S-DePARS

S-DePARS is the power flow based counterpart to I-DePARS and is a generalization of the reduced power flow equations and fast decoupled algorithm for radial systems first proposed in [12].

This approach is restricted to transformers of types 1, 5, and 9 which have the same connection on both primary and secondary. For an ideal transformer, there is no change in power from the primary to the second-ary so an identity block is used for the corresponding diagonal block of the system Jacobian. Unfortunately, for the other transformer connection types there is no simple way to approximate the Jacobian, so they have not been included in this formulation.

s_i

x_i

∂

∂F_i

 

 

 ^–1 s_j

∑

 

 

=

s_j

The approximate Jacobian for S-DePARS is identical to that of I-DePARS. The update step in the Newton method can therefore be solved by a block forward substitution.

6.4 Convergence Analysis

In this section, it will be shown that the fast decoupled algorithms described here fall into the class of inexact Newton methods discussed in [14]. In the classical Newton’s method shown in Table 6.1, the update step

at iteration i is the solution to

. (6.49)

In the inexact Newton methods, the corresponding equation is

, (6.50)

where the size of the residual is restricted so that the relative residual is bound by some forcing sequence. Specifically,

, (6.51)

where is a forcing sequence which is uniformly less than one. Here denotes an arbitrary norm in . Notice that the special case where

gives the exact Newton’s method.

A sequence of iterates produced by the inexact Newton method is locally and linearly convergent. This result is stated and proved as Theorem 2.3 in [14]. Using the present notation, this theorem asserts the following:

s^{( )i}

J^{( )i} s^{( )i} = –F^{( )i}

J^{( )i} s^{( )i} = –F^{( )i} +r^{( )i} r^{( )i}

r^{( )i} F^{( )i}

---≤η^{( )i} η^{( )i}

{ }

. IRⁿ

η^{( )i} ≡0

x^{( )i}

{ }

Assume that . There exists such that, if , then the sequence of inexact Newton iterates converges to . Moreover, the convergence is linear in the sense that

(6.52)

where .

For the fast decoupled methods presented here, the update step can be expressed as the solution to

, (6.53)

where is the constant approximation, shown in Figure 6.4,¹² to the Jaco-bian . With the definition

, (6.54)

(6.53) can be rewritten in the form of the inexact Newton equation of (6.50),

(6.55)

where the residual is .

To prove that DePARS is, in fact, an inexact Newton method, and therefore locally and linearly convergent, it remains to be shown that there is some sequence , uniformly less than one, which bounds the rela-tive residual. This relarela-tive residual can now be expressed as

. (6.56)

12 Or, for I-DePARS and S-DePARS, the transpose of the matrix shown in Figure 6.4.

η^{( )i} ≤η^max< <t 1 ε>0

For a network with only grounded wye to grounded wye transformers some conclusions can be drawn about the size of this residual under the follow-ing assumptions:

• All voltage magnitudes are close to 1 per unit.

• All transformer tap ratios are close to one.

• All per unit network parameters¹³ are small compared to voltage magnitudes (i.e. they are ).

Given these assumptions, is of the form shown in Figure 6.4,¹⁴ where all diagonal terms are equal to or nearly equal to one and the off-diagonal non-zero terms are equal to minus one. The matrix in the residual term is the difference between the true Jacobian, whose structure is shown in Figure 6.3, and . This elements of this matrix are all small compared to one. This implies that

, (6.57)

hence can be chosen such that

, (6.58)

thereby completing the proof that DePARS is a locally and linearly conver-gent inexact Newton method.

13 This includes all line and transformer impedances, line charging admittances, shunt admittances, constant Z load admittances, constant PQ load power injections, and cogen-erator power injections.

