Optimal allocation of HVDC interconnections for exchange of energy and reserve capacity services

Abstract

The increasing shares of stochastic renewables bring higher uncertainty in power system operation and underline the need for optimal utilization of flexibility. However, the European market structure that separates energy and reserve capacity trading is prone to inefficient utilization of flexible assets, such as the HVDC interconnections, since their capacity has to be ex-ante allocated between these services. Stochastic programming models that co-optimize day-ahead energy schedules with reserve procurement and dispatch, provide endogenously the optimal transmission allocation in terms of minimum expected system cost. However, this perfect temporal coordination of trading floors cannot be attained in practice under the existing market design. To this end, we propose a decision-support tool that enables an implicit temporal coupling of the different trading floors using as control parameters the inter-regional transmission capacity allocation between energy and reserves and the area reserve requirements. The proposed method is formulated as a stochastic bilevel program and cast as mixed-integer linear programming problem, which can be efficiently solved using a Benders decomposition approach that improves computational tractability. This model bears the anticipativity features of a transmission allocation model based on a pure stochastic programming formulation, while being compatible with the current market structure. Our analysis shows that the proposed mechanism reduces the expected system cost and thus can facilitate the large-scale integration of intermittent renewables.

Appendices

Appendix A

This Appendix provides the complete set of KKT conditions for the reserve capacity and day-ahead market clearing problems (8d) and (8e), respectively, that appear in the lower level of the preemptive allocation model (9). The dual multipliers of inequality constraints are listed to their right with complementarity relationships indicated by the $\perp$ symbol. For equality constraints, the corresponding dual multipliers are indicated after a colon.

The KKT conditions of reserve capacity auction (8d) are:

$$\begin{aligned}&0 \le \ {R}^{+}_{i} - \sum _{a} r_{ia}^{+} \perp \nu _i^{\text {R}^+} \ge 0, \quad \forall i, \end{aligned}$$

(20a)

$$\begin{aligned}&0 \le \ {R}^{-}_{i} - \sum _{a} r_{ia}^{-} \perp \nu _i^{\text {R}^-} \ge 0, \quad \forall i, \end{aligned}$$

(20b)

$$\begin{aligned}&0 \le \sum _{i} r_{ia}^{+} - RR_a^{+} \perp \nu _a^{\text {RR}^+} \ge 0, \quad \forall a, \end{aligned}$$

(20c)

$$\begin{aligned}&0 \le \sum _{i} r_{ia}^{-} - RR_a^{-} \perp \nu _a^{\text {RR}^-} \ge 0, \quad \forall a, \end{aligned}$$

(20d)

$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_s(e)}^I} r_{ia_r(e)}^{+} \perp \xi _e^{\text {r}+} \ge 0, \quad \forall e, \end{aligned}$$

(20e)

$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_r(e)}^I} r_{ia_s(e)}^{+} \perp \xi _e^{\text {s}+} \ge 0, \quad \forall e, \end{aligned}$$

(20f)

$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_s(e)}^I} r_{ia_r(e)}^{-} \perp \xi _e^{\text {r}-} \ge 0, \quad \forall e, \end{aligned}$$

(20g)

$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_r(e)}^I} r_{ia_s(e)}^{-} \perp \xi _e^{\text {s}-} \ge 0, \quad \forall e, \end{aligned}$$

(20h)

$$\begin{aligned}&0 \le C_i^+ + \nu _i^{\text {R}^+} - \nu _a^{\text {RR}^+} + \xi _e^{\text {r+}} \varPsi ^I_{a_s(e)} + \xi _e^{\text {s+}} \varPsi ^I_{a_r(e)} \perp r_{ia}^{+} \ge 0, \forall i, \forall a, \end{aligned}$$

(20i)

$$\begin{aligned}&0 \le C_i^- + \nu _i^{\text {R}^-} - \nu _a^{\text {RR}^-} + \xi _e^{\text {r}-} \varPsi ^I_{a_s(e)} + \xi _e^{\text {s}-} \varPsi ^I_{a_r(e)} \perp r_{ia}^{-} \ge 0, \forall i, \forall a, \end{aligned}$$

(20j)

where the mapping $\varPsi ^I_{a_s(e)}$ ($\varPsi ^I_{a_r(e)}$) is equal to 1 if unit i is located in the sending (receiving) end of link e and 0 otherwise.

