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Strategic Bid Problem

This problem is defined at period $t$ and scenario $\omega$ for an asset owner $i$.

Sets

Lists new or modified sets, not present in the centralized operation or price taker subproblems.

  • $I$: Set of asset owners.
  • $J^V(n, \tau)$: Set of vertices in the convex hull of the hypograph of the asset owner's revenue curve. One curve is defined for each network node $n$ and subperiod $\tau$.

The revenue curve

To obtain $J^V(n, \tau)$, a preprocessing step is required, where the inputs are:

  • The other asset owners' offers $(P_{i, n, \tau}(\omega), Q_{i, n, \tau}(\omega)) \forall j \in I | j \neq i$.
  • The demand $D_{j, \tau, \omega} \forall j \in J^D(n)$.
  • The deficit cost $C_\delta$.

Parameters

Lists new or modified parameters, not present in the centralized operation or price taker subproblems.

  • $E^R_{v, n, \tau}(\omega)$: Revenue of vertex $v \in J^V(n, \tau)$.
  • $E^Q_{v, n, \tau}(\omega)$: Quantity of vertex $v \in J^V(n, \tau)$.
  • $P_{i, n, \tau}(\omega)$: Price offer of asset owner $i$ on network node $n$ during subperiod $\tau$ and scenario $\omega$.
  • $Q_{i, n, \tau}(\omega)$: Quantity offer of asset owner $i$ on network node $n$ during subperiod $\tau$ and scenario $\omega$.

Variables

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

  • $\lambda_{v, n, \tau}$: Convex combination coefficients for vertex $v \in J^V(n, \tau)$.

Subproblem Constraints

The following constraints are defined for a subproblem at period $t$ and scenario $\omega$ for an asset owner $i$.

Revenue Curve: convex hull representation

A configuration parameter named ``aggregate buses for strategic bidding'' is defined to aggregate the other asset owners' offers $(P_{i, n, \tau}(\omega), Q_{i, n, \tau}(\omega))$ and the demand into a single bus to calculate the revenue curve. The equations below are presented for both the aggregated and non-aggregated cases. When the parameter is set to $true$, the following elements lose their bus dimension:

  • The parameters $E^R_{v, n, \tau}(\omega)$ and $E^Q_{v, n, \tau}(\omega)$
  • The set $J^V(n, \tau)$
  • The variables $\lambda_{v, n, \tau}$

Non-aggregated revenue curve

\[ \sum_{v \in J^V(n, \tau)}{E^Q_{v, n, \tau}(\omega) \lambda_{v, n, \tau}} = e_{n, \tau} \quad \forall n \in N, \tau \in B(t)\]

\[ \sum_{v \in J^V(n, \tau)}{\lambda_{v, n, \tau}} = 1 \quad \forall n \in N, \tau \in B(t)\]

\[ \lambda_{v, n, \tau} \geq 0 \quad \forall v \in J^V(n, \tau), n \in N, \tau \in B(t)\]

Aggregated revenue curve

\[ \sum_{v \in J^V(\tau)}{E^Q_{v, \tau}(\omega) \lambda_{v, n, \tau}} = \sum_{n \in N}e_{n, \tau} \quad \forall \tau \in B(t)\]

\[ \sum_{v \in J^V(\tau)}{\lambda_{v, \tau}} = 1 \quad \forall \tau \in B(t)\]

\[ \lambda_{v, \tau} \geq 0 \quad \forall v \in J^V(\tau), \tau \in B(t)\]

The remaining constraints are copied from the price taker problem.

Asset owner's total generation

\[ e_{n, \tau} = \sum_{j \in J^T_i(n)}{g^T_{j, \tau}} + \sum_{j \in J^H_i(n)}{\rho_j (u_{j, \tau})} + \sum_{j \in J^R_i(n)}{g^R_{j, \tau}} + \sum_{j \in J^B_i(n)}{g^B_{j, \tau}} \\ \quad \forall n \in N, \tau \in B(t)\]

Hydro Balance

Intra-period balance

\[ v_{j, \tau+1} = v_{j, \tau} - u_{j, \tau} - z_{j, \tau} + \sum_{n \in J^H_U(j)}{u_{n, \tau}} + \sum_{n \in J^H_Z(j)}{z_{n, \tau}} + a_{j, \tau} \quad \forall j \in J^H_i, \tau \in B(t)\]

Inter-period balance

\[ v^{S_{in}}_j = v_{j, 1} \quad \forall j \in J^H_i\]

\[ v^{S_{out}}_j = v_{j, |B(t)| + 1} \quad \forall j \in J^H_i\]

Renewable Balance

\[ g^R_{j, \tau} + y^r_{j, \tau} = G^R_{j, \tau}(\omega) \quad \forall j \in J^R_i, \tau \in B(t)\]

Battery Unit Balance

Intra-period balance

\[ s^b_{j, \tau+1} = s^b_{j, \tau} - g^B_{j, \tau} \quad \forall j \in J^B_i, \tau \in B(t)\]

Inter-period balance

\[ s^{B_{in}}_j = s^b_{j, 1} \quad \forall j \in J^B_i\]

\[ s^{B_{out}}_j = s^b_{j, |B(t)| + 1} \quad \forall j \in J^B_i\]

Hydro Bounds

Volume bounds

\[ 0 \leq v_{j, \tau} \leq V_j, \quad \forall j \in J^H_i, \tau = 1, ..., |B(t)| + 1\]

Other bounds

\[ 0 \leq u_{j, \tau} \leq U_j, \quad 0 \leq z_{j, \tau} , \quad \forall j \in J^H_i, \tau \in B(t)\]

Thermal Bounds

\[ 0 \leq g^T_{j, \tau} \leq G^T_j, \quad \forall j \in J^T_i, \tau \in B(t)\]

Renewable bounds

\[ 0 \leq g^R_{j, \tau} \leq G^R_j, \quad 0 \leq y^r_{j, \tau} \leq G^R_j, \quad \forall j \in J^R_i\]

Battery Unit bounds

\[ -G^B_j \leq g^B_{j, \tau} \leq G^B_j, \quad 0 \leq s^b_{j, \tau} \leq S^B_j, \quad \forall j \in J^B_i, \tau \in B(t)\]

Objective Function

The objective function is similar to the price taker problem, but replaces the exogenous spot price $\pi_{n, \tau}(\omega)$ with the convex revenue representation. The equation presented below is for the non-aggregated case.

\[ min{ \sum_{\tau \in B(t)}{\left( - \sum_{n \in N}{\left( \sum_{v \in J^V(n, \tau)}{\lambda_{v, n, \tau} E^R_{v, n, \tau}(\omega)} \right)} + \sum_{j \in J^T_i}{C^T_j g^T_{j, \tau}} + \sum_{j \in J^R_i}{C^R_j y^r_{j, \tau}} \right)} }\]

When aggregating buses to calculate the revenue curve, the objective function becomes:

\[ min{ \sum_{\tau \in B(t)}{\left( - \sum_{v \in J^V(\tau)}{\lambda_{v, \tau} E^R_{v, \tau}(\omega)} + \sum_{j \in J^T_i}{C^T_j g^T_{j, \tau}} + \sum_{j \in J^R_i}{C^R_j y^r_{j, \tau}} \right)} }\]