14 Figure 6.6 for I-DePARS and S-DePARS.

1

« J˜

∆J^{( )i}

J˜

∆J^{( )i} s^{( )i} J˜s^{( )i}

--- 1« η^{( )i}

∆J^{( )i} s^{( )i} J˜s^{( )i}

---<η^{( )i} <1

6.5 Comments

As with the backward/forward sweep methods of Chapter 5, the fast decoupled methods are applicable to a wide range of radial distribution networks. The one modeling limitation of the general DePARS formulation is that it does not include type 4 ungrounded wye to grounded wye trans-formers. One variation, S-DePARS, is further restricted to transformers of types 1, 5, and 9, all of which have identical connection and grounding on both primary and secondary sides.

As with BFS-PARS, the power flow based variations typically require more computation per iteration than their current based counterparts.

Considering this difference and the fact that S-DePARS is limited to only three types of transformer connections, VI-DePARS and I-DePARS appear to be the most attractive of the four fast decoupled methods. VI-DePARS has the added advantage of readily available initial values for the indepen-dent variable x.

As with NR-PARS and BFS-PARS, the amount of computation required per iteration is proportional to the number of buses. For a con-stant number of iterations for convergence, the computational complexity increases linearly with the size of the system, making DePARS effective for very large radial distribution networks.

127

Power Flow Algorithms for W eakly M eshed Systems (PAWMS)

The previous chapters have dealt with electric distribution systems with a radial topological structure. The algorithms developed and pre-sented in these chapters are specific to networks which do not contain any loops. This chapter investigates an approach for extending these radial algorithms to handle systems with a limited number of loops. The result-ing class of algorithms will be referred to as Power flow Algorithms for Weakly Meshed Systems, or PAWMS.

The approach taken by PAWMS requires that the meshed system be converted to a radial structure by breaking each of the loops. The current or power injections at each breakpoint are adjusted in order to balance the voltages at either side using a compensation method [30]. Variations of this method have been presented in [23], [19], and [20] for single-phase net-works, and more recently in [11] for three-phase systems. The approach presented here generalizes these to work with the various three-phase radial power flow algorithms presented in this dissertation.

Apart from handling cases with loops, the compensation method also makes it possible to solve systems with more than one voltage controlled bus. These can be secondary sources with both voltage magnitude and angle specified, or PV buses which specify voltage magnitude and real power injection. First, the method of dealing with loops is described in detail, followed by the modifications necessary for secondary sources and PV buses.

7.1 Detailed Solution Algorithm

Since PAWMS is based on a radial power flow solver, the weakly meshed system must first be converted to a radial structure. This is done by choosing a bus for each loop to serve as the breakpoint. Figure 7.1 shows a system with a loop containing bus k. This loop can be broken by splitting bus k to create a new artificial bus . The solution of the original meshed network is equivalent to the solution of the resulting radial system under the constraints that and the current injection at bus is the negative of the injection at bus k.

The actual creation of the artificial buses to break the loops is per-formed once during the initialization process. The algorithm, after initial-ization, consists of two steps which are repeated until convergence is achieved, as shown in Table 7.1. First, the breakpoint voltages are updated via a radial power flow method. Then, the current injections, at bus k and at bus , are adjusted according to the breakpoint voltages in order to eliminate any mismatch. The adjustment is based on the sensitiv-ity of the breakpoint voltage mismatch to changes in the breakpoint cur-rent injections. This sensitivity is approximated by the breakpoint

k′

V_k = V_k′ k′

I_BPj I

– _BPj k′

impedance matrix , which is a constant linear approximation to the sensitivity matrix. Since it is constant, it is necessary to form and factor

only once during the initialization of the algorithm.

Table 7.1 Power Flow Algorithms for Weakly Meshed Systems PAWMS - The Algorithm

Break loops.

Form and factor breakpoint impedance matrix.

Initialize breakpoint injections, initialize PARS.

1 Update PARS, compute breakpoint voltage mismatch.

2 Update breakpoint injections.

Repeat steps 1 and 2 until convergence is achieved.

Figure 7.1 Loop Breakpoint

r e s t o f n e t w o r k

bus kk′ bus k bus k

loop j

I_BPj I_BPj I_BPj

+– primary

loop j

+– primary

bus q

r e s t o f n e t w o r k

bus q

negative path positive path source source

Z_BP

Z_BP