Market Clearing Problem

This problem is defined at period $t$ and scenario $\omega$.

Sets

Lists new sets, not present in the centralized operation subproblems.

$K(i, n)$: Set of segment offers at node $n$ for the asset owner $i$.
$K^M(i)$: Set of bid profiles for the asset owner $i$.
$\mathcal{K}_m(i)$: Complementary grouping $m$ for the asset owner $i$.
$J^{VR}$: Set of virtual reservoirs.
$J^H_VR$: Set of hydro units associated with some virtual reservoir.
$J^H_{VR}(r)$: Set of hydro units associated with virtual reservoir $r$.
$I^{VR}(r)$: Set of asset owners associated with virtual reservoir $r$.
$K^{VR}(r, i)$: Set of segment offers at virtual reservoir $r$ for the asset owner $i$.
$P^{WG}(r)$: Set of waveguide points for virtual reservoir $r$.

Parameters

We add the following parameters to the list of parameters of the strategic subproblem:

Flexible Bids

$P_{i, n, \tau, k}(\omega)$: Price offer of asset owner $i$ on network node $n$ during subperiod $\tau$, segment $k$ and scenario $\omega$.
$Q_{i, n, \tau, k}(\omega)$: Quantity offer of asset owner $i$ on network node $n$ during subperiod $\tau$, segment $k$ and scenario $\omega$.

Profile Bids

$P^M_{i, n, k}(\omega)$: Price offer of asset owner $i$ on network node $n$, profile $k$ and scenario $\omega$.
$Q^M_{i, n, \tau, k}(\omega)$: Quantity offer of asset owner $i$ on network node $n$ during subperiod $\tau$, profile $k$ and scenario $\omega$.
$\mathcal{p}(k)$: Parent profile of profile $k$.
$X_{i, k}(\omega)$: Minimum activation level of profile $k$ of asset owner $i$ on network node $n$ and scenario $\omega$.

Virtual Reservoirs

$P^{VR}_{r, i, k}(\omega)$: Price offer of asset owner $i$ on virtual reservoir $r$ for segment $k$ at scenario $\omega$.
$Q^{VR}_{r, i, k}(\omega)$: Quantity offer of asset owner $i$ on virtual reservoir $r$ for segment $k$ at scenario $\omega$.
$v^{WG}_{r, h, p}$: Hydro $h$ volume at waveguide point $p$ of virtual reservoir $r$.
$E^{in}_{r,i}$: Energy account of asset owner $i$ on virtual reservoir $r$ at beginning of period.
$\zeta_{r,h}$: Factor that converts the water volume at hydro unit $h$, where $h \in J^H_{VR}(r)$, into energy.
$e^{inflow}_r$: Additional energy from inflow water, at virtual reservoir $r$.
$\gamma^{VR}_{r,i}$: Inflow shares of asset owner $i$ at virtual reservoir $r$. $\sum_{i \in I^{VR}(r)} \gamma^{VR}_{r,i} = 1 \; \forall r \in J^{VR}$.

Variables

Lists new variables, not present in the centralized operation subproblems.

Flexible Bids

$\lambda_{i, n, \tau, k}$: Linear combination coefficients for segment offer $k$ of asset owner $i$ on network node $n$ during subperiod $\tau$.
$q_{i, n, \tau, k}$: Total energy generated by asset owner $i$ on network node $n$ during subperiod $\tau$ related to the segment offer $k$.

Multi-hour Bids

$\lambda^M_{i, k}$: Convex combination coefficients for profile $k$ of asset owner $i$.
$\lambda^X_{i, k}$: Activation of profile $k$ of asset owner $i$.
$q^M_{i, n, \tau, k}$: Total energy generated by asset owner $i$ on network node $n$ during subperiod $\tau$ related to the profile $k$.

Virtual Reservoirs

$q^{VR}_{r, i, k}$: Total energy generated by asset owner $i$ on virtual reservoir $r$ related to segment offer $k$.
$\lambda^{WG}_{r, p}$: Convex combination coefficients for the waveguide point $p$ of virtual reservoir $r$.
$\delta^{WG}_{r, h}$: Hydro $h$ volume distance to waveguide points of virtual reservoir $r$.
$E^{out}_{r,i}$: Energy account of asset owner $i$ at virtual reservoir $r$ at the end of the period.

Subproblem Constraints

Offer bounds

Flexible Bids

\[ 0 \leq \lambda_{i, n , \tau, k} \leq 1 \quad \forall i \in I, n \in N, \tau \in B(t), k \in K(i, n) \\\]

Profile Bids

\[ 0 \leq \lambda^M_{i, k} \leq 1 \quad \forall i \in I, k \in K^M(i) \\\]

\[ \lambda^X_{i, k} \in \{0, 1\} \quad \forall i \in I, k \in K^M(i) \\\]

Virtual Reservoirs

\[ 0 \leq q^{VR}_{r, i, k} \leq Q^{VR}_{r, i, k}(\omega) \quad \forall r \in J^{VR}, i \in I^{VR}(r), k \in K^{VR}(r, i) \\\]

Bids Segment Curve

Flexible Bids

\[ q_{i, n, \tau, k} = \lambda_{i, n , \tau, k} Q_{i, n, \tau, k}(\omega) \quad \forall i \in I, n \in N, \tau \in B(t), k \in K(i, n) \\\]

