum _{t'=l_c+1}^{t-1}\min (P_B,P^{\max }-W_{t'}) \ge \min (P_B,P^{\max }-W_{t})$. Hence, the sum in (61) for t from $l_c+1$ to $\overline{t}-1$ is equal to $\sum _{t=l_c+1}^{\overline{t}-1}E_t\min (P_B,P^{\max }-W_{t})$.

When $t=\overline{t}$, we have $\overline{d}_c \ge {\Delta }\sum _{t'=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_{t'})$ and $\overline{d}_c{/\Delta } - \sum _{t'=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_{t'}) \le \min (P_B,P^{\max }-W_{\overline{t}})$. Hence, the term in (61) for $t=\overline{t}$ is equal to $E_{\overline{t}}\big ( (\overline{d}_c{/\Delta } - \sum _{t'=l_c+1}^{\overline{t}-1} \min (P_B,P^{\max }-W_{t'})\big )$.

When $t > \overline{t}$, we have $\overline{d}_c \le {\Delta }\sum _{t'=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_{t'})\le {\Delta }\sum _{t'=l_c+1}^{t-1}\min (P_B,P^{\max }-W_{t'})$, which implies that each term of the sum in (61) for t from $\overline{t}+1$ to T is equal to $E_t$ $\min (P_B,P^{\max }-W_t,\overline{d}_c{/\Delta }-\overline{d}_c{/\Delta }) = 0$.

Hence, we can rewrite $f^B_c$ by splitting the sum over t into three parts, as follows:

$$\begin{aligned}&f^B_c(x_t,u_t^D,u_t^B)=\sum _{t=l_c+1}^{\overline{t}-1}E_t \min (P_B,P^{\max }-W_t) +E_{\overline{t}}\big (\overline{d}_c{/\Delta }\\&\quad -\sum _{t=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_t)\big ) +\sum _{t=\overline{t}+1}^{T}0 \end{aligned}$$

This yields:

$$\begin{aligned} f^B_c(x_t,u_t^D,u_t^B)=&(E_{\overline{t}})\overline{d}_c{/\Delta } +~\sum _{t=l_c+1}^{\overline{t}-1}(E_t-E_{\overline{t}}) \min (P_B,P^{\max }-W_t) \end{aligned}$$

(62)

Note that $\sum _{t=l_c+1}^{\overline{t}-1}(E_t-E_{\overline{t}}) \min (P_B,P^{\max }-W_t)$ is a constant, and hence $f^B_c$ is a linear function of $\overline{d}_c$. The same holds for any $\overline{t}$ such that $l_c+1 \le \overline{t} \le T$. Furthermore, the union of the intervals $[{\Delta }\sum _{t=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_t),{\Delta }\sum _{t=l_c+1}^{\overline{t}} \min (P_B,P^{\max }-W_t)]$ for $l_c+1 \le \overline{t} \le T$ covers all the possible values of $\overline{d}_c$, as $l_c+1 \le t^B_c \le T$. As a consequence, since all parts of $f^B_c$ are linear, $f^B_c$ is in fact a piecewise linear function of $\overline{d}_c$, that we will simply denote by $f^B_c(\overline{d}_c)$. We can even show that $f^B_c(\overline{d}_c)$ is continuous with respect to $\overline{d}_c$.

Indeed, assume that $\overline{d}_c = {\Delta }\sum _{t=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_t)$ for some $\overline{t}$, which implies that $\overline{d}_c \in [{\Delta }\sum _{t=l_c+1}^{\overline{t}-1}\min (P_B,P^{\max }-W_t), {\Delta }\sum _{t=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_t)]$ on the one hand, and that $\overline{d}_c \in [{\Delta }\sum _{t=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_t),{\Delta }\sum _{t=l_c+1}^{\overline{t}+1}\min (P_B,P^{\max }-W_t)]$ on the other hand:

• (i) to begin with, $f^B_c(\overline{d}_c)$ is equal to $E_{\overline{t}}\sum _{t=l_c+1}^{\overline{t}}\min (P_B, P^{\max }-W_t) +\sum _{t=l_c+1}^{\overline{t}-1}(E_t-E_{\overline{t}}) \min (P_B,P^{\max }-W_t) = E_{\overline{t}}\min (P_B,P^{\max }-W_{\overline{t}})+\sum _{t=l_c+1}^{\overline{t}-1}E_t \min (P_B,P^{\max }-W_t) = \sum _{t=l_c+1}^{\overline{t}}E_t \min (P_B,P^{\max }-W_t)$.

