Unbalanced Power Flow Analysis in Distribution Systems Using TRX Matrix: Implementation Using DIgSILENT Programming Language

Abstract

In this chapter, a generalized model to solve multigrounded distribution networks under DIgSILENT platform taking into account current flows and voltages across neutral and ground paths is presented. The fundamental idea developed in this work is solving the 5-wire power flow problem using a unique matrix called TRX that includes all feeder characteristics related to phase, neutral, and equivalent ground paths. Proposed power flow routines are run from MATLAB engine using DIgSILENT databases. The method is suitable to be applied on unbalanced aerial and underground distribution systems with distributed generation. Proposed methodology has been successfully applied in an illustrative test system.

Appendices

Appendix A—Distribution Line Model

The nature of the distribution line requires an exact model in order to identify the self- and mutual impedances of the conductor taking into account the ground path for the unbalanced currents. Figure 4.6 shows a schematic distribution line section between nodes i and j (branch i–j). It consists of 4 wires: three phases (a, b, c) and neutral (n) with their respective self- and mutual impedance terms.

Fig. 4.6

Four-wire multigrounded distribution segment line

To include multigrounded neutrals, an equivalent conductor (g) that emulates the effect of ground return is required. The earth is a conductor of non-uniform conductivity. Thus, the ground current distribution is not uniform. The efforts aim to calculate the impedance of the phase and neutral conductors with ground-return effect. Researchers have faced and overcome this problem using different assumptions and methods. Carson’s series-based asymptotic approximation equations [3] are generally accepted as the best, considering the following assumptions: The conductors are parallel to the ground, and the earth is a solid with a plane surface, infinite in extent, and of uniform conductivity [16]. Advanced approaches have been also proposed to avoid truncation errors due to cases with wide separation among conductors, frequency higher than power frequency, or low earth resistivity [17]. The key issue is how to model equivalent conductor that emulates the effect of ground return in order to obtain self- and mutual ground impedances $\widehat{Z}_{gg}^{ij}$ and $\widehat{Z}_{qg}^{ij}$ for $q = a,b,c,n$.

According to Anderson [18], the extended primitive impedance matrix $\widehat{{\mathbf{Z}}}^{{{\mathbf{ij}}}}$ in Ω per unit length is given by

$$\widehat{{\mathbf{Z}}}^{{{\mathbf{ij}}}} = \left[ {\begin{array}{*{20}c} {\widehat{Z}_{aa}^{ij} } & {\widehat{Z}_{ab}^{ij} } & {\widehat{Z}_{ac}^{ij} } & {\widehat{Z}_{an}^{ij} } & {\widehat{Z}_{ag}^{ij} } \\ {\widehat{Z}_{ba}^{ij} } & {\widehat{Z}_{bb}^{ij} } & {\widehat{Z}_{bc}^{ij} } & {\widehat{Z}_{bn}^{ij} } & {\widehat{Z}_{bg}^{ij} } \\ {\widehat{Z}_{ca}^{ij} } & {\widehat{Z}_{cb}^{ij} } & {\widehat{Z}_{cc}^{ij} } & {\widehat{Z}_{cn}^{ij} } & {\widehat{Z}_{cg}^{ij} } \\ {\widehat{Z}_{na}^{ij} } & {\widehat{Z}_{nb}^{ij} } & {\widehat{Z}_{nc}^{ij} } & {\widehat{Z}_{nn}^{ij} } & {\widehat{Z}_{ng}^{ij} } \\ {\widehat{Z}_{ga}^{ij} } & {\widehat{Z}_{gb}^{ij} } & {\widehat{Z}_{gc}^{ij} } & {\widehat{Z}_{gn}^{ij} } & {\widehat{Z}_{gg}^{ij} } \\ \end{array} } \right]$$

(4.12)

Self- and mutual impedances in Ω per unit length with no ground effect of all phases and neutral are given by Eqs. 4.13 and 4.14, respectively:

$$\widehat{Z}_{qq}^{ij} = r_{q} + j4\pi \times 10^{ - 4} f\xi \ln \frac{1}{{GMR_{q} }}\quad q = a,b,c,n$$

(4.13)

$$\widehat{Z}_{ql}^{ij} = j4\pi \times 10^{ - 4} f\xi \ln \frac{1}{{D_{ql} }}\quad q,l = a,b,c,n$$

(4.14)

where

GMR _q :

Geometric mean radius of conductor q in feet

r _q :

AC resistance of conductor q in Ω/mile

ξ :

Constant equal to 1.609, if $\widehat{Z}_{qq}^{ij}$ and $\widehat{Z}_{ql}^{ij}$ are given in Ω/mile, and equal to 1.000, if $\widehat{Z}_{qq}^{ij}$ and $\widehat{Z}_{ql}^{ij}$ are given in Ω/km

D _ql :

Distance between conductor q and l in feet

f :

Frequency in Hz

Ciric [9] proposed a model where ground self-impedance $\widehat{Z}_{gg}^{ij}$ is explicitly represented. In this proposal, values of $\widehat{Z}_{gg}^{ij}$ are given in Ω per unit length and identified as the frequency-dependent terms in the modified Carson’s equations according to the following expression:

$$\widehat{Z}_{gg}^{ij} = \pi f\xi \times10^{ - 4} \left[ {\pi - j8 \times 0.03868 + j4\ln \frac{2}{{5.6198 \times 10^{ - 3} }}} \right]$$

(4.15)

Equation 4.15 is presented as a frequency-dependent complex number with fixed resistive and inductive parts. However, according to [18], it can be demonstrated that inductive ground behavior depends on both frequency and geometric assumptions. In this case, Eq. 4.15 could be considered as a particular case. If the skin effect is ignored, self-impedance in Ω per unit length of the equivalent ground conductor can be written as follows [18]:

$$\widehat{Z}_{gg}^{ij} = \pi^{2} \xi \times 10^{ - 4} f + j4\pi \xi \times 10^{ - 4} f\left[ {\ln \frac{2s}{{D_{sd} }} - 1} \right]$$

(4.16)

where D _sd is the GMR of ground equivalent conductor and s is the length of wire, both in the same units. In Eq. 4.16, $\pi^{2} \xi \times 10^{ - 4} f = 0.0953$ and $4\pi \xi \times 10^{ - 4} f = 0.12134$ in Ω/mile. It must be noticed that Eqs. 4.15 and 4.16 should be equivalent, and therefore, there exist several values of D _sd and s in order to accomplish both equations.

