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Update README.md
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@colonetti
colonetti authored Feb 29, 2024
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Expand Up @@ -40,24 +40,24 @@ $$

$$
PTDF = \begin{bmatrix}
0 & -0.6152& -0.5636& -0.5121& -0.3848& -0.4121& -0.503 & -0.503 & -0.4939& -0.4667& -0.4394& -0.4121& -0.4121& -0.4121\\
0 & -0.3848& -0.4364& -0.4879& -0.6152& -0.5879& -0.497 & -0.497 & -0.5061& -0.5333& -0.5606& -0.5879& -0.5879& -0.5879\\
0 & 0.0515& -0.5636& -0.1788& -0.0515& -0.0788& -0.1697& -0.1697& -0.1606& -0.1333& -0.1061& -0.0788& -0.0788& -0.0788\\
0 & 0.103 & -0.1273& -0.3576& -0.103 & -0.1576& -0.3394& -0.3394& -0.3212& -0.2667& -0.2121& -0.1576& -0.1576& -0.1576\\
0 & 0.2303& 0.1273& 0.0242& -0.2303& -0.1758& 0.0061& 0.0061& -0.0121& -0.0667& -0.1212& -0.1758& -0.1758& -0.1758\\
0 & 0.0515& 0.4364& -0.1788& -0.0515& -0.0788& -0.1697& -0.1697& -0.1606& -0.1333& -0.1061& -0.0788& -0.0788& -0.0788\\
0 & 0.1273& 0.2545& 0.3818& -0.1273& -0.0182& 0.3455& 0.3455& 0.3091& 0.2 & 0.0909& -0.0182& -0.0182& -0.0182\\
0 & 0.0091& 0.0182& 0.0273& -0.0091& -0.0727& -0.6182& -0.6182& -0.2636& -0.2 & -0.1364& -0.0727& -0.0727& -0.0727\\
0 & 0.0182& 0.0364& 0.0545& -0.0182& -0.1455& -0.2364& -0.2364& -0.5273& -0.4 & -0.2727& -0.1455& -0.1455& -0.1455\\
0 & -0.0273& -0.0545& -0.0818& 0.0273& -0.7818& -0.1455& -0.1455& -0.2091& -0.4 & -0.5909& -0.7818& -0.7818& -0.7818\\
0 & -0.0273& -0.0545& -0.0818& 0.0273& 0.2182& -0.1455& -0.1455& -0.2091& -0.4 & -0.5909& 0.2182& 0.2182& 0.2182\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1. & -1. & -1. \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1. & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.0091& 0.0182& 0.0273& -0.0091& -0.0727& 0.3818& 0.3818& -0.2636& -0.2 & -0.1364& -0.0727& -0.0727& -0.0727\\
0 & 0.0273& 0.0545& 0.0818& -0.0273& -0.2182& 0.1455& 0.1455& 0.2091& -0.6 & -0.4091& -0.2182& -0.2182& -0.2182\\
0 & 0.0273& 0.0545& 0.0818& -0.0273& -0.2182& 0.1455& 0.1455& 0.2091& 0.4 & -0.4091& -0.2182& -0.2182& -0.2182\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1. & -1. \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1.\\
0.615& 0 & 0.052& 0.103& 0.23 & 0.203& 0.112& 0.112& 0.121& 0.148& 0.176& 0.203& 0.203& 0.203\\
0.385& 0 & -0.052& -0.103& -0.23 & -0.203& -0.112& -0.112& -0.121& -0.148& -0.176& -0.203& -0.203& -0.203\\
-0.052& 0 & -0.615& -0.23 & -0.103& -0.13 & -0.221& -0.221& -0.212& -0.185& -0.158& -0.13 & -0.13 & -0.13 \\
-0.103& 0 & -0.23 & -0.461& -0.206& -0.261& -0.442& -0.442& -0.424& -0.37 & -0.315& -0.261& -0.261& -0.261\\
-0.23 & 0 & -0.103& -0.206& -0.461& -0.406& -0.224& -0.224& -0.242& -0.297& -0.352& -0.406& -0.406& -0.406\\
-0.052& 0 & 0.385& -0.23 & -0.103& -0.13 & -0.221& -0.221& -0.212& -0.185& -0.158& -0.13 & -0.13 & -0.13 \\
-0.127& 0 & 0.127& 0.255& -0.255& -0.145& 0.218& 0.218& 0.182& 0.073& -0.036& -0.145& -0.145& -0.145\\
-0.009& 0 & 0.009& 0.018& -0.018& -0.082& -0.627& -0.627& -0.273& -0.209& -0.145& -0.082& -0.082& -0.082\\
-0.018& 0 & 0.018& 0.036& -0.036& -0.164& -0.255& -0.255& -0.545& -0.418& -0.291& -0.164& -0.164& -0.164\\
0.027& 0 & -0.027& -0.055& 0.055& -0.755& -0.118& -0.118& -0.182& -0.373& -0.564& -0.755& -0.755& -0.755\\
0.027& 0 & -0.027& -0.055& 0.055& 0.245& -0.118& -0.118& -0.182& -0.373& -0.564& 0.245& 0.245& 0.245\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-0.009& 0 & 0.009& 0.018& -0.018& -0.082& 0.373& 0.373& -0.273& -0.209& -0.145& -0.082& -0.082& -0.082\\
-0.027& 0 & 0.027& 0.055& -0.055& -0.245& 0.118& 0.118& 0.182& -0.627& -0.436& -0.245& -0.245& -0.245\\
-0.027& 0 & 0.027& 0.055& -0.055& -0.245& 0.118& 0.118& 0.182& 0.373& -0.436& -0.245& -0.245& -0.245\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
\end{bmatrix}
$$

