Distflow based state estimation for power distribution networks

Abstract

A new state estimation method for electrical power distribution systems using the Distflow formulation and the Weighted Least Square method to determine the steady-state operating point is presented. In order to reduce the number of measurements needed for state estimation analysis, a special set of state variables is defined. The proposed methodology is shown to be able to successfully determine the operating conditions of a electrical power distribution system with high automation levels. The proposed approach is tested on the IEEE-37 and IEEE-123 bus test system, reducing the number of state variables up to 60% when compared with conventional state estimation method.

Structure of Jacobian matrix H

Fig. 2

Magnitudes upstream of section ij of an EPDS

In the same way that the backward sweep formulation shown in (1–2), the power flows $P_{ij}$ and $Q_{ij}$ can be calculated as show in Fig. 2:

$$\begin{aligned}&P_{ij}=\sum _{li \in \varOmega L} \left( P_{li} - P_{li}^L \right) - \sum _{ \begin{array}{c} im \in \varOmega L \\ m \ne j \end{array}} P_{im} - P_i^D + P_i^G \end{aligned}$$

(11)

$$\begin{aligned}&Q_{ij}=\sum _{li \in \varOmega L} \left( Q_{li} - Q_{li}^L \right) - \sum _{ \begin{array}{c} im \in \varOmega L \\ m \ne j \end{array}} Q_{im} - Q_i^D + Q_i^G \end{aligned}$$

(12)

where,

$$\begin{aligned}&P_{li}^L = R_{li}\left( \frac{P_{li}^2 + Q_{li}^2}{V_l^2} \right) \end{aligned}$$

(13)

$$\begin{aligned}&Q_{li}^L = X_{li}\left( \frac{P_{li}^2 + Q_{li}^2}{V_l^2} \right) \end{aligned}$$

(14)

Thus, according to the Backward/Forward Distflow formulation, it is possible to obtain the following terms from $H({\hat{x}})$ directly:

$$\begin{aligned}&\frac{\partial P_{ij}}{\partial P_i^G} = \frac{\partial Q_{ij}}{\partial Q_i^G} = \frac{\partial P_{ij}}{\partial P_j^D} = \frac{\partial Q_{ij}}{\partial Q_j^D} = 1 \end{aligned}$$

(15)

$$\begin{aligned}&\frac{\partial P_{ij}}{\partial P_j^G} = \frac{\partial Q_{ij}}{\partial Q_j^G} = \frac{\partial P_{ij}}{\partial P_i^D} = \frac{\partial Q_{ij}}{\partial Q_i^D} = -1 \end{aligned}$$

(16)

$$\begin{aligned}&\frac{\partial P_{ij}}{\partial Q_i^G} = \frac{\partial Q_{ij}}{\partial P_i^G} = \frac{\partial P_{ij}}{\partial Q_j^G} = \frac{\partial Q_{ij}}{\partial P_j^G} = 0 \end{aligned}$$

(17)

$$\begin{aligned}&\frac{\partial P_{ij}}{\partial Q_i^D} = \frac{\partial Q_{ij}}{\partial P_i^D} = \frac{\partial P_{ij}}{\partial Q_j^D} = \frac{\partial Q_{ij}}{\partial P_j^D} = 0 \end{aligned}$$

(18)

However, due to the lack of equations that relate the magnitudes to the state variables of the system, the chain rule is used to determine the elements of H [17].

1.1 Voltage measurements

The elements of H correspondent to the measured voltage $V_j^2$ are calculated as shown below.