• (ii) then, it is also equal to $E_{\overline{t}+1} \sum _{t=l_c+1}^{\overline{t}}\min (P_B,P^{\max }-W_t) +\sum _{t=l_c+1}^{\overline{t}}(E_t-E_{\overline{t}+1}) \min (P_B,P^{\max }-W_t)= \sum _{t=l_c+1}^{\overline{t}} E_t \min (P_B,P^{\max }-W_t)$.

As we have just shown that the value of $f^B_c(x_t,u_t^D,u_t^B)$ for each c no longer depends on the ones of the variables $x_t$, $u_t^D$, and $u_t^B$ once the value $\overline{d}_c$ is known, we can rewrite $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B) - f_c^B(x_t,u_t^D,u_t^B)\big )$ as follows:

$$\begin{aligned}&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B) - f_c^B(x_t,u_t^D,u_t^B)\big )\\ =&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B) - f_c^B(\overline{d}_c)\big )\\ =&- f_c^B(\overline{d}_c) + \max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big ). \end{aligned}$$

The second part of the proof will show that, once the values $\overline{y}_c$ of all the variables $y_c$ are known, $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ for each $c \in \mathcal {C}$ is a function of the values $\overline{d}_c$ and $\overline{d}_{c^-}$ (while $f_c^S(x_t,u_t^D,u_t^B)$ is not), where $c^-$ is the curtailment with $\overline{y}_{c^-}=1$ that directly precedes c in the solution. Hence, once the values $\overline{d}_{c^-}$ and $\overline{d}_c$ are fixed, so is the value of $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$. Intuitively, $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ is in fact equal to the optimal economic gain associated with a curtailment c to be performed, which is composed of the savings induced by not buying energy for consumption, and of the reward received when performing a curtailment.

If, for each $t \in \{f_c,\dots ,l_c\}$, we define $G'_t = \frac{E_t + R_t}{\Delta }$ if the OTR reward policy is considered, and $G'_t = \frac{E_t + R_{f_c}}{\Delta }$ if the FTR reward policy is considered, then we can rewrite the value of $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ as follows:

$$\begin{aligned}&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )\\ =&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (\sum _{t=f_c}^{l_c}G'_t(x_t-x_{t+1})\big ). \end{aligned}$$

From Constraints (13), (14) and (12), we obtain that $W_t-D^{\max } \le u^D_t \le \min ( p_c^{\max },W_t)$ and that $u^B_t=0$ for all time periods during curtailment c. Therefore, from Constraints (8) and (10), we derive that $x_t-x_{t+1} \ge {\Delta }\max (\min (W_t,D^{\min }),W_t-p_c^{\max })$. Let us define such a lower bound on $x_t-x_{t+1}$ for each time period $t \in \{f_c,\dots ,l_c\}$ as $d^{\min }_t$. In addition, from Constraints (6), (12) and (10), we derive that $x_t-x_{t+1} \le {\Delta }\min (W_t,D^{\max })$. Let us define such an upper bound on $x_t-x_{t+1}$ for each time period $t \in \{f_c,\dots ,l_c\}$ as $d^{\max }_t$.

From Constraints (57), the battery discharge can be written as $\overline{d}_c=x_{f_c}-x_{l_c+1}$. Hence, from Constraints (9), we have that $x_{f_c}=B^{\max }$ and $x_{l_c+1}=B^{\max }-\overline{d}_c$.

