Bilevel programming solution algorithms for optimal price-bidding of energy producers in multi-period day-ahead electricity markets with non-convexities

Abstract

We consider the problem of devising optimal price-offers (bids) for an energy producer participating in a multi-period day-ahead electricity market which exhibits non-convexities due to the discrete nature of the generation units’ commitments and quantities. The problem definition assumes perfect knowledge of the market’s technical characteristics, as well as of the bidding offers of the remaining producers. The problem is formulated as a bilevel optimization model with integer decision variables and linear constraint sets at both levels. The producer acts as the upper-level decision maker, aiming to find the optimal bidding offers that will maximize his individual profit upon clearing of the market, while an independent system operator acts as the lower-level decision maker, aiming to ensure satisfaction of the demand for energy at the minimum total bid-cost. Utilizing the theoretical properties of this problem, we develop both a heuristic as well as an exact algorithmic solution methodology for tackling it. More effective between the two naturally turns out to be the heuristic approach, which works iteratively, optimizing a single price-offer at each iteration, given that the remaining ones are kept fixed at their current values. We present experimental results demonstrating that it provides high quality solutions, while exhibiting reasonable computational requirements. We also demonstrate how the underlying theory can be utilized for the generation of valid inequalities to a suitable relaxation of the original formulation, in which the so-called bilevel feasibility of the obtained solution is not guaranteed. These inequalities are exploited within a cutting-plane framework by the exact solution approach for identifying the global optimum of the problem.

Appendix: Relaxing bilevel feasibility in the case of uniform pricing

The procedure we adopt for relaxing bilevel feasibility in the case of uniform pricing necessitates the introduction of two binary variables w_i,t and v_i,t for each generation unit i and time period t. Variable w_i,t takes the value 1 if and only if q_i,t > m_i, and 0 otherwise, while variable v_i,t takes the value 1 if and only if q_i,t < M_i, and 0 otherwise. We can modify accordingly constraints (11)–(14) to ensure that w_i,t and v_i,t correctly depict these two conditions as follows:

$$q_{i,t} \le \, \left( {M_{i} - m_{i} } \right)w_{i,t} + m_{i}$$

(18)

$$q_{i,t} \ge \, \left( {m_{i} + 1} \right)w_{i,t}$$

(19)

$$q_{i,t} \le M_{i} - v_{i,t}$$

(20)

$$q_{i,t} \ge M_{i} \left( {1 \, - v_{i,t} } \right)$$

(21)

When w_i,t = 1, constraint (19) sets q_i,t strictly greater than m_i, while constraint (18) becomes redundant. On the other hand, when w_i,t = 0, constraint (18) sets q_i,t less or equal to m_i, while constraint (19) becomes redundant. Similarly, when v_i,t = 0, constraint (21) sets q_i,t equal to M_i, while constraint (20) is redundant. When v_i,t = 1, on the other hand, constraint (20) sets q_i,t strictly less than M_i, while constraint (21) becomes redundant. After the introduction of these constraints, Rule 1 for time period t is enforced by adding the following two constraints for each unit i, in which K is a sufficiently large number:

$$\lambda_{t} \le p_{i,t} + (2 - w_{i,t} - v_{i,t} )K$$

(22)

$$\lambda_{t} \ge p_{i,t} - (2 - w_{i,t} - v_{i,t} )K$$

(23)

If m_i < q_i,t < M_i, then w_i,t = v_i,t = 1, so these two constraints set the smp of time period t equal to the price-offer of unit i; in any other case, they are both redundant. In order to express Rule 2, we additionally introduce a binary variable u_j,t for each unit j > 1 and time period t, which takes the value 1 if and only if p_1,t < p_j,t, and 0 otherwise. Correct values for variables u_j,t are ensured through the introduction of the following two constraints which are analogous to constraints (13) and (14):

$$p_{1,t} + \, 1 \, \le \, \left( {C_{1} - p_{j,t} + \, 1} \right)\left( {1 \, - u_{j,t} } \right) \, + p_{j,t}$$

(24)

$$p_{1,t } \ge c_{1} + \, \left( {p_{j,t} - c_{1} } \right)\left( {1 \, - u_{j,t} } \right)$$

(25)

For each unit j > 1 and time period t, we also introduce a binary variable b_j,t which takes the value 1 if and only if both q_j,t = m_j (i.e., z_j,t = 1 and w_j,t = 0) and p_j,t ≤ p_1,t (i.e., u_j,t = 0) hold, and 0 otherwise, as well as a binary variable g_j,t, which takes the value 1 if and only if both q_1,t = m₁ (i.e., z_1,t = 1 and w_1,t = 0) and p_j,t > p_1,t (i.e., u_j,t = 1) hold, and 0 otherwise. Correct values for variables b_j,t and g_j,t are ensured through the following eight constraints:

$$b_{j,t} \ge z_{j,t} - w_{j,t} - u_{j,t}$$

(26)

$$b_{j,t} \le z_{j,t}$$

(27)

$$b_{j,t} \le \, 1 \, - w_{j,t}$$

(28)

$$b_{j,t} \le \, 1 \, - u_{j,t}$$

(29)

$$g_{j,t} \ge z_{1,t} + u_{j,t} - w_{1,t} - \, 1$$

(30)

$$g_{j,t} \le z_{1,t}$$

(31)

$$g_{j,t} \le u_{j,t}$$

(32)

$$g_{j,t} \le \, 1 \, - w_{1,t}$$

(33)