$$\begin{aligned} \frac{\partial V_j^2}{\partial V_0^2}= & {} \left( \frac{\partial V_j^2}{\partial V_{i}^2}\right) \left( \frac{\partial V_{i}^2}{\partial V_{(i-1)}^2}\right) \cdots \left( \frac{\partial V_{(k+1)}^2}{\partial V_{0}^2}\right) \end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_m^G}= & {} \left( \frac{\partial V_j^2}{\partial P_{ij}}\right) \left( \frac{\partial P_{ij}}{\partial P_{(i-1)i}}\right) \cdots \left( \frac{\partial P_{m(m+1)}}{\partial P_{m}^G}\right) \end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_m^D}= & {} \left( \frac{\partial V_j^2}{\partial P_{ij}}\right) \left( \frac{\partial P_{ij}}{\partial P_{(i-1)i}}\right) \cdots \left( \frac{\partial P_{m(m+1)}}{\partial P_{m}^D}\right) \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_m^G}= & {} \left( \frac{\partial V_j^2}{\partial Q_{ij}}\right) \left( \frac{\partial Q_{ij}}{\partial Q_{(i-1)i}}\right) \cdots \left( \frac{\partial Q_{m(m+1)}}{\partial Q_{m}^G}\right) \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_m^D}= & {} \left( \frac{\partial V_j^2}{\partial Q_{ij}}\right) \left( \frac{\partial Q_{ij}}{\partial Q_{(i-1)i}}\right) \cdots \left( \frac{\partial Q_{m(m+1)}}{\partial Q_{m}^D}\right) \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_n^G}= & {} \left( \frac{\partial V_j^2}{\partial P_{ij}}\right) \left( \frac{\partial P_{ij}}{\partial P_{j(j+1)}}\right) \cdots \left( \frac{\partial P_{(n-1)n}}{\partial P_{n}^G}\right) \end{aligned}$$

(24)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_n^D}= & {} \left( \frac{\partial V_j^2}{\partial P_{ij}}\right) \left( \frac{\partial P_{ij}}{\partial P_{j(j+1)}}\right) \cdots \left( \frac{\partial P_{(n-1)n}}{\partial P_{n}^D}\right) \end{aligned}$$

(25)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_n^G}= & {} \left( \frac{\partial V_j^2}{\partial Q_{ij}}\right) \left( \frac{\partial Q_{ij}}{\partial Q_{j(j+1)}}\right) \cdots \left( \frac{\partial Q_{(n-1)n}}{\partial Q_{n}^G}\right) \end{aligned}$$

(26)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_n^D}= & {} \left( \frac{\partial V_j^2}{\partial Q_{ij}}\right) \left( \frac{\partial Q_{ij}}{\partial Q_{j(j+1)}}\right) \cdots \left( \frac{\partial Q_{(n-1)n}}{\partial Q_{n}^D}\right) \end{aligned}$$

(27)

Therefore, the high computational effort necessary to evaluate the elements of H for each section of the system is clear. This disadvantage can be mitigated as follows:

*:

Considering that in per unit $Z_{ij}^2 \ll 1$, $\forall _{ij} \in \varOmega L$, we have:

$$\begin{aligned} \frac{\partial V_j^2}{\partial V_{i}^2}= & {} 1 + Z_{ij}^2 \left( \frac{P_{ij}^2 + Q_{ij}^2}{V_{i}^4} \right) \approx 1 \end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_{ij}}= & {} -2R_{ij} + 2P_{ij}Z_{ij}^2 \approx -2R_{ij} \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_{ij}}= & {} -2X_{ij} + 2Q_{ij}Z_{ij}^2 \approx -2X_{ij} \end{aligned}$$

(30)

*:

Considering that in per unit the terms: $R_{ij}\frac{P_{ij}}{V_i^2}\ll 1$, $R_{ij}\frac{P_{ij}}{V_j^2}\ll 1$, $X_{ij}\frac{Q_{ij}}{V_i^2}\ll 1$ and $X_{ij}\frac{Q_{ij}}{V_j^2}\ll 1$, $\forall _{ij} \in \varOmega L$, we have:

$$\begin{aligned} \frac{\partial P_{ij}}{\partial P_{j(j+1)}}= & {} 1 + 2R_{j(j+1)}\left( \frac{P_{j(j+1)}}{V_{j+1}^2}\right) \approx 1 \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial P_{ij}}{\partial P_{(i-1)i}}= & {} 1 - 2R_{(i-1)i}\left( \frac{P_{(i-1)i}}{V_{i-1}^2}\right) \approx 1 \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial Q_{ij}}{\partial Q_{j(j+1)}}= & {} 1 + 2X_{j(j+1)}\left( \frac{Q_{j(j+1)}}{V_{j+1}^2}\right) \approx 1 \end{aligned}$$

(33)

$$\begin{aligned} \frac{\partial Q_{ij}}{\partial Q_{(i-1)i}}= & {} 1 - 2X_{(i-1)i}\left( \frac{Q_{(i-1)i}}{V_{i-1}^2}\right) \approx 1 \end{aligned}$$

(34)

Thus, the elements of H shown in (19–27) are calculated as follows:

$$\begin{aligned} \frac{\partial V_j^2}{\partial V_0^2}\approx & {} 1 \end{aligned}$$