This implies that the value of $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (\sum _{t=f_c}^{l_c}G'_t(x_t-x_{t+1})\big )$ can be rewritten as follows (and hence depends on the values of the variables $x_t$ only):

$$\begin{aligned}&\max \sum _{t=f_c}^{l_c}G'_t(x_t-x_{t+1}),{} & {} \nonumber \\&\hspace{0.5cm} s.t.{} & {} \nonumber \\&\hspace{0.5cm} d^{\min }_t \le x_t-x_{t+1} \le d^{\max }_t{} & {} \forall t \in \{f_c,\dots ,l_c\}. \end{aligned}$$

(63)

$$\begin{aligned}&\hspace{0.5cm} x_{f_c}=B^{\max }, \ x_{l_c+1}=B^{\max }-\overline{d}_c.{} & {} \end{aligned}$$

(64)

$$\begin{aligned}&\hspace{0.5cm} x_t \in [B^{\min }, B^{\max }]{} & {} \forall t \in \{f_c,\dots ,l_c+1\}. \end{aligned}$$

(65)

Note that, from Constraints (63), we must have that $\overline{d}_c = x_{f_c} - x_{l_c+1} = \sum _{t=f_c}^{l_c}(x_t-x_{t+1}) \ge \sum _{t=f_c}^{l_c}d^{\min }_t$ and $\overline{d}_c \le \sum _{t=f_c}^{l_c}d^{\max }_t$. In addition, from Constraints (65), we have that $\overline{d}_c = x_{f_c} - x_{l_c+1} \le B^{\max }-B^{\min }$. Thus, $\overline{d}_c$ must belong to the interval $[\sum _{t=f_c}^{l_c}d^{\min }_t,\min (\sum _{t=f_c}^{l_c}d^{\max }_t,B^{\max }-B^{\min })]$.

Since $\overline{d}_c$ belongs to such an interval, we have from Constraints (64) that $x_{l_c+1} = B^{\max }-\overline{d}_c \ge B^{\min }$. Hence, for each $t \in \{f_c, \dots , l_c\}$, since $d^{\min }_t > 0$, the value of $x_t$ is also greater than $B^{\min }$ from Constraints (63). Similarly, for each $t \in \{f_c+1, \dots , l_c+1\}$, since $d^{\min }_t > 0$ and $x_{f_c}=B^{\max }$ from Constraints (64), the value of $x_t$ is smaller than $B^{\max }$ from Constraints (63). Thus, Constraints (65) are always satisfied for each time period $t \in \{f_c, \dots , l_c+1\}$, and can be relaxed if the value of $\overline{d}_c$ belongs to such an interval.

Moreover, if we consider an optimal solution $x'$ to the problem obtained by relaxing Constraints (64) while considering that $x_{f_c}-x_{l_c+1}=\sum _{t=f_c}^{l_c}(x_t-x_{t+1})=\overline{d}_c$, a solution x of same value that respects Constraints (64) can be obtained as follows:

$$\begin{aligned}&x_{f_c} = B^{\max }. \end{aligned}$$

(66)

$$\begin{aligned}&x_{t+1} = x_t + (x'_{t+1}-x'_{t}){} & {} \forall t \in \{f_c,\dots ,l_c\}. \end{aligned}$$

(67)

Indeed, we have $x_{l_c+1} = x_{f_c} - \sum _{t=f_c}^{l_c}(x_t-x_{t+1}) = B_{\max } - \sum _{t=f_c}^{l_c}(x'_t-x'_{t+1}) = B_{\max } - \overline{d}_c$, as desired. Consequently, by setting $\Delta _t = x_t-x_{t+1}$ for all $t \in \{f_c,\dots ,l_c\}$, we obtain the following equivalent problem:

$$\begin{aligned}&\max \sum _{t=f_c}^{l_c}G'_t\Delta _t,{} & {} \nonumber \\&\hspace{0.5cm} s.t.{} & {} \nonumber \\&\hspace{0.5cm} \sum _{t \in \{f_c,\dots ,l_c\}}\Delta _t=\overline{d}_c.{} & {} \end{aligned}$$

(68)

$$\begin{aligned}&\hspace{0.5cm} \Delta _t \in [d^{\min }_t,d^{\max }_t]{} & {} \forall t \in \{f_c,\dots ,l_c\}. \end{aligned}$$

(69)