Rule 2 for time period t is then expressed through the introduction of the following set of constraints. Constraints (34)–(35) are introduced only once as they pertain to unit 1, while constraints (36)–(37) are introduced once for each unit i > 1.

$$\lambda_{t} \le p_{1,t} + K(1 - z_{1,t} + w_{1,t} ) + \sum\limits_{j > 1 \, } {Kb_{j,t} } + \sum\limits_{j > 1 \, } {K(w_{j,t} + v_{j,t} - 1)}$$

(34)

$$\lambda_{t} \ge p_{1,t} - K(1 - z_{1,t} + w_{1,t} ) - \sum\limits_{j > 1} {Kb_{j,t} } - \sum\limits_{j > 1 \, } {K(w_{j,t} + v_{j,t} - 1)}$$

(35)

$$\lambda_{t} \le p_{i,t} + K(1 - z_{i,t} + w_{i,t} ) + \sum\limits_{{j > 1,j \ne i: \, p_{j,t} < p_{i,t} }} {K(z_{j,t} - w_{j,t} )} + Kg_{i,t} + \sum\limits_{j \ne i \, } {K(w_{j,t} + v_{j,t} - 1)}$$

(36)

$$\lambda_{t} \ge p_{i,t} - K(1 - z_{i,t} + w_{i,t} ) - \sum\limits_{{j > 1,j \ne i: \, p_{j,t} < p_{i,t} }} {K(z_{j,t} - w_{j,t} ) - Kg_{i,t} } - \sum\limits_{j \ne i \, } {K(w_{j,t} + v_{j,t} - 1)}$$

(37)

Constraint (36) sets λ_t less than or equal to p_i,t plus a summation of non-negative terms each of which involves a multiplication with the sufficiently large number K. Similarly, constraint (37) sets λ_t greater than or equal to p_i,t minus the summation of the same exact terms. Thus, if all these terms are equal to 0, then λ_t is set equal to p_i,t by these two constraints; otherwise, they are both redundant. The term (1 − z_i,t + w_i,t) is equal to 0 if unit i produces at its technical minimum in time period t; otherwise, it is equal to 1. The term (z_j,t − w_j,t) is equal to 1 if unit j produces at its technical minimum in time period t; otherwise, it is equal to 0. The term g_i,t is equal to 1 if unit 1 has smaller price-offer than unit i and produces at its technical minimum in time period t; otherwise, it is equal to 0. Finally, the term (w_j,t + v_j,t − 1) is equal to 1 if the output of unit j is strictly between its technical minimum and its technical maximum in time period t; otherwise, it is equal to 0. Thus, these two constraints set λ_t equal to the price-offer of unit i in time period t if unit i produces at its technical minimum, and in addition there is no other unit with smaller price-offer producing at its technical minimum, and no other unit producing strictly between its technical minimum and its technical maximum in the same time period; in any other case, these constraints are both redundant. Constraints (34)–(35) are similar to constraints (36)–(37), pertaining to the unit of the strategic producer. One can easily verify that these constraints determine the smp value correctly even when more than one units are simultaneously marginal according to Rule 2, both in the case that one of them is the strategic unit as well as in the case that it is not.

In order to express Rule 3, we additionally introduce a binary variable a_j,t for each unit j > 1 and time period t, which takes the value 1 if and only if both q_j,t = M_j (i.e., v_j,t = 0) and p_j,t > p_1,t (i.e., u_j,t = 1) hold, and 0 otherwise, as well as a binary variable h_j,t, which takes the value 1 if and only if both q_1,t = M₁ (i.e., v_1,t = 0) and p_j,t ≤ p_1,t (i.e., u_j,t = 0) hold, and 0 otherwise. Correct values for variables a_j,t and h_j,t are ensured through the following six constraints:

$$a_{j,t} \ge u_{j,t} - v_{j,t}$$

(38)

$$a_{j,t} \le u_{j,t}$$

(39)

$$a_{j,t} \le \, 1 \, - v_{j,t}$$

(40)

$$h_{j,t} \ge \, 1 \, - v_{1,t} - u_{j,t}$$

(41)

$$h_{j,t} \le \, 1 \, - v_{1,t}$$

(42)

$$h_{j,t} \le \, 1 \, - u_{j,t}$$

(43)

Rule 3 for time period t is then expressed through the introduction of the following set of constraints. Constraints (44)–(45) are introduced only once as they pertain to unit 1, while constraints (46)–(47) are introduced once for each unit i > 1.

