# Voltage sensitivity coefficients for unbalanced radial distribution feeders

I am wondering if my implementation of voltage sensitivity coefficients for unbalanced three-phase-four-wire radial distribution feeders is correct.

We got $$\\bar{V}\$$ for phase(s) of all the buses and $$\\bar{Y}\$$, the admittance matrix of the feeder. $$\\mathcal{S}\$$ is the set for phases of the slack bus, and $$\\mathcal{N}\$$ is the set for phase(s) of all the buses. For real and reactive power injections $$\P\$$, $$\Q\$$, all wye-connected，the following relationship can be obtained using first order Taylor expansion: $$d |\bar{V}_{i}| = \sum_{j \in \mathcal{N}} \frac{\partial |\bar{V}_{i}|}{\partial P_j} \cdot d P_j + \sum_{j \in \mathcal{N}} \frac{\partial |\bar{V}_i|}{\partial Q_j} \cdot d Q_j \quad \forall i \in \mathcal{N}$$ where we should get $$\\partial \bar{V}_{i} / \partial P_{j} \$$ first, and then use: $$\frac{\partial\left|\bar{V}_{i}\right|}{\partial P_{l}}=\frac{1}{\left|\bar{V}_{i}\right|} \operatorname{Re}\left(\underline{V}_{i} \frac{\partial \bar{V}_{i}}{\partial P_{l}}\right)$$

The calculation for $$\\partial \bar{V}_{i} / \partial Q_j\$$ is similar.

In [christakou2013efficient], to calculate $$\\partial V_{i} / \partial P_j\$$, $$\(\forall i, j \in \mathcal{N} \cap \mathcal{S}) \$$, we need to solve the following set of linear equations: \begin{aligned} \frac{\partial \bar{V}_{i}}{\partial P_{k}} &= 0 \quad \forall i \in \mathcal{S}, \forall k \in \mathcal{S} \cup \mathcal{N} \\ \frac{\partial \underline{V}_{i}}{\partial P_{k}} &= 0 \quad \forall i \in \mathcal{S}, \forall k \in \mathcal{S} \cup \mathcal{N} \\ \frac{\partial \bar{V}_{i}}{\partial P_{k}} &= 0 \quad \forall i \in \mathcal{N}, \forall k \in \mathcal{S} \\ \frac{\partial \underline{V}_{i}}{\partial P_{k}} &= 0 \quad \forall i \in \mathcal{N}, \forall k \in \mathcal{S} \\ \frac{\partial \underline{V}_{i}}{\partial P_{i}} \sum_{j \in \mathcal{S} \cup \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j} \bar{V}_j) + \mathrm{Im}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_i} + \mathrm{Re}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_i} &= 1 \quad \forall i \in \mathcal{N} \\ \frac{\partial \underline{V}_{i}}{\partial P_{i}} \sum_{j \in \mathcal{S} \cup \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j} \bar{V}_j) + \mathrm{Im}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_i} + \mathrm{Re}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_i} &= 0 \quad \forall i \in \mathcal{N} \\ \frac{\partial \underline{V}_{i}}{\partial P_k} \sum_{j \in \mathcal{S} \cup \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j} \bar{V}_j) + \mathrm{Im}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_k} + \mathrm{Re}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_k} &= 0 \quad \forall i, k \in \mathcal{N}, k \neq i \\ \frac{\partial \underline{V}_{i}}{\partial P_k} \sum_{j \in \mathcal{S} \cup \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j} \bar{V}_j) + \mathrm{Im}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Re}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_k} + \mathrm{Re}(\underline{V}_i) \sum_{j \in \mathcal{N}} \mathrm{Im}(\bar{Y}_{i j}) \frac{\partial \bar{V}_{j}}{\partial P_k} &= 0 \quad \forall i, k \in \mathcal{N}, k \neq i \end{aligned} where $$\\underline{V}\$$ is the conjugate of $$\\bar{V}\$$. I am not sure about this equation set, because it is written in a simplified version in the paper. In [christakou2013efficient], there are only two equations: \begin{align} \frac{\partial \bar{V}_{i}}{\partial P_{l}} &= 0, \quad \forall i \in \mathcal{S} \\ \frac{\partial \underline{V}_{i}}{\partial P_{l}} \sum_{j \in \mathcal{S} \cup \mathcal{N}} \bar{Y}_{i j} \bar{V}_{j}+\underline{V}_{i} \sum_{j \in \mathcal{N}} \bar{Y}_{i j} \frac{\partial \bar{V}_{j}}{\partial P_{l}} &= \mathbb{1}_{\{i=l\}} \end{align}