This problem can be solved in polynomial time using a greedy algorithm that considers a list L of time periods t ordered decreasingly by $G'_t$. In particular, in such a list, $t_1 \in L$ represents the time period $t \in \{f_c,\dots ,l_c\}$ for which $G'_t$ is the largest. An optimal solution for this problem can be obtained by defining $\Delta _{t_i}=d^{\min }_{t_i} + \delta _{t_i}$ for all $t_i \in L$, where $\delta _{t_i} \ge 0$. Intuitively, the values of $\Delta _{t_i}$ are set to $d^{\max }_{t_i}$ in the order defined by L. However, Constraint (68) imposes a total amount of $\overline{d}_c$, and hence this will be possible up to a given time period $t_k$ for which $\delta _{t_k} = \overline{d}_c - \sum _{t_j \in L | j\ge k}d^{\min }_{t_j} - \sum _{t_j \in L | j<k} d^{\max }_{t_j}$, and, for the subsequent periods $t_j$, i.e., such that $j>k$, we will have $\delta _{t_j}=0$. Formally, we have:

$$\begin{aligned} \delta _{t_i} = \max (0,\min (d^{\max }_{t_i}-d^{\min }_{t_i}, \overline{d}_c - \sum _{t_j \in L | j \ge i}d^{\min }_{t_j} - \sum _{t_j \in L | j<i} d^{\max }_{t_j})), \ \ \ \ \ \forall t_i \in L. \end{aligned}$$

Note that $\delta _{t_i}=0$ if $\sum _{t_j \in L | j \ge i}d^{\min }_{t_j} + \sum _{t_j \in L | j<i} d^{\max }_{t_j} \ge \overline{d}_c$, and $\delta _{t_i} = \min (d^{\max }_{t_i}-d^{\min }_{t_i}, \overline{d}_c - \sum _{t_j \in L | j \ge i}d^{\min }_{t_j} - \sum _{t_j \in L | j<i} d^{\max }_{t_j})$ otherwise. Hence, $\delta _{t_i}$ can be rewritten as follows:

$$\begin{aligned} \delta _{t_i} = \min (d^{\max }_{t_i}-d^{\min }_{t_i}, \overline{d}_c - \min (\overline{d}_c,\sum _{t_j \in L | j \ge i}d^{\min }_{t_j} + \sum _{t_j \in L | j<i} d^{\max }_{t_j})), \ \ \ \ \ \forall t_i \in L. \end{aligned}$$

Note that, given an optimal solution $(\Delta _t)_{t \in \{f_c,\dots ,l_c\}}$, the values of $x_t$ that respect Constraints (63) and (64) can be obtained as follows:

$$\begin{aligned}&x_{f_c}=B^{\max }.{} & {} \\&x_{t+1} = x_{t}-\Delta _t{} & {} \forall t \in \{f_c,\dots ,l_c\}. \end{aligned}$$

Again, note that this implies $x_{l_c+1} = x_{f_c} - \sum _{t=f_c}^{l_c}(x_t-x_{t+1}) = B_{\max } - \sum _{t=f_c}^{l_c}\Delta _t = B_{\max } - \overline{d}_c$. Finally, we can rewrite $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ as follows:

$$\begin{aligned}&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )\nonumber \\ =&\sum _{t_i \in L}G'_{t_i}d^{\min }_{t_i} + \sum _{t_i \in L}G'_{t_i}\Big [ \min \big (d^{\max }_{t_i}-d^{\min }_{t_i}, \overline{d}_c-\min (\overline{d}_c, \sum _{t_j \in L | j \ge i}d^{\min }_{t_j} + \sum _{t_j \in L | j<i} d^{\max }_{t_j})\big ) \Big ].{} & {} \end{aligned}$$

(70)

To simplify the writing, let us define $d^{\min }$ as the lower bound imposed on $\overline{d}_c$, computed as $d^{\min }=\sum _{t=f_c}^{l_c}d^{\min }_t$, and $d^{\max }$ as the upper bound, computed as $d^{\max }=\min (\sum _{t=f_c}^{l_c}d^{\max }_t,B^{\max }-B^{\min })$. Now, let us fix a time period $t_{i^*} \in L$, and assume that $\overline{d}_c \in [d^{\min }+ \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}), d^{\min } +\sum _{t_j \in L | j \le i^*} (d^{\max }_{t_j}-d^{\min }_{t_j})]$, where $d^{\min } + \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) = \sum _{t_j \in L | j \ge i^*} d^{\min }_{t_j}+\sum _{t_j \in L | j < i^*} d^{\max }_{t_j}$ and $d^{\min } + \sum _{t_j \in L | j \le i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) = \sum _{t_j \in L | j > i^*} d^{\min }_{t_j}+\sum _{t_j \in L | j \le i^*} d^{\max }_{t_j}$. Then, Eq.  (70) can be decomposed into three parts:

• for all $t_i$ in L such that $i<i^*$,

• for $t_i=t_{i^*}$,

• and for all $t_i$ in L such that $i>i^*$.

