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


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.

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A :

Set of areas

\(a_r(e)\) :

Receiving-end area of link e

\(a_s(e)\) :

Sending-end area of link e

E :

Set of inter-area links

I :

Set of dispatchable power plants

J :

Set of stochastic power plants

\(L^{\text {AC}}\) :

Set of AC transmission lines

\(L^{\text {DC}}\) :

Set of HVDC transmission lines

N :

Set of network nodes

\( {\mathcal {M}}_{n}^{I}\) :

Set of dispatchable power plants i located at node n

\( {\mathcal {M}}_{n}^{J}\) :

Set of stochastic power plants j located at node n

\( {\mathcal {M}}_{a}^I \) :

Set of dispatchable power plants i located in area a

\( \varLambda _{a_r(e)}^{a_s(e)}\) :

Set of AC and HVDC lines connecting areas \(a_s(e)\) and \(a_r(e)\) across link e

\( {\mathcal {S}}\) :

Set of stochastic power production scenarios

\({\overline{W}}_{j}\) :

Expected power production of stochastic power plant j [MW]

\(\pi _{s}\) :

Probability of occurrence of scenario s

\(A_{\ell n}\) :

Line-to-bus incidence matrix

\(B_{\ell }\) :

Absolute value of the susceptance of AC line \(\ell \)

\(D_{n}\) :

Demand at node n [MW]

\(C_{i}\) :

Energy offer price of power plant i [$/MWh]

\(C^{+/-}_{i}\) :

Up/down reserve capacity offer price of power plant i [$/MW]

\(C^{\text {sh}}\) :

Value of involuntarily shed load [$/MWh]

\(P_{i}\) :

Capacity of dispatchable power plant i [MW]

\(R^{+/-}_{i}\) :

Up/down reserve capacity offer quantity of power plant i [$/MW]

\(RR_a^{+/-}\) :

Up/down reserve capacity requirements of area a [MW]

\(T_{e/\ell }\) :

Transmission capacity of link e/line \(\ell \) [MW]

\(\hat{W}_{j}\) :

Capacity of stochastic power plant j [MW]

\(W_{js}\) :

Power production by stochastic power plant j in scenario s [MW]

\(\delta _n\) :

Voltage angle at node n at day-ahead stage [rad]

\({\tilde{\delta }}_{ns}\) :

Voltage angle at node n in scenario s [rad]

\(\chi _{e/\ell }\) :

Percentage of inter-area interconnection capacity of link e/line \(\ell \) allocated to reserves exchange

\(f_{\ell }\) :

Power flow in AC line \(\ell \) at day-ahead stage [MW]

\({\tilde{f}}_{\ell s}\) :

Power flow in AC line \(\ell \) in scenario s [MW]

\(l^{\text {sh}}_{ns}\) :

Load shedding at node n in scenario s [MW]

\({p}_{i}\) :

Day-ahead schedule of dispatchable power plant i [MW]

\({p}_{is}^{+/-}\) :

Up/down regulation provided by dispatchable power plant i in scenario s [MW]

\(r^{+/-}_{ia}\) :

Up/down reserve capacity scheduled for dispatchable power plant i, procured by area a [MW]

\({w}_{j}\) :

Day-ahead schedule of stochastic power plant j [MW]

\(w^{\text {spill}}_{js}\) :

Power spilled by stochastic power plant j in scenario s [MW]

\(z_{\ell }\) :

Power flow in HVDC line \(\ell \) at day-ahead stage [MW]

\({\tilde{z}}_{\ell s}\) :

Power flow in HVDC line \(\ell \) in scenario s [MW]


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The authors would like to thank the editor and the three anonymous referees for their constructive comments as well as their valuable suggestions that certainly increased the value of this paper. S. Delikaraoglou is partly supported by the EU FP7 Project “Best Paths”, under Grant agreement no. 612748. P. Pinson is partly supported by the Danish Strategic Council for Strategic Research through the project “5s - Future Electricity Markets”, under Grant agreement no. 12-132636/DSF.

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Correspondence to Stefanos Delikaraoglou.


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:

$$\begin{aligned}&0 \le \ {R}^{+}_{i} - \sum _{a} r_{ia}^{+} \perp \nu _i^{\text {R}^+} \ge 0, \quad \forall i, \end{aligned}$$
$$\begin{aligned}&0 \le \ {R}^{-}_{i} - \sum _{a} r_{ia}^{-} \perp \nu _i^{\text {R}^-} \ge 0, \quad \forall i, \end{aligned}$$
$$\begin{aligned}&0 \le \sum _{i} r_{ia}^{+} - RR_a^{+} \perp \nu _a^{\text {RR}^+} \ge 0, \quad \forall a, \end{aligned}$$
$$\begin{aligned}&0 \le \sum _{i} r_{ia}^{-} - RR_a^{-} \perp \nu _a^{\text {RR}^-} \ge 0, \quad \forall a, \end{aligned}$$
$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_s(e)}^I} r_{ia_r(e)}^{+} \perp \xi _e^{\text {r}+} \ge 0, \quad \forall e, \end{aligned}$$
$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_r(e)}^I} r_{ia_s(e)}^{+} \perp \xi _e^{\text {s}+} \ge 0, \quad \forall e, \end{aligned}$$
$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_s(e)}^I} r_{ia_r(e)}^{-} \perp \xi _e^{\text {r}-} \ge 0, \quad \forall e, \end{aligned}$$
$$\begin{aligned}&0 \le \chi _e T_e - \sum _{i \in {\mathcal {M}}_{a_r(e)}^I} r_{ia_s(e)}^{-} \perp \xi _e^{\text {s}-} \ge 0, \quad \forall e, \end{aligned}$$
$$\begin{aligned}&0 \le C_i^+ + \nu _i^{\text {R}^+} - \nu _a^{\text {RR}^+} + \xi _e^{\text {r+}} \varPsi ^I_{a_s(e)} + \xi _e^{\text {s+}} \varPsi ^I_{a_r(e)} \perp r_{ia}^{+} \ge 0, \forall i, \forall a, \end{aligned}$$
$$\begin{aligned}&0 \le C_i^- + \nu _i^{\text {R}^-} - \nu _a^{\text {RR}^-} + \xi _e^{\text {r}-} \varPsi ^I_{a_s(e)} + \xi _e^{\text {s}-} \varPsi ^I_{a_r(e)} \perp r_{ia}^{-} \ge 0, \forall i, \forall a, \end{aligned}$$

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:

$$\begin{aligned}&\sum _{j \in {\mathcal {M}}_n^J} w_j + \sum _{i\in {\mathcal {M}}_n^I}{p}_{i} - d_n - \sum _{\ell \in L^{\text {AC}}} A_{\ell n} f_{\ell } - \sum _{\ell \in L^{\text {DC}}} A_{\ell n} z_{\ell } = 0 : \lambda _n \; \text {free}, \quad \forall n, \end{aligned}$$
$$\begin{aligned}&0 \le p_i - \sum _{a} {\hat{r}}_{ia}^{-} \perp \nu ^{\text {PR}-}_i \ge 0, \quad \forall i, \end{aligned}$$
$$\begin{aligned}&0 \le \sum _{a} {\hat{r}}_{ia}^{+} - p_i \perp \nu ^{\text {PR}+}_i \ge 0, \quad \forall i, \end{aligned}$$
$$\begin{aligned}&0 \le w_j - {\overline{W}}_{j} \perp \nu ^{\text {W}}_j \ge 0, \quad \forall j, \end{aligned}$$
$$\begin{aligned}&f_{\ell } = B_{\ell } \sum _n A_{\ell n} \delta _n : \lambda _n^{\text {F}} \; \text {free}, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$
$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } + f_{\ell } \perp \underline{\xi }_{\ell }^{\text {AC}} \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$
$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } - f_{\ell } \perp {\overline{\xi }}_{\ell }^{\text {AC}} \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$
$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } + z_{\ell } \perp \underline{\xi }_{\ell }^{\text {DC}} \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$
$$\begin{aligned}&0 \le (1-\chi _{\ell }) \ {T}_{\ell } - z_{\ell } \perp {\overline{\xi }}_{\ell }^{\text {DC}} \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$
$$\begin{aligned}&\delta _1 = 0 : \lambda ^{\text {ref}} \; \text {free}, \end{aligned}$$
$$\begin{aligned}&0 \le C_i + \sum _{i\in {\mathcal {M}}_n^I} \lambda _n + \nu ^{\text {PR}+}_i - \nu ^{\text {PR}-}_i \perp p_i \ge 0, \quad \forall i, \end{aligned}$$
$$\begin{aligned}&0 \le \sum _{j \in {\mathcal {M}}_n^J} \lambda _n + \nu ^{\text {W}}_j \perp w_j \ge 0, \quad \forall j, \end{aligned}$$
$$\begin{aligned}&0 \le - \sum _n A_{\ell n} \lambda _n + \lambda _n^{\text {F}} - \underline{\xi }_{\ell }^{\text {AC}} + {\overline{\xi }}_{\ell }^{\text {AC}} \perp f_{\ell } \ge 0, \quad \forall \ell \in L^{\text {AC}}, \end{aligned}$$
$$\begin{aligned}&0 \le - \sum _n A_{\ell n} \lambda _n - \underline{\xi }_{\ell }^{\text {DC}} + {\overline{\xi }}_{\ell }^{\text {DC}} \perp z_{\ell } \ge 0, \quad \forall \ell \in L^{\text {DC}}, \end{aligned}$$
$$\begin{aligned}&0 = - \sum _{\ell \in L^{\text {AC}}} B_{\ell } A_{\ell n} \lambda _n^{\text {F}} \perp \delta _{n} \; \text {free}, \forall n \backslash n = 1, \end{aligned}$$
$$\begin{aligned}&0 = - \sum _{\ell \in L^{\text {AC}}} B_{\ell } A_{\ell n} \lambda _n^{\text {F}} + \lambda ^{\text {ref}} \perp \delta _{1} \; \text {free}, n = 1. \end{aligned}$$

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)


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

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Delikaraoglou, S., Pinson, P. Optimal allocation of HVDC interconnections for exchange of energy and reserve capacity services. Energy Syst 10, 635–675 (2019).

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  • Electricity markets
  • High voltage direct current (HVDC)
  • Transmission capacity allocation
  • Reserve capacity
  • Stochastic programming
  • Bilevel programming