DEPRECATED  This module is not supported as of Hassayampa (Version 3.0)

# Introduction

The network module implements a balanced three-phase positive sequence power flow solver using the Gauss-Seidel method (Gauss 1809), as described by Kundur (1993).

# Balanced Three-Phase Steady-state Transmission Network Flow Solution Using Gauss-Seidel Method

The Gauss-Seidel (GS) method is a linear iterative method (as opposed to the Newton-Raphson (NR) method, which uses a quadratic iterative method). The only differences between the two methods are the rate of convergence and the robustness of convergence when the starting state is bad. Because GridLAB-D is a quasi-steady time-series simulation, most state changes are small and result in only one or two iterations in either method. However, the GS method can be more easily implemented using parallel processing systems (such as multicore systems). In addition, the GS method is more likely to solve for new conditions that are far from the current solution. However, the GS method can have difficulty computing flows under high-transfer conditions.

The GS method uses an iterative approach proposed by von Seidel (1874). The GS method iterates using the first-order approximation of the Taylor expansion, and the NR method uses a second-order approximation. In the GS method the fundamental equation for the kth node can written as

${\displaystyle {\frac {P_{k}+\jmath Q_{k}}{{\overline {V}}_{K}^{*}}}=Y_{kk}{\overline {V}}_{k}+\sum \limits _{i=1,i\neq k}^{n}{{\overline {Y}}_{ki}{\overline {V}}_{i}}}$

The GS method is often criticized as inferior. This is categorically not true. The GS method has strengths and weaknesses when compared to the NR method. Depending on the circumstances, one method may be preferred over the other. But both, indeed all, valid methods produce the same answers. In fact, many commercial power flow solvers implement both methods concurrently, recognizing the necessity to exploit the appropriate method under any given circumstance. Hence, the implementation of the GS method does not make GridLAB-D's solution method inferior. It is simply a recognition that the circumstances of the power flow solution needed in GridLAB-D led to the choice of GS as the default power flow solver. Other power flow solvers can and should be implemented to address different circumstances. We encourage users and developers to consider doing so.

Important note: The solution method requires that the best known voltage be used at all times. This means that each time a voltage is updated, all the branch YV contributions to adjacent busses must be immediately updated.

## Bus Solutions

The node is one of the major components of the method used for solving a power flow network. In essence, the distribution network can be seen as a series of nodes and links. A node’s primary responsibility is to act as an aggregation point for the links that are attached to it, and to update the current and voltage values that will be used in the calculations done in the links.

Three types of nodes are defined. Nodes are simply a basic object that exports the voltages for each phase. Triplex nodes export voltages for three lines off a single split phase. All other node objects can have one or more phases defined.

The types of nodes that are supported are the following:

• PQ buses are nodes that have both constant real and reactive power injections.
• PV buses are for nodes that have constant real power injection but can control reactive power injection.
• SWING buses are nodes that are designated to absorb the residual error and are also used for generators that can control both real and reactive power injections.

### PQ Bus

The PQ bus is the most commonly found bus type in electric network models. PQ buses are nodes where both the real power (P) and reactive power (Q) are given. In these cases, the updated voltage at a node is found from an existing (non-zero) voltage using

${\displaystyle {\overline {V}}_{k}\gets {\frac {P_{k}-\jmath Q_{k}}{{\overline {Y}}_{k}({\overline {V}}_{k}^{*}-\sum \limits _{i=1,i\neq k}^{n}{{\overline {Y}}_{ki}{\overline {V}}_{i}})}}}$

where ${\displaystyle {\overline {Y}}_{k}\gets G_{k}+\jmath B_{k}+\sum \limits _{i=1,i\neq k}^{n}{{\overline {Y}}_{ki}}}$

### PV Bus

The PV bus is the next most common bus type in electric network model. PV buses are nodes where the real power (P) is given, but the reactive power (Q) must be determined at each iteration. In these cases, the updated voltage for a node is found by calculating (3.2) If Q exceeds the Q limits [Qmin, Qmax], then the angle is fixed to the corresponding limit and the bus is solved as a PQ bus.