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$$
\begin{align}
{-}10 \leq 0.103 \cdot \left(p_2 + s_2 - 1 \right) -0.1273 \cdot \left(p_3 + s_3 - 1 \right) -0.3576 \cdot \left(s_4 - 1 \right) -0.103 \cdot \left(s_5 - 1 \right) -0.1576 \cdot \left(p_4 + s_6 - 1 \right) -0.3394 \cdot p_5 -0.3212 \cdot \left(s_9 - 1 \right) -0.2667\cdot \left(s_{10} - 1 \right) -0.2121\cdot \left(s_{11} - 1 \right) -0.1576\cdot \left(s_{12} - 1 \right) -0.1576\cdot \left(s_{13} - 1 \right) -0.1576 \cdot \left(s_{14} - 1 \right) \leq 10 & \qquad \text{(l = 4)}\\
{-}10 \leq -0.103 \cdot p_1 -0.23 \cdot (p_3 + s_3 - 1) -0.461 \cdot (s_4 - 1 ) -0.206 \cdot (s_5 - 1) -0.261 \cdot (p_4 + s_6 - 1) -0.442 \cdot p_5 -0.424 \cdot (s_9 - 1) -0.37 \cdot (s_{10} - 1) -0.315 \cdot (s_{11} - 1) -0.261 \cdot (s_{12} - 1) -0.261 \cdot (s_{13} - 1) -0.261 \cdot (s_{14} - 1) \leq 10 & \qquad \text{(l = 4)}\\
{-}10 \leq - p_5 \leq 10 & \qquad \text{(l = 13)}\\
{-}10 \leq 0.0091 \cdot \left(p_2 + s_2 - 1 \right) + 0.0182 \cdot \left(p_3 + s_3 - 1 \right) + 0.0273 \cdot \left(s_4 - 1 \right) -0.0091 \cdot \left(s_5 - 1 \right) -0.0727 \cdot \left(p_4 + s_6 - 1 \right) + 0.3818 \cdot p_5 -0.2636 \cdot \left(s_9 - 1 \right) -0.2\cdot \left(s_{10} - 1 \right) -0.1364\cdot \left(s_{11} - 1 \right) -0.0727\cdot \left(s_{12} - 1 \right) -0.0727\cdot \left(s_{13} - 1 \right) -0.0727 \cdot \left(s_{14} - 1 \right) \leq 10 & \qquad \text{(l = 14)}\\
{-}10 \leq 0.0273 \cdot \left(p_2 + s_2 - 1 \right) + 0.0545 \cdot \left(p_3 + s_3 - 1 \right) + 0.0818 \cdot \left(s_4 - 1 \right) -0.0273 \cdot \left(s_5 - 1 \right) -0.2182 \cdot \left(p_4 + s_6 - 1 \right) + 0.1455 \cdot p_5 + 0.2091 \cdot \left(s_9 - 1 \right) + 0.4\cdot \left(s_{10} - 1 \right) -0.4091\cdot \left(s_{11} - 1 \right) -0.2182\cdot \left(s_{12} - 1 \right) -0.2182\cdot \left(s_{13} - 1 \right) -0.2182 \cdot \left(s_{14} - 1 \right) \leq 10 & \qquad \text{(l = 16)}\\
{-}10 \leq -0.009 \cdot p_1 + 0.009 \cdot (p_3 + s_3 - 1) + 0.018 \cdot (s_4 - 1 ) -0.018 \cdot (s_5 - 1) -0.082 \cdot (p_4 + s_6 - 1) + 0.373 \cdot p_5 -0.273 \cdot (s_9 - 1) -0.209 \cdot (s_{10} - 1) -0.145 \cdot (s_{11} - 1) -0.082 \cdot (s_{12} - 1) -0.082 \cdot (s_{13} - 1) -0.082 \cdot (s_{14} - 1) \leq 10 & \qquad \text{(l = 14)}\\
{-}10 \leq -0.027 \cdot p_1 + 0.027 \cdot (p_3 + s_3 - 1) + 0.055 \cdot (s_4 - 1 ) -0.055 \cdot (s_5 - 1) -0.245 \cdot (p_4 + s_6 - 1) + 0.118 \cdot p_5 + 0.182 \cdot (s_9 - 1) + 0.373 \cdot (s_{10} - 1) -0.436 \cdot (s_{11} - 1) -0.245 \cdot (s_{12} - 1) -0.245 \cdot (s_{13} - 1) -0.245 \cdot (s_{14} - 1) \leq 10 & \qquad \text{(l = 16)}\\
\end{align}
$$