(35)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_m^G} = - \frac{\partial V_i^2}{\partial P_m^D}\approx & {} 2R_{ij} \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_m^G} = - \frac{\partial V_i^2}{\partial Q_m^D}\approx & {} 2X_{ij} \end{aligned}$$

(37)

$$\begin{aligned} \frac{\partial V_j^2}{\partial P_n^G} = - \frac{\partial V_i^2}{\partial P_n^D}\approx & {} -2R_{ij} \end{aligned}$$

(38)

$$\begin{aligned} \frac{\partial V_j^2}{\partial Q_n^G} = - \frac{\partial V_i^2}{\partial Q_n^D}\approx & {} -2X_{ij} \end{aligned}$$

(39)

1.2 Power flow measurements

Considering that in per unit the terms $R_{ij}\left( \frac{P_{ij}^2+Q_{ij}^2}{V_i^4}\right) \ll 1$, e $X_{ij}\left( \frac{P_ij^2+Q_ij^2}{V_i^4}\right) \ll 1$, we have:

$$\begin{aligned} \frac{\partial P_{ij}}{\partial V_0^2} = R_{i(i-1)}\left( \frac{\partial P_{i(i-1)}^2 + Q_{i(i-1)}^2}{\partial V_{(i-1)}^4}\right) \approx 0 \end{aligned}$$

(40)

$$\begin{aligned} \frac{\partial Q_{ij}}{\partial V_0^2} = X_{i(i-1)}\left( \frac{\partial P_{i(i-1)}^2 + Q_{i(i-1)}^2}{\partial V_{(i-1)}^4}\right) \approx 0 \end{aligned}$$

(41)

1.3 Power injection measurements

$$\begin{aligned} \frac{\partial P_i^G}{\partial P_m^G} = \frac{\partial Q_i^G}{\partial Q_m^G} = \frac{\partial P_i^D}{\partial P_m^D} = \frac{\partial Q_i^D}{\partial Q_m^D} = \left\{ \begin{array}{cc} 1, &{} if i=m \\ &{} \\ 0, &{} if i \ne m \end{array}\right. \end{aligned}$$

(42)

$$\begin{aligned} \frac{\partial P_i^G}{\partial Q_m^G} = \frac{\partial P_i^G}{\partial Q_m^D} = \frac{\partial P_i^G}{\partial V_0^2} = \frac{\partial Q_i^G}{\partial P_m^G} = \frac{\partial Q_i^G}{\partial P_m^D} = \frac{\partial Q_i^G}{\partial V_0^2} = 0 \end{aligned}$$

(43)

$$\begin{aligned} \frac{\partial P_i^D}{\partial Q_m^G} = \frac{\partial P_i^D}{\partial Q_m^D} = \frac{\partial P_i^D}{\partial V_0^2} = \frac{\partial Q_i^D}{\partial P_m^G} = \frac{\partial Q_i^D}{\partial P_m^D} = \frac{\partial Q_i^D}{\partial V_0^2} = 0 \end{aligned}$$

(44)

1.4 Current measurements

$$\begin{aligned} \frac{\partial I_{ij}^2}{\partial P_m^G}= & {} -\frac{\partial I_{ij}^2}{\partial P_m^D} \approx \frac{2P_{ij}}{V_j} \end{aligned}$$

(45)

$$\begin{aligned} \frac{\partial I_{ij}^2}{\partial Q_m^G}= & {} -\frac{\partial I_{ij}^2}{\partial Q_m^D} \approx \frac{2Q_{ij}}{V_j} \end{aligned}$$

(46)

$$\begin{aligned} \frac{\partial I_{ij}^2}{\partial P_n^G}= & {} -\frac{\partial I_{ij}^2}{\partial P_n^D} \approx -\frac{2P_{ij}}{V_j} \end{aligned}$$

(47)

$$\begin{aligned} \frac{\partial I_{ij}^2}{\partial Q_n^G}= & {} -\frac{\partial I_{ij}^2}{\partial Q_n^D} \approx -\frac{2Q_{ij}}{V_j} \end{aligned}$$

(48)

$$\begin{aligned} \frac{\partial I_{ij}^2}{\partial V_0^2}\approx & {} \frac{P_{ij}^2+Q_{ij}^2}{V_i^2} \end{aligned}$$

(49)