DC Approximation (lossless approximation) Part I

• \(L_i:\) load node \(i\)

• \(P_g:\) power generator \(g\), such that: \(P_g^{min}\le P_g\le P_g^{max}\)

• \(\theta_i:\) phase \(i\)

• \(\sum_{i=1}^{n}L_i=\sum_{g}P_g.\)

• \(\Pi_i:\) Power injection

• \(\Pi_i=\sum_{g\in G_i}P_g-L_i,\)

• where \(G_i\) is the set of generators of node \(i\).

• \(P_{ij}\) power flow from node \(i\) to node \(j\): \(P_{ij}=-P_{ji}\)

DC ApproximationPart II

• Kirchhoff Flow

• \(\Pi_i=\sum_{i\sim j}P_{ij},\)

• \(i\sim j\) means that \(i\) is a neighbor to \(j\).

• \(b_{ij}\) susceptance from node \(i\) to node \(j\).

• \(P_{ij}=b_{ij}(\theta_i-\theta_j)\)

• \(\Pi_i=\sum_{i\sim j}b_{ij}(\theta_{i}-\theta_{j}),\)

• Thermal limit of the line:

• \(\begin{aligned}-P^{max}_{ij}&\le b_{ij}(\theta_{i}-\theta_{j})\le P^{max}_{ij}\\-P^{max}_{ij}&\le P_{ij}\le P^{max}_{ij}\end{aligned}\)

OPF: Optimal Power Flow + PTDF: Power Transfer Distribution Factors Part I

\[ \begin{aligned} \min&\Biggr\{\sum_{g}c_g P_g\Biggr\}\\ \text{s.t}\\ \forall{i}&:~\Pi_i=\sum_{i\sim j}b_{ij}(\theta_{i}-\theta_{j})\\ \forall{i\sim j}&:-P^{max}_{ij}\le b_{ij}(\theta_{i}-\theta_{j})\le P^{max}_{ij}\text{ thermal limit}\\(\forall{i\sim j}&:-\theta_{\text{Max}}\le \theta_i-\theta_j\le \theta_{\text{Max}}), \end{aligned} \]

• \(c_g\) represents the cost of producing energy at generator \(g\).

OPF + PTDF Part II

• \(\Pi_i=\sum_{i\sim j}b_{ij}(\theta_{i}-\theta_{j})\) equivalent to:

• \(\begin{aligned}I_{1\times n}&=B_{n\times n}\Theta_{1\times n}\\\text{s.t}&\begin{cases}I_{1\times n}=\begin{bmatrix}\Pi_1\\\Pi_2\\\vdots\\\Pi_n\end{bmatrix}\\\Theta_{1\times n}=\begin{bmatrix}\theta_1\\\theta_2\\\vdots\\\theta_n\end{bmatrix}\\B_{n\times n}=[B_{ij}]_{n\times n}=\begin{cases}-b_{ij}&\text{ if }i\sim j\\\sum_{j\sim i}b_{ij}&\text{ if }i= j\\0&\text{ if }i\nsim j\end{cases}\end{cases}\end{aligned}.\)

• \(n=n_{nodes}\).

• \(B\) the susceptance matrix: singular matrix.

• \(\lambda=0\).

• \(v=\begin{bmatrix}1\\1\\\vdots\\1\end{bmatrix}\) eigenvector of \(B\).

• \(B\hat{\Theta}=B(\Theta+cv)=B(\Theta)+cB(v)=B(\Theta)+0=B(\Theta)=I\).

OPF + PTDF Part III

• \(A\in\mathbb{R}^{n_{nodes}\times n_{lines}}\)

• \(n_{nodes}:=\text{number of nodes}\)

• \(n_{lines}:=\text{number of lines}\)

• \([A]_i^k=A_{ik}=\begin{cases}1&\text{if node }i\text{ is the "from" node of line } k\\-1&\text{if node }i\text{ is the "to" node of line } k\\0 &\text{ otherwise}\end{cases}.\)

• \(\text{diag}\{b_k\}\)

• \(\text{diag}\{b_k\}_{n_{lines}\times n_{lines}}=\begin{bmatrix}b_{1}&0&\cdots&0\\0&b_{2}&\cdots&0\\0&0&\ddots&\vdots\\0&0&\cdots&b_{n_{lines}}\end{bmatrix}\)

• \(B=A\text{diag}\{b_k\}A^*\)

OPF + PTDF Part IV

\[ I=B\Theta=(A\text{diag}\{b_k\}A^*)\Theta \]

• Assume we know \(I\).