In the first case, for any time period $t_i$ in L such that $i<i^*$, we have that $\overline{d}_c \ge d^{\min } + \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) \ge d^{\min } + \sum _{t_j \in L | j<i} (d^{\max }_{t_j}-d^{\min }_{t_j})$, which implies that $\min (\overline{d}_c,d^{\min } + \sum _{t_j \in L | j<i} (d^{\max }_{t_j}-d^{\min }_{t_j})) = d^{\min } + \sum _{t_j \in L | j<i} (d^{\max }_{t_j}-d^{\min }_{t_j})$. Moreover, we have that $\overline{d}_c \ge d^{\min } + \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) \ge d^{\min } + \sum _{t_j \in L | j\le i} (d^{\max }_{t_j}-d^{\min }_{t_j})$, since $i\le i^*-1$, which implies that $\overline{d}_c-d^{\min } - \sum _{t_j \in L | j<i} (d^{\max }_{t_j}-d^{\min }_{t_j}) \ge d^{\max }_{t_i}-d^{\min }_{t_i}$. Then, from Eq.  (70), we have $\Delta _{t_i} = x_{t_i}-x_{{t_i}+1} = d^{\min }_{t_i} + d^{\max }_{t_i}-d^{\min }_{t_i} = d^{\max }_{t_i}$.

When $t_i=t_{i^*}$, we have $\overline{d}_c \ge d^{\min } + \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j})$ and $\overline{d}_c \le d^{\min } + \sum _{t_j \in L | j\le i^*} (d^{\max }_{t_j}-d^{min}_{t_j})$, i.e., $\overline{d}_c - d^{\min } - \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) \le d^{\max }_{t_{i^*}}-d^{\min }_{t_{i^*}}$. Hence, from Eq.  (70), we have $\Delta _{t_{i^*}} = x_{t_{i^*}}-x_{{t_{i^*}}+1} = d^{\min }_{t_{i^*}} + \overline{d}_c - d^{\min } - \sum _{t_j \in L | j<i^*} (d^{\max }_{t_j}-d^{\min }_{t_j})$.

In the third case, for any time period $t_i$ in L such that $i>i^*$, we have $\overline{d}_c \le d^{\min } + \sum _{t_j \in L | j\le i^*} (d^{\max }_{t_j}-d^{\min }_{t_j}) \le d^{\min } + \sum _{t_j \in L | j<i} (d^{\max }_{t_j}-d^{\min }_{t_j})$, which implies from Eq.  (70) that $\Delta _{t_i}=x_{t_i}-x_{{t_i}+1}=d^{\min }_{t_i}$.

Thus, we can rewrite $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ by splitting the sum over $t_i$ into three parts, as follows:

$$\begin{aligned}&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )\nonumber \\ =&\sum _{t_i \in L | i<i^*}G'_{t_i}d^{\max }_{t_i} +G'_{t_{i^*}} \Big ( d^{\min }_{t_{i^*}} + \overline{d}_c - d^{\min } - \sum _{t_i \in L | i<i^*} (d^{\max }_{t_i}-d^{\min }_{t_i} )\Big ) +\sum _{t_i \in L | i > i^*}G'_{t_i}d^{\min }_{t_i}. \hspace{-0.6cm}{} & {} \end{aligned}$$

(71)

This yields:

$$\begin{aligned}&\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )\nonumber \\ =&\sum _{t_i \in L | i<i^*}G'_{t_i}d^{\max }_{t_i} +G'_{t_{i^*}}\overline{d}_c -G'_{t_{i^*}} \Big ( d^{\min } +\sum _{t_i \in L | i<i^*} (d^{\max }_{t_i}-d^{\min }_{t_i}) \Big ) +\sum _{t_i \in L | i \ge i^*}G'_{t_i}d^{\min }_{t_i}.{} & {} \end{aligned}$$

