Strategic bidding in price coupled regions

Abstract

With the emerging deregulated electricity markets, a part of the electricity trading takes place in day-ahead markets where producers and retailers place bids in order to maximize their profit. We present a price-maker model for strategic bidding from the perspective of a producer in Price Coupled Regions (PCR) considering a capacitated transmission network between local day-ahead markets. The aim for the bidder is to establish a production plan and set its bids taking into consideration the reaction of the market. We consider the problem as deterministic, that is, the bids of the competitors are known in advance. We are facing a bilevel optimization problem where the first level is a Unit Commitment problem, modeled as a Mixed Integer Linear Program (MILP), and the second level models a market equilibrium problem through a Linear Program. The problem is first reformulated as a single level problem. Properties of the optimal spot prices are studied to obtain an extended formulation that is linearized and tightened using new valid inequalities. Several properties of the spot prices allow to reduce significantly the number of binary variables. Two novel heuristics are proposed, the first applicable in PCR, the second for general formulations with Special Ordered Sets (SOS) of type 1. Our computational experiments highlights the risk of a loss for the bidder if some aspects usually not considered in the literature, such as Price Coupled Regions, or an accurate UC problem, are not taken into account. They also show that the reformulation techniques, combined with new valid inequalities, allow to solve much larger instances than the current state-of-the-art. Finally, our experiments also show that the proposed heuristics deliver very high quality solutions in a short computation time.

Appendices

BP-MILP formulation

$$\begin{aligned} \max \quad&{\sum _{n \in N}\left( \sum _{t \in T} \sum _{i \in I^t_n}{\tilde{\lambda }}^t_i P^t_{in} \right) - c(p_n)} \nonumber \\ \text {s.t.} \quad&p_n \in P_n&n \in N \nonumber \\&\sum _{b \in B_n} Q^t_b X^t_{ib} - \sum _{s \in S_n} Q^t_s X^t_{is} \nonumber \\&\quad + \sum _{m \in \varTheta _m} ({\overline{F}}^t_{inm} - {\underline{F}}^t_{imn})= P^t_{in}&t\in T,n \in N, i \in I^t \nonumber \\&\sum _{i \in I^t_n}{\tilde{\lambda }}^t_i(z^t_{in}-X^t_{ib}) - \pi ^t_b(1 - x^t_b) \ge 0&\quad t \in T, n \in N, b \in B_n \nonumber \\&- \sum _{i \in I^t_n}{\tilde{\lambda }}^t_i(z^t_{in} - X^T_{is}) + \pi ^t_s( 1- x^t_s) \ge 0&\quad t \in T, n \in N, s \in S_n \nonumber \\&-\sum _{i \in I^t_n}{\tilde{\lambda }}^t_iX^t_{ib} + \pi ^t_bx^t_b \ge 0&\quad t \in T, n \in N, b \in B_n \nonumber \\&\sum _{i \in I^t_n}{\tilde{\lambda }}^t_iX^t_{is} - \pi ^t_sx^t_s \ge 0&\quad t \in T, n \in N, s \in S_n \nonumber \\&\sum _{i \in I^t_n}{\tilde{\lambda }}^t_iz^t_{in} - \sum _{i \in I^t_m}{\tilde{\lambda }}^t_iz^t_{im} + r^t_{nm} - r^t_{mn} = 0&t \in T, nm \in A \nonumber \\&x^t_b = 0&\quad t \in T, n \in N, b \in B_n : \pi ^t_b< {\tilde{\lambda }}^t_{{\underline{i}}^t_n} \nonumber \\&x^t_b = 1&\quad t \in T, n \in N, b \in B_n : \pi ^t_b> {\tilde{\lambda }}^t_{{\overline{\imath }}^t_n} \nonumber \\&x^t_s = 0&\quad t \in T, n \in N, b \in S_n : \pi ^t_s > {\tilde{\lambda }}^t_{{\overline{\imath }}^t_n} \nonumber \\&x^t_s = 1&\quad t \in T, n \in N, b \in S_n : \pi ^t_s < {\tilde{\lambda }}^t_{{\underline{i}}^t_n} \nonumber \\&C^{max}_{nm}r^t_{nm} = \sum _{i \in I^t_n}{\tilde{\lambda }}^t_i{\underline{F}}^t_{inm} - \sum _{i \in I^t_n}{\tilde{\lambda }}^t_i{\overline{F}}^t_{inm}&t \in T, nm \in A \nonumber \\&\sum _{i \in I^t_n} z^t_{in} = 1&t \in T, n \in N \nonumber \\&\sum _{i \in I^t_n} P^t_{in} = p^t_n&t \in T, n \in N \nonumber \\&\sum _{i \in I^t_n} X^t_{ib} = x^t_b&t \in T, n \in N, b \in B \nonumber \\&\sum _{i \in I^t_n} X^t_{is} = x^t_s&t \in T, n \in N, s \in S \nonumber \\&\sum _{i \in I^t_n} {\overline{F}}^t_{inm} = f^t_{nm}&t \in T, nm \in A \nonumber \\&\sum _{i \in I^t_n} {\underline{F}}^t_{inm} = f^t_{nm}&t \in T, nm \in A \nonumber \\&0 \le P^t_{in}\le {\overline{Q}}^t_nz^t_{in}&t \in T, n \in N, i \in I^t_n \nonumber \\&P^t_{in}\le p^t_{n}&t \in T, n \in N, i \in I^t_n \nonumber \\&P^t_{in} \ge p^t_n - {\overline{Q}}^t_n(1-z^t_{in})&t \in T, n \in N, i \in I^t_n \nonumber \\&0 \le X^t_{ib} \le z^t_{in}&t \in T, n \in N, b \in B_n, i \in I^t_n \nonumber \\&X^t_{ib} \ge x^t_b + z^t_{in} - 1&t \in T, n \in N, b \in B_n, i \in I^t_n \nonumber \\&0 \le X^t_{is} \le z^t_{in}&t \in T, n \in N, s \in S_n, i \in I^t_n \nonumber \\&X^t_{is} \ge x^t_s + z^t_{in} - 1&t \in T, n \in N, s \in S_n i \in I^t_n \nonumber \\&0 \le {\overline{F}}^t_{inm}\le C^{max}_{nm}z^t_{in}&t \in T, nm \in A, i \in I^t_n \nonumber \\&{\overline{F}}^t_{inm} \ge f^t_{nm} - C^{max}_{nm}(1 - z^t_{in})&t \in T, nm \in A, i \in I^t_n \nonumber \\&0 \le {\underline{F}}^t_{inm}\le C^{max}_{nm}z^t_{im}&t \in T, nm \in A, i \in I^t_n \nonumber \\&{\underline{F}}^t_{inm} \ge f^t_{nm} - C^{max}_{nm}(1 - z^t_{im})&t \in T, nm \in A, i \in I^t_n \nonumber \\&0 \le x^t_b \le 1&t \in T, n \in N, b \in B_n \nonumber \\&0 \le x^t_s \le 1&t \in T, n \in N, s \in S_n \nonumber \\&0 \le f^t_{nm} \le C^{max}_{nm}&t \in T, nm \in A \nonumber \\&0 \le r^t_{nm}&t\in T, nm \in A \nonumber \\&z^t_{in} \in \{0,1\}&t \in T, n \in N, i \in I^t_n \end{aligned}$$