Profile Bids

\[ q^M_{i, n, \tau, k} = \lambda^M_{i, k} Q^M_{i, n, \tau, k}(\omega) \quad \forall i \in I, n \in N, \tau \in B(t), k \in K^M(i) \\\]

Complementarity Constraints

\[ \sum_{k \in K}{\lambda^M_{i, k}} \leq 1 \quad \forall K \in \mathcal{K}_m(i), i \in I, m \in M \\\]

Precedence Relationship

\[ \lambda^M_{i, k} \leq \lambda^M_{i, \mathcal{p}(k)} \quad \forall i \in I, k \in K^M(i) \\\]

Minimum Acceptance

\[ \lambda^X_{i, k} X_{i, k}(\omega) \leq \lambda^M_{i, k} \leq \lambda^X_{i, k} \quad \forall i \in I, k \in K^M(i) \\\]

\[E^{out}_{r,i} = E^{in}_{r,i} + e^{inflow}_r \cdot \gamma^{VR}_{r,i} - \sum_{k \in K^{VR}(r, i)}{q^{VR}_{r, i, k}} \quad \forall i \in I^{VR}(r), \forall r \in J^{VR} \\ E^{out}_{r,i} \ge 0 \quad \forall i \in I^{VR}(r), \forall r \in J^{VR}\]

Physical-Virtual Coupling

The physical-virtual coupling can be done by generation:

\[ \sum_{\tau \in B(t)}{\sum_{h \in J^H_{VR}(r)}{(u_{h, \tau} + s_{h, \tau}) \cdot \rho_h \cdot C_{hm^3/h \rightarrow m^3/s}}} = \sum_{i \in I^{VR}(r)}{\sum_{k \in K^{VR}(r, i)}{q^{VR}_{r, i, k}}} \quad \forall r \in J^{VR}\]

or by volume:

\[ \sum_{h \in J^H_{VR}(r)} v_{h, \tau^{end}} \cdot \zeta_{r,h} = \sum_{i \in I^{VR}(r)} \left(E^{out}_{r,i} - \sum_{k \in K^{VR}(r,i)} q^{VR}_{r,i,k}\right) \quad \forall r \in J^{VR} \]

Waveguide Convex Combination

\[ \sum_{p \in P^{WG}(r)}{\lambda^{WG}_{r, p}} = 1 \quad \forall r \in J^{VR}\]

Waveguide Volume Distance

\[ \delta^{WG}_{r, h} \geq v_{h, \tau^{end}} - \sum_{p \in P^{WG}(r)}{\lambda^{WG}_{r, p} v^{WG}_{r, h, p}} \quad \forall r \in J^{VR}, h \in J^H_{VR}(r) \\ \delta^{WG}_{r, h} \geq \sum_{p \in P^{WG}(r)}{\lambda^{WG}_{r, p} v^{WG}_{r, h, p}} - v_{h, \tau^{end}} \quad \forall r \in J^{VR}, h \in J^H_{VR}(r) \]

Demand Balance

\[ \sum_{i \in I} \sum_{k \in K(i, n)}{q_{i, n, \tau, k}} + \sum_{i \in I} \sum_{k \in K^M(i, n)}{q^M_{i, n, \tau, k}} + \sum_{h \in J^H_{VR}}{g^H_{h, \tau}} + \sum_{l \in L^{in}(n)}{f_{l, \tau}} - \sum_{l \in L^{out}(n)}{f_{l, \tau}} + \sum_{j \in J^D(n)}{\delta_{j, \tau}} \\ = \sum_{j \in J^D(n)}{D_{j, \tau, \omega}} \quad \forall n \in N, \tau \in B(t)\]

Transmission Bounds

\[ -F_{j, \tau} \leq f_{j, \tau} \leq F_{j, \tau}, \quad \forall j \in L, \tau \in B(t)\]

Demand Deficit Bounds

\[ 0 \leq \delta_{j, \tau}, \quad \forall j \in J^D, \tau \in B(t)\]

Convex combination bounds

\[ 0 \leq \lambda^{WG}_{r, p} \leq 1 \quad \forall r \in J^{VR}, p \in P^{WG}(r)\]

Objective Function

Flexible Bids

\[ min{ \sum_{\tau \in B(t)}{ \sum_{n \in N}{ \sum_{i \in I} \sum_{k \in K(i, n)}{P_{i, n, \tau, k}(\omega) q_{i, n, \tau, k}} } } }\]

Multi-hour Bids

\[ min{ \sum_{\tau \in B(t)}{ \sum_{n \in N}{ \sum_{i \in I} \sum_{k \in K^M(i)}{P^M_{i, k}(\omega) q^M_{i, n, \tau, k}} } } }\]

Virtual Reservoirs

\[ min{ \sum_{r \in J^{VR}}{ \sum_{i \in I_{VR}(r)}{ \sum_{k \in K^{VR}(r, i)}{P^{VR}_{r, i, k}(\omega) q^{VR}_{r, i, k}} } } }\]

Hydro Units related to Virtual Reservoirs

\[ min{ \sum_{\tau \in B(t)} \sum_{h \in J^{H}_{VR}}{ C^H g^H_{h, \tau} } }\]

Waveguide distance penalty

\[ min{ \quad \varepsilon \cdot \sum_{r \in J^{VR}}{ \sum_{h \in J^H_{VR}(r)} \delta^{WG}_{r, h} } }\]

Hydro constraints penalty

\[ min{ \sum_{\tau \in B(t)}{ \sum_{j \in J^H} C^\eta \eta_{j, \tau} } }\]