$$\lambda_{t} \le p_{1,t} + Kv_{1,t} + \sum\limits_{j > 1 \, } {K(z_{j,t} + v_{j,t} - 1)} + \sum\limits_{j > 1 \, } K \alpha_{j,t}$$

(44)

$$\lambda_{t} \ge p_{1,t} - Kv_{1,t} - \sum\limits_{j > 1 \, } {K(z_{j,t} + v_{j,t} - 1)} - \sum\limits_{j > 1 \, } K a_{j,t}$$

(45)

$$\lambda_{t} \le p_{i,t} + Kv_{i,t} + \sum\limits_{j \ne i \, } {K(z_{j,t} + v_{j,t} - 1)} + \sum\limits_{{j > 1,j \ne i,p_{j,t} > p_{i,t} \, }} K (1 - v_{j,t} ) + Kh_{i,t}$$

(46)

$$\lambda_{t} \ge p_{i,t} - Kv_{i,t} - \sum\limits_{j \ne i \, } {K(z_{j,t} + v_{j,t} - 1)} - \sum\limits_{{j > 1,j \ne i,p_{j,t} > p_{i,t} \, }} K (1 - v_{j,t} ) - Kh_{i,t}$$

(47)

Constraint (46) sets λ_t less than or equal to p_i,t plus a summation of non-negative terms each of which involves a multiplication with the sufficiently large number K. Similarly, constraint (47) sets λ_t greater than or equal to p_i,t minus the summation of the same exact terms. Thus, if all these terms are equal to 0, then λ_t is set equal to p_i,t by these two constraints; otherwise, they are both redundant. The term v_i,t is equal to 0 if unit i produces at its technical maximum in time period t; otherwise, it is equal to 1. The term (z_j,t + v_j,t − 1) is equal to 1 if the energy quantity of unit j in time period t is positive but strictly smaller than its technical maximum; otherwise, it is equal to 0. The term (1 − v_j,t) is equal to 1 if unit j produces at its technical maximum in time period t; otherwise, it is equal to 0. Finally, the term h_i,t is equal to 1 if unit 1 has greater or equal price-offer than unit i and produces at its technical maximum in time period t; otherwise, it is equal to 0. Thus, these two constraints set λ_t equal to the price-offer of unit i in time period t if unit i produces at its technical maximum, and in addition there is no other unit whose energy dispatch is positive but strictly smaller than its technical maximum, and no other unit with larger price-offer (for j > 1, j ≠ i) or larger or equal price-offer (for j = 1) producing at its technical maximum; in any other case, they are redundant. Constraints (44)–(45) are similar to constraints (46)–(47), pertaining to the unit of the strategic producer. One can easily verify in this case, too, that these constraints determine the smp value correctly even when more than one units are simultaneously marginal according to Rule 3, both in the case that one of them is the strategic unit as well as in the case that it is not.

With the introduction of the above constraints, the complete model for relaxing bilevel feasibility in the case of a uniform pricing clearing scheme is formulated as follows:

$$\mathop {\text{Max}}\limits_{{p_{1,t} }} \, F_{1} = \sum\limits_{t = 1}^{T} {\left( {\lambda_{t} - c_{1} } \right)q_{1,t} }$$

$${\text{s}} . {\text{t}} .\quad c_{1} \le p_{1,t} \le C_{1},\quad t = 1, \ldots ,T$$

$$\sum\limits_{i \in I} {q_{i,t} } = d_{t} \,, \, t = 1, \ldots ,T$$

$$m_{i,t} z_{i,t} \le q_{i,t} \le M_{i,t} z_{i,t} ,\quad i \in I,\quad t = 1, \ldots ,T$$

$$y_{i,t} \ge z_{i,t} - z_{i,t - 1} ,\quad i \in I,\quad t = 1, \ldots ,T$$

$${\text{constraints}}\,\,\, ( 1 8 )- (23),\quad i \in I,\quad \, t = 1, \ldots ,T$$

$${\text{constraints}}\quad (24){-}(33),\,\,(36){-}(43),\,\,(46) - (47),\quad i \in I,\quad \, i > 1,\quad \, t = 1, \ldots ,T$$

$${\text{constraints}}\quad ( 3 4 ){-}(35),\,\,(44){-}(45),\quad t = 1, \ldots ,T$$

$$p_{1,t} \in Z^{ + } ,q_{i,t} \in Z^{ + } \quad ,i \in I, \quad t = 1, \ldots ,T$$

$$y_{i,t} ,z_{i,t} ,w_{i,t} ,v_{i,t} \quad {\text{binary}},\quad i \in I,\quad t = 1, \ldots ,T$$

$$u_{i,t} ,b_{i,t} ,g_{i,t} ,a_{i,t} ,h_{i,t} \,\,\,{\text{binary}},\quad i \in I,\quad \, i > 1,\quad t = 1, \ldots ,T.$$