(72)

Hence, we obtain that, for each c, $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ is a function of $\overline{d}_c$ and $\overline{d}_{c^-}$, that we will simply denote by $f_c^S(\overline{d}_c, \overline{d}_{c^-})$, where $c^-$ is the curtailment with $\overline{y}_{c^-}=1$ that directly precedes c in the optimal solution. Note that the dependence in $\overline{d}_c$ is linear, but the one in $\overline{d}_{c^-}$ is not, as $\overline{d}_{c^-}$ is implicitly used in the computation of $p_c^{\max }$, which in turn is used in the computation of $d_t^{\min }$ for each $t \in \{f_c, \dots , l_c\}$.

However, in the case where the computation of the variables $p_c^{\max }$ only depends on the values $\overline{y}_c$ themselves, things are different. Indeed, in this case, all terms in Eq.  (72) are constant, except the term $G'_{t_{i^*}}\overline{d}_c$. Thus, $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B)\big )$ is then a linear function of $\overline{d}_c$, and the same holds for any $t_{i^*} \in L$. Furthermore, the union of the intervals $[d^{\min }+ \sum _{t_i \in L | i<i^*} (d^{\max }_{t_i}-d^{\min }_{t_i}), d^{\min } +\sum _{t_i \in L | i \le i^*} (d^{\max }_{t_i}-d^{\min }_{t_i})]$ for all $t_{i^*} \in L$ covers all the possibles values of $\overline{d}_c$ in the range $[d^{\min },d^{\max }]$. As a consequence, since all parts of the function $\max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,z_t,u_t^D,u_t^B)\big )$ are linear, it is in fact, in this special case, a piecewise linear function of $\overline{d}_c$ in the range $[d^{\min },d^{\max }]$, that we will simply denote by $f_c^S(\overline{d}_c)$. We can even show that $f_c^S(\overline{d}_c)$ is continuous with respect to $\overline{d}_c$ in such a range. Indeed, let us assume that $\overline{d}_c = d^{\min }+ \sum _{t_i \in L | i\le i^*}(d^{\max }_{t_i}-d^{\min }_{t_i})$ for some $t_{i^*} \in L$, which implies that $\overline{d}_c \in [d^{\min }+ \sum _{t_i \in L | i<i^*} (d^{\max }_{t_i}-d^{\min }_{t_i}), d^{\min } +\sum _{t_i \in L | i \le i^*} (d^{\max }_{t_i}-d^{\min }_{t_i})]$ and that $\overline{d}_c \in [d^{\min }+ \sum _{t_i \in L | i\le i^*} (d^{\max }_{t_i}-d^{\min }_{t_i}), d^{\min } +\sum _{t_i \in L | i \le i^*+1} (d^{\max }_{t_i}-d^{\min }_{t_i})]$:

• (i) to begin with, $f_c^S(\overline{d}_c)$ is equal to $\sum _{t_i \in L | i<i^*}G'_{t_i}d^{\max }_{t_i} +G'_{t_{i^*}}(d^{\min }+ \sum _{t_i \in L | i\le i^*}(d^{\max }_{t_i}-d^{\min }_{t_i})) -G'_{t_{i^*}} \Big ( d^{\min } +\sum _{t_i \in L | i<i^*} (d^{\max }_{t_i}-d^{\min }_{t_i}) \Big ) +\sum _{t_i \in L | i \ge i^*}G'_{t_i}d^{\min }_{t_i}$ $=$ $\sum _{t_i \in L | i<i^*}G'_{t_i}d^{\max }_{t_i} +G'_{t_{i^*}}(d^{\max }_{t_{i^*}}-d^{\min }_{t_{i^*}}) +\sum _{t_i \in L | i \ge i^*}G'_{t_i}d^{\min }_{t_i}$ $=$ $\sum _{t_i \in L | i\le i^*}G'_{t_i}d^{\max }_{t_i} +\sum _{t_i \in L | i > i^*}G'_{t_i}d^{\min }_{t_i}$.