Expand Down Expand Up @@ -170,13 +170,21 @@ $$
$$

$$
\begin{align}
{-}10 \leq 0.103 \cdot \left(p_2 + s_2 - 1 \right) -0.1273 \cdot \left(p_3 + s_3 - 1 \right) -0.3576 \cdot \left(s_4 - 1 \right) -0.103 \cdot \left(s_5 - 1 \right) -0.1576 \cdot \left(p_4 + s_6 - 4 \right) -0.3394 \cdot p_5 -0.3212 \cdot \left(s_9 - 1 \right) -0.2667\cdot \left(s_{10} - 1 \right) -0.2121\cdot \left(s_{11} - 1 \right) \leq 10 & \qquad \text{(l = 4)}\\
{-}10 \leq 0.0091 \cdot \left(p_2 + s_2 - 1 \right) + 0.0182 \cdot \left(p_3 + s_3 - 1 \right) + 0.0273 \cdot \left(s_4 - 1 \right) -0.0091 \cdot \left(s_5 - 1 \right) -0.0727 \cdot \left(p_4 + s_6 - 4 \right) + 0.3818 \cdot p_5 -0.2636 \cdot \left(s_9 - 1 \right) -0.2\cdot \left(s_{10} - 1 \right) -0.1364\cdot \left(s_{11} - 1 \right) \leq 10 & \qquad \text{(l = 14)}\\
{-}10 \leq 0.0273 \cdot \left(p_2 + s_2 - 1 \right) + 0.0545 \cdot \left(p_3 + s_3 - 1 \right) + 0.0818 \cdot \left(s_4 - 1 \right) -0.0273 \cdot \left(s_5 - 1 \right) -0.2182 \cdot \left(p_4 + s_6 - 4 \right) + 0.1455 \cdot p_5 + 0.2091 \cdot \left(s_9 - 1 \right) + 0.4\cdot \left(s_{10} - 1 \right) -0.4091\cdot \left(s_{11} - 1 \right) \leq 10 & \qquad \text{(l = 16)}\\
\end{align}
$$

PTDF = \begin{bmatrix}
0 & -0.615 & -0.23 & -0.103 & -0.13 & -0.221 & -0.212 & -0.185 & -0.158\\
0 & -0.23 & -0.461 & -0.206 & -0.261 & -0.442 & -0.424 & -0.37 & -0.315\\
0 & 0.385 & -0.23 & -0.103 & -0.13 & -0.221 & -0.212 & -0.185 & -0.158\\
0 & 0.127 & 0.255 & -0.255 & -0.145 & 0.218 & 0.182 & 0.073 & -0.036\\
0 & 0.009 & 0.018 & -0.018 & -0.082 & -0.627 & -0.273 & -0.209 & -0.145\\
0 & 0.018 & 0.036 & -0.036 & -0.164 & -0.255 & -0.545 & -0.418 & -0.291\\
0 & -0.027 & -0.055 & 0.055 & -0.755 & -0.118 & -0.182 & -0.373 & -0.564\\
0 & -0.027 & -0.055 & 0.055 & 0.245 & -0.118 & -0.182 & -0.373 & -0.564\\
0 & 0.009 & 0.018 & -0.018 & -0.082 & 0.373 & -0.273 & -0.209 & -0.145\\
0 & 0.027 & 0.055 & -0.055 & -0.245 & 0.118 & 0.182 & -0.627 & -0.436\\
0 & 0.027 & 0.055 & -0.055 & -0.245 & 0.118 & 0.182 & 0.373 & -0.436\\
0 & -0.155 & -0.309 & -0.691 & -0.609 & -0.336 & -0.364 & -0.445 & -0.527\\
\end{bmatrix}
$$



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