66 lines
2.6 KiBLFS
Markdown
66 lines
2.6 KiBLFS
Markdown
# The Mathematical Model
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This document provides a complex number based mathematical model.
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## Sets and Parameters
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### Sets
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$$
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\begin{aligned}
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& N \text{ : buses} \\
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& R \text{ : reference buses} \\
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& E, E^R \text{ : branches, forward and reverse orientation} \\
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& G, G_i \text{ : generators and generators at bus } i \\
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& L, L_i \text{ : loads and loads at bus } i \\
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& S, S_i \text{ : shunts and shunts at bus } i \\
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\end{aligned}
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$$
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### Parameters
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$$
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\begin{aligned}
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& S^{gl}_k, S^{gu}_k \quad \forall k \in G \text{ : generator complex power bounds} \\
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& c_{2k}, c_{1k}, c_{0k} \quad \forall k \in G \text{ : generator cost components} \\
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& v^l_i, v^u_i \quad \forall i \in N \text{ : voltage bounds} \\
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& S^d_k \quad \forall k \in L \text{ : load complex power consumption} \\
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& Y^s_k \quad \forall k \in S \text{ : bus shunt admittance} \\
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& Y_{ij}, Y^c_{ij}, Y^c_{ji} \quad \forall (i,j) \in E \text{ : branch }\pi\text{-section parameters} \\
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& T_{ij} \quad \forall (i,j) \in E \text{ : branch complex transformation ratio} \\
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& s^u_{ij} \quad \forall (i,j) \in E \text{ : branch apparent power limit} \\
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& i^u_{ij} \quad \forall (i,j) \in E \text{ : branch current limit} \\
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& \theta^{\Delta l}_{ij}, \theta^{\Delta u}_{ij} \quad \forall (i,j) \in E \text{ : branch voltage angle difference bounds}
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\end{aligned}
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$$
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## Mathmatical Model
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A complete mathematical model is as follows,
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### Variables
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$$
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\begin{aligned}
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& S^g_k \quad \forall k\in G \text{ : generator complex power dispatch} \\
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& V_i \quad \forall i\in N \text{ : bus complex voltage} \\
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& S_{ij} \quad \forall (i,j) \in E \cup E^R \text{ : branch complex power flow} \\
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\\
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\end{aligned}
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$$
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### Formulation
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$$
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\begin{aligned}
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\text{minimize:} &
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\sum_{k \in G} c_{2k} (\Re(S^g_k))^2 + c_{1k}\Re(S^g_k) + c_{0k} \\
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\text{subject to:} & \\
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& \angle V_{r} = 0 \quad \forall r \in R \\
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& S^{gl}_k \leq S^g_k \leq S^{gu}_k \quad \forall k \in G \\
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& v^l_i \leq |V_i| \leq v^u_i \quad \forall i \in N \\
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& \sum_{k \in G_i} S^g_k - \sum_{k \in L_i} S^d_k - \sum_{k \in S_i} (Y^s_k)^* |V_i|^2 = \sum_{(i,j)\in E_i \cup E_i^R} S_{ij} \quad \forall i\in N \\
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& S_{ij} = (Y_{ij} + Y^c_{ij})^* \frac{|V_i|^2}{|T_{ij}|^2} - Y^*_{ij} \frac{V_i V^*_j}{T_{ij}} \quad \forall (i,j)\in E \\
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& S_{ji} = (Y_{ij} + Y^c_{ji})^* |V_j|^2 - Y^*_{ij} \frac{V^*_i V_j}{T^*_{ij}} \quad \forall (i,j)\in E \\
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& |S_{ij}| \leq s^u_{ij} \quad \forall (i,j) \in E \cup E^R \\
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& |S_{ij}| \leq |V_i| i^u_{ij} \quad \forall (i,j) \in E \cup E^R \\
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& \theta^{\Delta l}_{ij} \leq \angle (V_i V^*_j) \leq \theta^{\Delta u}_{ij} \quad \forall (i,j) \in E
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\end{aligned}
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$$
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