• (ii) then, it is also equal to $\sum _{t_i \in L | i \le i^*}G'_{t_i}d^{\max }_{t_i} +G'_{t_{i^*+1}}(d^{\min }+ \sum _{t_i \in L | i\le i^*}(d^{\max }_{t_i} - d^{\min }_{t_i})) - G'_{t_{i^*+1}}\Big (d^{\min } +$ $\sum _{t_i \in L | i\le i^*} (d^{\max }_{t_i} - d^{\min }_{t_i}) \Big ) +\sum _{t_i \in L | i \ge i^*+1}G'_{t_i}d^{\min }_{t_i}$ $=$ $\sum _{t_i \in L | i \le i^*} G'_{t_i}d^{\max }_{t_i} + \sum _{t_i \in L | i \ge i^*+1}G'_{t_i}d^{\min }_{t_i}$ $=$ $\sum _{t_i \in L | i\le i^*}G'_{t_i}d^{\max }_{t_i} +\sum _{t_i \in L | i > i^*}G'_{t_i}d^{\min }_{t_i}$.

Summing up, we have managed to prove that, for each c, we have the following result:

$$\begin{aligned} \max _{((x_t)_{t},(u_t^D)_{t},(u_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})}\big (f_c^S(x_t,u_t^D,u_t^B) - f_c^B(x_t,u_t^D,u_t^B)\big ) = f_c^S(\overline{d}_c, \overline{d}_{c^-}) - f_c^B(\overline{d}_c). \end{aligned}$$

Let $f_c^G(\overline{d}_c, \overline{d}_{c^-}) = f_c^S(\overline{d}_c, \overline{d}_{c^-}) - f_c^B(\overline{d}_c)$ for each c. Intuitively, for any curtailment c such that $\overline{y}_c=1$, the value of $f_c^G$ gives the best total economic gain associated with c. Moreover, as we have discussed throughout the proof, we must have $((\overline{y}_c)_{c \in \mathcal {C}},(\overline{d}_c)_{c \in \mathcal {C}}) \in \mathcal {F}_{opt}$. In particular, for each c, $\overline{d}_c$ must belong to $]{\Delta }\sum _{t=l_c+1}^{t^B_c-1} \min (P_B,P^{\max }-W_{t}), {\Delta }\sum _{t=l_c+1}^{t^B_c} \min (P_B,P^{\max }-W_{t})]$ because of the function $f_c^B$, and to $[d^{\min },d^{\max }]$ because of the function $f_c^S$. Conversely, for any $((\overline{y}_c)_{c \in \mathcal {C}},(\overline{d}_c)_{c \in \mathcal {C}}) \in \mathcal {F}_{opt}$, we have also shown how to compute $((\overline{x}_t)_{t},(\overline{u}_t^D)_{t},(\overline{u}_t^B)_{t}) \in \mathcal {F}^{*}_c(\overline{y},\overline{d})$. In other words, we have essentially proved that $\mathcal {F}^{*} = \mathcal {F}_{opt}$, which, together with the computation of $f_c^G(\overline{d}_c, \overline{d}_{c^-})$, implies the first part of the proposition.

Recall that, if the computation of the values of the variables $p_c^{\max }$ only depends on the values $\overline{y}_c$ themselves, then $f_c^S(\overline{d}_c, \overline{d}_{c^-}) = f_c^S(\overline{d}_c)$, and hence we have $f_c^G(\overline{d}_c) = f_c^S(\overline{d}_c) - f_c^B(\overline{d}_c)$ for each c. Moreover, in this case, since $f_c^B$ and $f_c^S$ are continuous piecewise linear functions of $\overline{d}_c$, $f_c^G$ is also a continuous piecewise linear function for $\overline{d}_c$ in such a range (Edelsbrunner et al. (1989)). The subtraction of two piecewise linear functions can be done in linear time in function of the number of parts of each function (Edelsbrunner et al. (1989)); this number is at most $T-l_c$ for $f_c^B$ and at most $l_c-f_c+1$ for $f_c^S$, and hence $f_c^G$ can be computed in O(T) time. $\square$