In this document, for the sake of simplicity, it is assumed $\widehat{Z}_{gg}^{ij}$has no inductive part. This arbitrary assumption is also applied by [18] to derive self- and mutual ground equivalent impedances. As a result, self-ground impedance in Ω per unit length used in the remainder of this document is given by the following expression.

$$\widehat{Z}_{gg}^{ij} = \pi^{2} \xi \times 10^{ - 4} f$$

(4.17)

Mutual impedance between phase and neutral conductors in Ω per unit length with respect to ground $\widehat{Z}_{qg}^{ij}$ may be directly obtained from simplified Carson’s equations as written in Kersting’s book [2].

A direct relation between Carson’s impedances $Z_{qq}^{ij}$ and primitive impedances $\widehat{Z}_{qq}^{ij}$ in Ω per unit length can be written as follows [18]:

$$Z_{qq}^{ij} - \widehat{Z}_{qq}^{ij} = Z_{ql}^{ij} - \widehat{Z}_{ql}^{ij} = \widehat{Z}_{gg}^{ij} - 2\widehat{Z}_{qg}^{ij}$$

(4.18)

$$\begin{aligned} Z_{qq}^{ij} & = r_{q} + \pi^{2} \xi \times 10^{ - 4} f + j4\pi \times 10^{ - 4} f\xi \left[ {\ln \frac{1}{{GMR_{q} }} \cdots } \right. \\ & \left. {\quad + 7.6728 + \frac{1}{2}\ln \frac{{\rho_{ij} }}{f}} \right]\quad q = a,b,c,n \\ \end{aligned}$$

(4.19)

$$\begin{aligned} Z_{ql}^{ij} & = \pi^{2} \xi \times 10^{ - 4} f + j4\pi \times 10^{ - 4} f\xi \left[ {\ln \frac{1}{{D_{ql} }} \cdots } \right. \\ & \left. {\quad + 7.6728 + \frac{1}{2}\ln \frac{{\rho_{ij} }}{f}} \right]\quad q,l = a,b,c,n \\ \end{aligned}$$

(4.20)

Then, from Eqs. 4.18–4.20, mutual ground impedances in Ω per unit length are given by

$$\widehat{Z}_{qg}^{ij} = 2\pi \xi \times 10^{ - 4} f\left[ {\frac{1}{2}\ln \frac{{\rho_{ij} }}{f} - 7.6728} \right]\;q = a,b,c,n$$

(4.21)

where ρ _ij is the earth resistivity at section line i–j in Ω-m.

The grounding resistance of a node j between neutral and ground is given in Ω:

$$\overline{Z}_{T}^{j} = \frac{{\rho_{ij} }}{2\pi l}\left[ {\ln \frac{4l}{{r_{\text{rod}} }} - 1} \right]$$

(4.22)

where l is the ground rod length in meters and r _rod is the radius of ground rod in meters. For commercial ground rods used in distribution systems 10 ft and 5/8 in, ground impedance can be approximated by $\overline{Z}_{T}^{j} \approx \rho_{ij} /3.35$ [19]. For instance, for ρ _ij  = 100 Ω-m, $\overline{Z}_{T}^{j} \approx 30\;\varOmega$

Finally, in order to complete the 4-wire multigrounded model for a given section i–j, primitive shunt admittance in μS per unit length is given by $\widehat{{\mathbf{Y}}}^{{{\mathbf{ij}}}} = j{\mathbf{B}}^{{{\mathbf{ij}}}} = jw{\mathbf{C}}^{{{\mathbf{ij}}}}$ where $w = 2\pi f$ is the angular frequency and C ^ij is the capacitance primitive matrix. Details about self- and mutual capacitance calculations can be found in [2].

$$\widehat{{\mathbf{Y}}}^{{{\mathbf{ij}}}} = \left[ {\begin{array}{*{20}c} {\widehat{Y}_{aa}^{ij} } & {\widehat{Y}_{ab}^{ij} } & {\widehat{Y}_{ac}^{ij} } & {\widehat{Y}_{an}^{ij} } & 0 \\ {\widehat{Y}_{ba}^{ij} } & {\widehat{Y}_{bb}^{ij} } & {\widehat{Y}_{bc}^{ij} } & {\widehat{Y}_{bn}^{ij} } & 0 \\ {\widehat{Y}_{ca}^{ij} } & {\widehat{Y}_{cb}^{ij} } & {\widehat{Y}_{cc}^{ij} } & {\widehat{Y}_{cn}^{ij} } & 0 \\ {\widehat{Y}_{na}^{ij} } & {\widehat{Y}_{nb}^{ij} } & {\widehat{Y}_{nc}^{ij} } & {\widehat{Y}_{nn}^{ij} } & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$

(4.23)

Appendix B—DPL5Wire Code

Script developed in DPL for exchanging files with MATLAB platform.

Appendix C—DPL Script for Displaying Final Results in DIgSILENT

Appendix D—Case Study: I/O Files Structure

See Tables 4.2 and 4.3.

Table 4.2 Casedata.txt file elements

Table 4.3 Casesolution.txt elements