• \(\Theta=B^{\dagger}I\)

• \(B^{\dagger}\) pseudo-inverse of \(B\), meaning that \(B^{\dagger}=B^{\dagger}BB^{\dagger}\).

• \(P_{ij}=b_{ij}(\theta_i-\theta_j)\)

• \(f_k=P_{ij}=b_{ij}(\theta_i-\theta_j)\)

  • \(\forall{i\sim j}:-P^{max}_{ij}\le b_{ij}(\theta_{i}-\theta_{j})\le P^{max}_{ij}\)

• \(\forall{k\text{ line }}:-f^{max}_{k}\le f_{k}\le f^{max}_{k}\).

OPF + PTDF Part V

• \(F=\begin{bmatrix}f_1\\f_2\\\vdots\\f_k\end{bmatrix}\):

\(F=\begin{bmatrix}f_1\\f_2\\\vdots\\f_k\end{bmatrix}=\begin{bmatrix}b_{1}&0&\cdots&0\\0&b_{2}&\cdots&0\\0&0&\ddots&\vdots\\0&0&\cdots&b_{n_{lines}}\end{bmatrix}A^*\begin{bmatrix}\theta_1\\\theta_2\\\vdots\\\theta_n\end{bmatrix}=\{b_k\}A^*\Theta\)

• \(F=\{b_k\}A^*\Theta=\{b_k\}A^*B^{\dagger}I=\{b_k\}A^*(A\text{diag}\{b_k\}A^*)^{\dagger}I\)

• Power Transfer Distribution Factors PTDF’s : \(\{b_k\}A^*B^{\dagger}\)

• \(\Phi=[\Phi_{ik}]=\{b_k\}A^*B^{\dagger}\) such that:

• \(f_k=\sum_{i}^{n_{lines}}\Phi_{ki}\Pi_i=[\Phi\cdot\Pi]\cdot e_k,\)

OPF + PTDF Part VI

• \(P=\begin{bmatrix}P_g\end{bmatrix}_{g\in G}\)

• \(\text{gen2bus}\)

• \(\text{gen2bus}P=\begin{bmatrix}\sum_{g\in G_1}P_g\\\sum_{g\in G_2}P_g\\\vdots\\\sum_{g\in G_{n_nodes}}P_g\end{bmatrix}\)

• \(\Pi_i=\sum_{g\in G_i}P_g-L_i.\)

• \(I=\text{gen2bus}\cdot P-L~,~L= \begin{bmatrix}L_1\\L_2\\\vdots\\L_{n_{nodes}}\end{bmatrix}.\)

• \(\text{gen2bus}\cdot P=I+L=B\Theta+L.\)

OPF + PTDF Part VII

• \(\forall{k\text{ line }}:-f^{max}_{k}\le f_{k}\le f^{max}_{k}\)

\[ \begin{aligned} \forall{k\text{ line }}&:-f^{max}_{k}\le \sum_{i}^{n_{lines}}\Phi_{ki}\Pi_i\le f^{max}_{k}\\ \forall{k\text{ line }}&:-f^{max}_{k}\le \sum_{i}^{n_{lines}}\Phi_{ki}\left(\sum_{g\in G_i}P_g-L_i\right)\le f^{max}_{k}\\ \forall{k\text{ line }}&:-f^{max}_{k}\le \sum_{i}^{n_{lines}}\Phi_{ki}\sum_{g\in G_i}P_g- \sum_{i}^{n_{lines}}\Phi_{ki}L_i\le f^{max}_{k}\\ \forall{k\text{ line }}&:-f^{max}_{k}+\sum_{i}^{n_{lines}}\Phi_{ki}L_i\le \sum_{i}^{n_{lines}}\Phi_{ki}\sum_{g\in G_i}P_g\le f^{max}_{k}+\sum_{i}^{n_{lines}}\Phi_{ki}L_i \end{aligned} \]

OPF + PTDF Part VIII

• Meaning that our minimization problem can be rewritten as:

\[ \begin{aligned} \min&\Biggr\{\sum_{g}c_g P_g\Biggr\}\\ \text{s.t}\\ \forall{g\text{ generator }}&:P_{g}^{min}\le P_g \le P_{g}^{max}\\ \forall{k\text{ line }}&:-f^{max}_{k}+\sum_{i}^{n_{lines}}\Phi_{ki}L_i\le \sum_{i}^{n_{lines}}\Phi_{ki}\sum_{g\in G_i}P_g\le f^{max}_{k}+\sum_{i}^{n_{lines}}\Phi_{ki}L_i~~~~\text{thermal limit.} \end{aligned} \]

Example