Besides, I am wondering if there are other methods. I read some method using impedance matrix, but I don't want delta-connected loads in [maharjan2020enhanced]. I know we cannot have inverted Jacobian matrix because Newton-Raphson algorithm should not be used. I don't want to use perturb-and-observe method, for example in [tamp2014sensitivity].

• Christakou, K., LeBoudec, J. Y., Paolone, M., & Tomozei, D. C. (2013). Efficient computation of sensitivity coefficients of node voltages and line currents in unbalanced radial electrical distribution networks. IEEE Transactions on Smart Grid, 4(2), 741-750.
• Maharjan, S., Khambadkone, A. M., & Peng, J. C. H. (2020). Enhanced Z-bus method for analytical computation of voltage sensitivities in distribution networks. IET Generation, Transmission & Distribution, 14(16), 3187-3197.
• Tamp, F., & Ciufo, P. (2014). A sensitivity analysis toolkit for the simplification of MV distribution network voltage management. IEEE Transactions on Smart Grid, 5(2), 559-568.

I implemented another method from [zhou2008simplified]. Though the method is designed for balanced networks, I tried anyway.

The set of equations is: \begin{aligned} \frac{\partial \bar{V}_{i}}{\partial P_{k}} &= \sum_{j \in \mathcal{N} / \{k\}} \Re \left[\frac{-\bar{Z}_{i j} \underline{S}_{j}}{\left(\underline{V}_{j}\right)^{2}} \right] \frac{\partial \underline{V}_{j}}{\partial P_{k}} + \Re \left[\frac{\bar{Z}_{i k} \underline{V}_{k}}{\left(\underline{V}_{k} \right)^{2}} \right] \quad \forall i, k \in \mathcal{N} \\ 0 &= \sum_{j \in \mathcal{N} / \{k\}} \Im \left[\frac{-\bar{Z}_{i j} \underline{S}_{j}}{\left(\underline{V}_{j}\right)^{2}} \right] \frac{\partial \underline{V}_{j}}{\partial P_{k}} + \Im \left[\frac{\bar{Z}_{i k} \underline{V}_{k}}{\left(\underline{V}_{k} \right)^{2}} \right] \quad \forall i, k \in \mathcal{N} \\ \end{aligned} where impedance matrix, $$\\bar{Z}\$$, and complex power injections, $$\\bar{S}\$$, are used, which can be calculated using: \begin{aligned} \bar{Z} &= \bar{Y}^{-1} \\ \bar{S} &= \bar{V} \odot \underline{I} \end{aligned}

But I couldn't get any solution so far.

I am implementing the method in [zad2018new], which I believe is based on fixed-point linearization (FPL). FPL is different from first-order Taylor expansion (FOT) used in [christakou2013efficient] and [zhou2008simplified], according to [bernstein2018load].

• Zhou, Q., & Bialek, J. (2008, July). Simplified calculation of voltage and loss sensitivity factors in distribution networks. In Proc. 16th Power Syst. Comput. Conf.(PSCC2008).
• Zad, B. B., Lobry, J., & Vallée, F. (2018). A New Voltage Sensitivity Analysis Method for Medium-Voltage Distribution Systems Incorporating Power Losses Impact. Electric Power Components and Systems, 46(14-15), 1540-1553.
• Bernstein, A., Wang, C., Dall’Anese, E., Le Boudec, J. Y., & Zhao, C. (2018). Load flow in multiphase distribution networks: Existence, uniqueness, non-singularity and linear models. IEEE Transactions on Power Systems, 33(6), 5832-5843.