(19a)

BP$^M$-MILP formulation

${\overline{Q}}^t_{jn} :$ maximum production capacity of generator j at node n in period t,

$P^t_{ijn}$ : $t \in T, n \in N, j \in J_n, i \in I^t_n$.

BP$^M$-MILP is obtained from BP-MILP, replacing constraints (19a) by

$$\begin{aligned} p_{nj} \in P^j_n ~~~~~ n \in N, j \in J_n, \end{aligned}$$

and by adding the following constraints:

$$\begin{aligned}&p^t_n = \sum _{j \in J_n} p^t_{jn}&t \in T, n \in N \\&P^t_{in} = \sum _{j \in J_n} P^t_{ijn}&t \in T, n \in N, i \in I^t_n \\&\sum _{i \in I^t_n}{\tilde{\lambda }}^t_iP^t_{ijn} - \pi ^t_{jn}p^t_{jn} \ge 0&\quad t \in T, n \in N, j \in J_n \\&\sum _{i \in I^t_n} P^t_{ijn} = p^t_{jn}&t \in T, n \in N, j \in J_n \\&0 \le P^t_{ijn}\le {\overline{Q}}^t_{jn}z^t_{in}&t \in T, n \in N, j \in J_n, i \in I^t_n \\&P^t_{ijn}\le p^t_{jn}&t \in T, n \in N, j \in J_n, i \in I^t_n \\&P^t_{ijn} \ge p^t_{jn} - {\overline{Q}}^t(1-z^t_{in})&t \in T, n \in N, j \in J_n, i \in I^t_n \end{aligned}$$

BP-$\{N\}$ formulation

$$\begin{aligned} \max \quad&{\sum _{t \in T} \sum _{i \in I^t}{\tilde{\lambda }}^t_i P^t_{i} - c(p)} \\ \text {s.t.} \quad&p \in P \\&\sum _{i \in I^t} z^t_{i} = 1&\quad t \in T \\&\sum _{i \in I^t} P^t_{i} = p^t&\quad t \in T \\&P^t_{i} \le r^t_{i}z^t_{i}&\quad t \in T, i \in I^t \\&P^t_{i} \ge r^t_{i+1}z^t_{i}&\quad t \in T, i \in I^t \\&0 \le P^t_{i}\le {\overline{Q}}^tz^t_{i}&\quad t \in T, i \in I^t \\&P^t_{i}\le p^t&\quad t \in T, i \in I^t \\&P^t_{i} \ge p^t - {\overline{Q}}^t(1-z^t_{i})&\quad t \in T, i \in I^t \\&z^t_{i} \in \{0,1\}&\quad t \in T, i \in I^t \end{aligned}$$