The KKT conditions of day-ahead market auction (8e) are:

$$\begin{aligned}&\sum _{j \in {\mathcal {M}}_n^J} w_j + \sum _{i\in {\mathcal {M}}_n^I}{p}_{i} - d_n - \sum _{\ell \in L^{\text {AC}}} A_{\ell n} f_{\ell } - \sum _{\ell \in L^{\text {DC}}} A_{\ell n} z_{\ell } = 0 : \lambda _n \; \text {free}, \quad \forall n, \end{aligned}$$

(21a)

$$\begin{aligned}&0 \le p_i - \sum _{a} {\hat{r}}_{ia}^{-} \perp \nu ^{\text {PR}-}_i \ge 0, \quad \forall i, \end{aligned}$$

(21b)

$$\begin{aligned}&0 \le \sum _{a} {\hat{r}}_{ia}^{+} - p_i \perp \nu ^{\text {PR}+}_i \ge 0, \quad \forall i, \end{aligned}$$

(21c)

$$\begin{aligned}&0 \le w_j - {\overline{W}}_{j} \perp \nu ^{\text {W}}_j \ge 0, \quad \forall j, \end{aligned}$$

(21d)

$$\begin{aligned}&f_{\ell } = B_{\ell } \sum _n A_{\ell n} \delta _n : \lambda _n^{\text {F}} \; \text {free}, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$

(21e)

$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } + f_{\ell } \perp \underline{\xi }_{\ell }^{\text {AC}} \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$

(21f)

$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } - f_{\ell } \perp {\overline{\xi }}_{\ell }^{\text {AC}} \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$

(21g)

$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } + z_{\ell } \perp \underline{\xi }_{\ell }^{\text {DC}} \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$

(21h)

$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } - z_{\ell } \perp {\overline{\xi }}_{\ell }^{\text {DC}} \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$

(21i)

$$\begin{aligned}&\delta _1 = 0 : \lambda ^{\text {ref}} \; \text {free}, \end{aligned}$$

(21j)

$$\begin{aligned}&0 \le C_i + \sum _{i\in {\mathcal {M}}_n^I} \lambda _n + \nu ^{\text {PR}+}_i - \nu ^{\text {PR}-}_i \perp p_i \ge 0, \quad \forall i, \end{aligned}$$

(21k)

$$\begin{aligned}&0 \le \sum _{j \in {\mathcal {M}}_n^J} \lambda _n + \nu ^{\text {W}}_j \perp w_j \ge 0, \quad \forall j, \end{aligned}$$

(21l)

$$\begin{aligned}&0 \le - \sum _n A_{\ell n} \lambda _n + \lambda _n^{\text {F}} - \underline{\xi }_{\ell }^{\text {AC}} + {\overline{\xi }}_{\ell }^{\text {AC}} \perp f_{\ell } \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$

(21m)

$$\begin{aligned}&0 \le - \sum _n A_{\ell n} \lambda _n - \underline{\xi }_{\ell }^{\text {DC}} + {\overline{\xi }}_{\ell }^{\text {DC}} \perp z_{\ell } \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$

(21n)

$$\begin{aligned}&0 = - \sum _{\ell \in L^{\text {AC}}} B_{\ell } A_{\ell n} \lambda _n^{\text {F}} \perp \delta _{n} \; \text {free}, \forall n \backslash n = 1, \end{aligned}$$

(21o)

$$\begin{aligned}&0 = - \sum _{\ell \in L^{\text {AC}}} B_{\ell } A_{\ell n} \lambda _n^{\text {F}} + \lambda ^{\text {ref}} \perp \delta _{1} \; \text {free}, n = 1. \end{aligned}$$

(21p)

Appendix B

See Tables 11 and 12.

Table 11 Power flows for the SeqM and StochM transmission allocation models (values in MW)

Table 12 Power flows for the PrM1 and PrM2 transmission allocation models (values in MW)

1.1 Nomenclature

The main notation used in this paper is treated as abbreviation style. Additional symbols are defined in the paper when needed.