Correction Step

In each iteration of PAWMS, the breakpoint injections are updated. In the basic PAWMS presented above, the radial power flow iteration immedi-ately following this breakpoint update uses, as its initial condition, the result of the iteration preceding the update. This method can be improved by updating this initial condition to reflect the changes in the breakpoint injections.

For NR-PARS and BFS-PARS, the initial condition is specified by the bus voltages. The voltages at the end of each lateral are sufficient to spec-ify the initial condition for VI-DePARS and VS-DePARS. For the other two fast decoupled methods, I-DePARS and S-DePARS, the initial condition is comprised of the currents or powers, respectively, injected into the begin-ning of each lateral.

Figure 7.4 Effect of Power vs. Current Injection on Convergence

Iterations 1e-9

1e-8 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

0 5 10 15 20 25

∆I = U(∆C'+j∆D')

∆S = (V1+V2)/2 * conj(∆I)

∆S = ∆C'-j∆D'

∆S = ∆C′–j∆D′

∆S V′_k+V_k′′ ---2

 

  ∆.* I′∗

=

∆I = U(∆C′+j∆D′)

First consider the correction for NR-PARS and BFS-PARS. This volt-age correction step adjusts the bus voltvolt-ages to reflect the change in the breakpoint injections according to the method presented in [20]. A back-ward/forward sweep of the radial network is performed with:

• all loads, shunts, and cogenerators disconnected

• breakpoint injections set to their incremental values (the change in injection just computed)

• source voltage set to zero

The bus voltages computed from this process are incremental values reflecting the change in bus voltages due to the change in breakpoint injec-tion. These incremental voltages are added to the bus voltages from the previous radial power flow iteration, resulting in a better starting point for the new radial power flow.

For VI-DePARS and VS-DePARS, each iteration consists of a mis-match evaluation followed by an update to the end voltages. The voltage correction step described above can be used to calculate the incremental voltages. In this case, it is only necessary to add the incremental values to the end voltages since they are used to update all other bus voltages in the succeeding mismatch evaluation.

When using I-DePARS and S-DePARS, the breakpoint injections are updated immediately following a function evaluation. Each iteration of the radial power flow then begins with the update to the independent vari-ables, based on the mismatch from the previous iteration, and ends with the evaluation of a new mismatch. It is convenient to switch this order from that used by the other two DePARS methods above to allow for a mis-match correction to adjust for the change in breakpoint injections.

The effect of a change in the injection at a particular breakpoint then is simply to increase the corresponding lateral’s mismatch by the amount of the change. The only other detail is that, for I-DePARS, incremental power injections are converted to their approximate current equivalents³ before adding them to the appropriate mismatch. Likewise, for S-DePARS, incremental current injections are converted to their approximate power equivalents.³

The effect of adding a correction step to PAWMS is to improve the overall convergence characteristics of the algorithm. Using the correction step typically reduces the total number of radial power flow iterations required to solve the weakly meshed power flow.

7.4 Comments

For a distribution system with a small number of loops, secondary sources, and PV buses, the size of the breakpoint impedance matrix is rela-tively small. The computation involved in forming and factoring the matrix is still relatively small compared to the work required for the solution of the radial system. However, as the number of loops grows, the computa-tional burden associated with the breakpoint impedance matrix grows.

Since is, in general, not necessarily sparse, at some point, as the num-ber of loops increases, the work associated with becomes so large that it is more efficient to use a general nodal approach, such as the traditional Newton-Raphson or Implicit Z_bus Gauss methods. For this reason, PAWMS is well suited to weakly meshed systems, but less well suited to the highly connected structure of a typical transmission network.

3 Based on balanced 1 per unit voltages.

Z_BP

Z_BP

Another issue raised by extending a radial power flow technique to handle weakly meshed systems is that of the existence and uniqueness of solutions. According to [13], a typical radial distribution network always has a unique feasible power flow solution. On the other hand, it is a well-known fact that a meshed transmission network may have many feasible steady-state equilibrium points or none at all. The solution found by PAWMS for a weakly meshed distribution network may therefore not be a unique feasible solution. Presumably, choosing a different initial value for the breakpoint injections could result in a different solution to the power flow problem. Furthermore, divergence of the algorithm in certain cases could be due to the lack of a feasible solution.

148

Simulation Results

All of the algorithms under consideration were implemented in a pro-gram written for MATLAB®.¹ The program reads the network data once from a text file and then stores it in binary format for later use. After read-ing the network data, either from the original text file or from the binary file, the network is traversed as described in Section 2.2.2.2, “Breadth-First Search”. During this traversal the nodes and laterals are indexed, data is verified for consistency, and sections are marked as grounded or ungrounded. Since it is common to all of the methods, this preprocessing step is omitted from the comparison of computational effort associated with each algorithm.

MATLAB® is a high-level interpreted language designed with matrix manipulation in mind. The version used for this implementation, version 4, includes sparse matrix storage and manipulation as built-in functions. These capabilities made it an attractive choice for quick imple-mentation and testing of ideas during the stage of algorithm development.

1 MATLAB® is a trademark of The MathWorks, Inc.

For the analysis of the algorithms, the interpreted nature of MATLAB®

makes it ideal for observing the behavior of individual parameters. On the other hand, an interpreted language is usually quite a bit slower in execu-tion than the compiled languages typically used for power flow, such as Fortran or C. For this reason, MATLAB® is probably not the language of choice for a program intended for use in industry. On a sufficiently fast workstation, however, computation time for this implementation was not a problem even with networks of more than 1000 buses.

The goal of this analysis of the results of the MATLAB® simulation is to draw some meaningful conclusions about the behavior of the algorithms in a compiled language, such as Fortran or C. This entails comparisons of the effectiveness of each of the algorithms presented in Chapter 4 through Chapter 7 relative to one another and relative to other relevant power flow algorithms.

For this purpose, computation time in MATLAB® is nearly meaning-less. In MATLAB®, solving a set of linear equations is a built-in function and therefore executes at approximately the same speed as Fortran or C. Sim-ple loops, however, are many times slower since each line is interpreted each time through the loop. Consequently, the run time of an algorithm in MATLAB® may be completely unrelated to the run time of the same algo-rithm in Fortran or C. The number of floating point operations (flops) required, though not a perfect measure, is chosen as a much better indica-tor of relative run time in a compiled language.

8.1 Summary of Algorithms Tested

The simulations performed involve 16 different algorithms which can be classified into the following four categories:

• traditional algorithms for the standard formulation (Newton-Raphson, Implicit Z_bus Gauss)

• network reduction methods (NR-PARS)

• backward/forward sweep methods (BFS-PARS)

• fast decoupled methods (DePARS)

These algorithms are summarized in Table 8.1. The first class, included for the sake of comparison, consists of standard power flow meth-ods applied to the traditional power flow formulation for general meshed systems. The remaining three classes are for radial networks only and are collectively referred to as PARS (Power flow Algorithms for Radial Sys-tems). Each variation of PARS can be extended, as described in Chapter 7, to solve the power flow for weakly meshed networks. When a particular version of PARS is used in conjunction with these extensions, the “PARS”

in the name of the algorithm is simply changed to “PAWMS” (Power flow Algorithms for Weakly Meshed Systems).

Since the two traditional distribution power flow algorithms tested, Newton-Raphson and Implicit Z_bus Gauss, have not been discussed in pre-vious chapters, each of them will be described briefly.