UC formulation from Ostrowski et al. (2012)

Formulation is given for a single node. Data:

• $J$: set of production units (generators)

• ${\underline{P}}_j, {\overline{P}}_j$: lower and upper bound on production level of unit $j$ over one time period

• $A_j$ : fixed cost for a generator is turned on

• $NL_j$: number of segments for production cost of unit $j$

• $F_{lj}$: unit cost on segment $l$ of unit $j$

• $T_{lj}$: upper bound on accumulated production up to segment $l$ of unit $j$

• $UT_j, DT_j$: minimum up and down times for unit $j$

• $K_j, C_j$: Ramping up/down costs

• $RU_j, RD_j$: ramping up/down rates

• $SU_j, SD_j$: start/stop rates

• $p^0_j$: initial production of unit $j$

• $L_j$: number of periods unit $j$ must be initially offline due to its minimum down time constraint

• $G_j$: number of periods unit $j$ must be initially online due to its minimum up time constraint

Variables:

• $p^t \ge 0$: energy bidding in period $t$ in the node.

• $p_j(t) \ge 0$: energy produced in period $t$ by unit $j$

• $c(p)$: total production cost for $\{p_j(t)\}_{\{j \in J, t \in T\}}$

• $\varTheta _l(j,t) \ge 0$: energy produced by $j$ in period $t$ on segment $l$

• $v_j(t) \in \{0,1\}$: indicates if unit $j$ is active at period $t$

• $y_j(t) \in \{0,1\}$: indicates if unit $j$ is started at period $t$

• $z_j(t) \in \{0,1\}$: indicates if unit $j$ is turned-off at period $t$

• $c^p_j(t)$: production cost of unit $j$ at period $t$

• $c^u_j(t)$: ramping up cost of unit $j$ at period $t$

• $c^d_j(t)$: ramping down cost of unit $j$ at period $t$

$$\begin{aligned}&\text {Link with market equilibrium variables:} \\&c(p) = \sum _{t \in T}\sum _{j \in J} c^p_j(t) + c^u_j(t) + c^d_j(t) \\&p^t \le \sum _{j \in J} p_j(t)&t \in T\\&\text {Production cost:} \\&c^p_j(t) = A_jv_j(t) + \sum _{l=1}^{NL_j} F_{lj}\varTheta _l(j,t)&j \in J, t \in T \\&p_j(t) = \sum _{l=1}^{NL_j} \varTheta _l(j,t) + {\underline{P}}_jv_j(t)&j \in J, t \in T \\&\varTheta _1(j,t) \le T_{1j}-{\underline{P}}_j&j \in J, t \in T \\&\varTheta _l(j,t) \le T_{lj} - T_{l-1 \, j}&j \in J, t \in T, l = 2 ,\dots , NL_j-1 \\&\varTheta _{NL_j}(j,t) \le {\overline{P}}_j - T_{NL_j-1 \, j}&j \in J, t \in T \\&\text {Start-up / shut-down costs:} \\&c^u_j(t) \ge K_j(v_j(t) - \sum _{n=1}^t v_j(t-n))&j \in J, t \in T \\&c^d_j(t) \ge C_j(v_j(t-1)-v_j(t))&j \in J, t \in T \\&v_j(t-1) - v_j^t + y_j(t) - z_j(t) = 0&j \in J, t \in T \\&\text {Production capacities:} \\&{\underline{P}}_jv_j(t) \le p_j(t) \le {\overline{p}}_j(t)&j \in J, t \in T \\&0 \le p_j(t) \le {\overline{P}}_jv_j(t)&j \in J, t \in T \\&\text {Ramping up and down :} \\&{{\overline{p}}_j(t) - p_j(t-1)\le RU_jv_j(t-1) + SU_jy_j(t)}&j \in J, t=2,\dots ,|T| \\&{\overline{p}}_j(1) - p^0_j\le RU_jv_j(0) + SU_jy_j(1)&\qquad j \in J \\&{p_j(t-1) - p_j(t) \le RD_jv_j(t) + SD_jz_j(t)}&j \in J, t=2,\dots ,|T| \\&{\overline{p}}^0_j - p_j(1) \le RD_jv_j(1) + SD_jz_j(1)&\qquad j \in J \\&\text {Minimum up / down times:} \\&\sum _{t' = t-UT_j+1}^t y_j(t') \le v_j(t)&\qquad j \in J, t \in G_j+1,\dots ,|T|\\&v_j(t) + \sum _{t' = t-DT_j+1}^t z_j(t') \le 1&j \in J, t \in L_j+1,\dots ,|T|\\&\sum _{t = 1}^{G_j} v_j(t) = G_j&j \in J\\&\sum _{t = 1}^{L_j} v_j(t) = 0&j \in J\\&c^p_j(t),c^u_j(t),c^d_j(t),\varTheta _l(j,t) \ge 0&j \in J, t \in T \\&v_j(t),y_j(t),z_j(t) \in \{0,1\}&j \in J, t \in T \end{aligned}$$