${\displaystyle Q_{k}\gets \Im \left[{\overline {V}}_{k}^{*}({\overline {Y}}_{k}{\overline {V}}_{k}+\sum \limits _{i=1,i\neq k}^{n}{{\overline {Y}}_{ki}{\overline {V}}_{i}})\right]}$

### SWING Bus

The SWING bus occurs at least once in any given island of a network models. Large models may have more than one SWING bus, particularly if areas of the network are only lightly coupled by relatively high-impedance links. SWING bus nodes are nodes where both the real power (P) and reactive power (Q) must be determined each iteration. In these cases, the updated voltage for a node is found by

${\displaystyle {\overline {S}}\gets {\overline {V}}_{k}^{*}({\overline {Y}}_{k}{\overline {V}}_{k}+\sum \limits _{i=1,i\neq k}^{n}{{\overline {Y}}_{ki}{\overline {V}}_{i}})}$

where ${\displaystyle P+\jmath Q={\overline {S}}}$

## Branch Solutions

The link object implements the general solution elements for branches using the GS method. The following sequence of operations is performed on each branch during a bottom-up sync event.

The effective admittance Y is calculated by including half of the line charging capacitance B (Kundur 1993):

${\displaystyle {\overline {Y}}_{k_{eff}}\gets {\overline {Y}}_{k}+\jmath {\frac {B}{2}}}$

The admittance coefficient is the inverse of the transformer turns ratio ${\displaystyle n}$, if any (Kundur 1993):

${\displaystyle c\gets {\frac {1}{n}}}$

${\displaystyle {\overline {Y}}_{c}\gets c{\overline {Y}}_{k_{eff}}}$

{\displaystyle {\begin{alignedat}{2}\sum {\overline {Y}}_{from}&\gets \sum {\overline {Y}}_{from}+{\overline {Y}}_{c}+(c-1){\overline {Y}}_{c}\\\sum {\overline {Y}}_{to}&\gets \sum {\overline {Y}}_{to}+{\overline {Y}}_{c}+(1-c){\overline {Y}}_{eff}\\\end{alignedat}}}

Compute the line current injections on the busses:

{\displaystyle {\begin{alignedat}{2}{\begin{alignedat}{2}\sum {\overline {I}}_{from}&\gets \sum {\overline {V}}_{to}{\overline {Y}}_{c}\\\sum {\overline {I}}_{to}&\gets \sum {\overline {V}}_{from}{\overline {Y}}_{c}\\\end{alignedat}}\end{alignedat}}}

Add the current injections to the busses:

{\displaystyle {\begin{alignedat}{2}\sum {\overline {YV}}_{from}&\gets \sum {\overline {YV}}_{from}+{\overline {I}}_{from}\\\sum {\overline {YV}}_{to}&\gets \sum {\overline {YV}}_{to}+{\overline {I}}_{to}\\\end{alignedat}}}

# Network module implementation

The network module implements the Gauss-Seidel solution method for balanced positive-sequence flow power models of transmission systems.

The module global variables are shown in Table 1.

Table 1. Network module properties
Property
Type
Default
Unit
Description
acceleration_factor double 1.4 pu The voltage update gain factor (usually between 1.4 and 1.7)
convergence_limit double 0.001 V The maximum allowable voltage change for iteration to halt
mvabase double 1.0 MVA The megaVolt-Amp basis to use in calculating powers
kvbase double 12.5 kV The kiloVolt basis to use calculating voltages
model_year int16 2000 CE The basis year of the model
model_case char8 "S" [WSF] The basis of the case (e.g., winter, summer, fall)
model_name char32 "(unnamed)" * The name of the model

## Node class

Network node objects represent busses in the transmission network. Three types of nodes are possible, and selecting using the type property:

1. PQ busses are general busses that have constant power (both real and reactive power are invariant);
2. PV busses are busses that have constant real power, but reactive power must be computed; and
3. SWING busses are busses for which both real and reactive power must be computed and there must be at least one SWING bus per network island.

The node class must clear the admittance and current injection accumulators on the pre-topdown pass and compute the new voltage on the bottom-up pass of synchronization. The voltage update pseudo code is as follows:  

 self-admittance <- admittance-accumulator + complex (conductance + j susceptance) if type is SWING power <= conjugate ( conjugate voltage x ( self-admittance x voltage + current-injection-accumulator ) ) new-voltage <= voltage voltage-change <= complex zero else if self-admittance non-zero if type is PV imaginary power <= imaginary ( voltage * ( self-admittance x voltage + current-injection-accumulator ) ) if imaginary power less-than minimum reactive-limit imaginary power <= minimum-reactive-limit new-voltage <= ( conjugate power / conjugate voltage - current-injection-accumulator ) / self-admittance else if imaginary power greater-than maximum reactive-limit imaginary power <= minimum-reactive-limit new-voltage <= ( conjugate power / conjugate voltage - current-injection-accumulator ) / self-admittance else new-voltage <= ( conjugate power / conjugate voltage - current-injection-accumulator ) / self-admittance magnitude new-voltage <= magnitude voltage end end voltage-change <= new-voltage - voltage voltage <= new-voltage end else new-voltage <= voltage voltage-change <= complex zero end for each branch in incident-branches-list current-change <= voltage-change x branch.admittance / branch.turns-ratio if node is branch.from-bus other-bus <= branch.to-bus else other-bus <= branch.from-bus end other-bus.current-injection-accumulator <+ current-change end if magnitude voltage-change greater-than convergence-limit time-step <= zero else time-step <= infinity end 

The network link object represents branches in the transmission network. The link class must compute the initial voltage estimates for the solution during the first bottom-up synchronization pass. The link need not update the voltage in subsequent iterations for the same time-step unless the admittance or turns-ratio has changed. Changes to the current-injection-accumulators resulting from bus voltage changes are handled by the node's bottom-up pass. The code for the link bottom-up synchronization is as follows:  

 effective-admittance <= admittance + j line-susceptance / 2 admittance-contribution <= effective-admittance / turns-ratio from-bus.self-admittance-accumulator <+ admittance-contribution + admittance-contribution x ( 1/turns-ratio - 1 ) from-bus.current-injection-accumulator <+ to-bus.voltage x admittance / turns-ratio to-bus.self-admittance-accumulator <+ admittance-contribution + effective-admittance x ( 1 - 1/turns-ratio ) to-bus.current-injection-accumulator <+ from-bus.voltage x admittance / turns-ratio net-current-flow <= ( from-bus.voltage - to-bus.voltage ) x admittance / turns-ratio 

# Derived Classes

All derived classes must update the appropriate properties of the link and node classes. The following derivations are recommended/anticipated:

## Transformers

The transformer class is derived from the link class, with the value of turns_ratio being non-unitary.

## Switches

The switch class is derived from the link class, with the value of the admittance being zero when the switch is open. Attention should be given the possibility that operating a switch can lead to islands, which would require additional SWING busses.

## Generators

The generator class is derived from the node class where the real power is positive. If the reactive power is fixed, the node type should be PQ and if the reactive power is variable, it should be PV

The load class is derived from the node class where the real power is negative. The node type should be PQ.

## Others

Consideration should be given to implementing the following

• capacitor banks
• phase shifters
• high-voltage DC lines
• relays
• metering devices and phasor measurement units

# Model check procedure

The network model check procedure requires the following properties be verified.

1. . Each link must be connected to a bus on both ends (from and to);
2. . Each link must have a non-zero admittance (i.e., is must not be normally open);
3. . Each node must have at least one incident link;
4. . Each flow area must have a SWING bus;
5. . Each no must be connected to a SWING bus or to a node that recursively connects to a SWING bus

If each of these can be confirmed for the entire model, then the check is successful. A warning may be emitted if a flow area has more than one SWING bus.

# Model import/export

## CDF Files

The CDF import routine reads an IEEE CDF file and loads the model into GridLAB-D.

The IEEE CDF file format is documented at http://www.ee.washington.edu/research/pstca/formats/cdf.txt. Additional information and sample data files are also available at http://www.ee.washington.edu/research/pstca/. For more information on IEEE CDF files, see http://www.google.com/search?hl=en&q=IEEE%20power%20flow%20file%20format%20CDF.

# References

Gauss CF. 1809. Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Perthes et Besser, Hamburg, Germany.

Kundur, P. 1993. Power System Stability and Control. McGraw Hill, New York.

Von Seidel, P.L. 1874. Über ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineare Gleichungen überhaupt, durch successive Annäherung aufzulösen. Abh. bayer Akad. Wiss, Germany.