4.5 Comments
5.3.8 V-S-PARS
V-S-PARS is the power flow based counterpart to V-I-PARS, using the backward sweep of VS-S-PARS and the forward sweep of V-VS-PARS. This method uses simple power flow summation for the backward sweep and voltage drop calculation for the forward sweep, which is also the basis for the methods proposed in [19] and [20].
5.4 Convergence Analysis
In this section, the convergence of V-I-PARS is considered for a distri-bution system with no ungrounded sections. Some necessary notation for the proof is given below in Table 5.5. With this notation, the backward sweep can be summarized by the following two equations:
(5.26) (5.27) The forward sweep can be summarized by the following two equations:
(5.28) (5.29) where the initial node voltages at each bus are set to the corresponding source voltage. Combining these four equations gives an expression for the
In( )k = In+YVn( )k + f V( n( )k ) Ib( )k = ATIn( )k
Vb(k 1+ ) = ZIb( )k
Vn(k 1+ ) = Vn( )0 – AVb(k 1+ )
†Or, equivalently, for all in .
Table 5.5 Notation for V-I-PARS Convergence Proof
Symbol Interpretation
set of branches between bus i and the source
set of buses supplied through branch i number of elements in
number of elements in
vector of node voltages at iteration k
vector of branch voltages drops at iteration k vector of branch currents at iteration k
vector of node current injections at iteration k
vector of node current injections from constant current loads vector of node current injections from constant PQ elements, as a function of voltage V
vector of node voltages at the solution impedance matrix for branch i
total admittance to ground at bus i, including line charging, shunt capacitors, and constant impedance loads
Z block diagonal matrix of branch impedances
Y block diagonal matrix of bus admittances to ground
A matrix with an identity block in each block row i and block
column j for all in †
complex power injection from constant PQ elements at node i P i( )
nodal voltages at iteration as a function of the same voltages at iteration k.
(5.30) If the node voltage vector at the solution is , the node voltage at iteration k can be expressed as a sum of the solution plus the error.
(5.31) Expressing the node voltage vector in this way for both iteration k and iteration , (5.30) becomes
. (5.32) Since the solution is a fixed point of (5.30), that is,
, (5.33)
(5.33) can be subtracted from (5.32) to give an expression for the error at iteration as a function of the error at iteration k.
(5.34) To prove linear convergence of V-I-PARS, it is sufficient to show that the ratio of the error magnitudes is smaller than one. Specifically, if it can be shown that
, (5.35)
then V-I-PARS converges linearly.
First, consider the case with no constant power devices in the net-work. In this case,
k 1+
Vn(k 1+ ) = Vn( )0 – AZAT[In+YVn( )k + f V( n( )k )] V*
Vn( )k = V* +∆Vn( )k
k 1+
V*+∆Vn(k 1+ )=Vn( )0 – AZAT[In+Y V( * +∆Vn( )k ) + f V( * +∆Vn( )k )]
V* = Vn( )0 – AZAT[In+YV* +f V( *)]
k 1+
∆Vn(k 1+ ) = –AZAT[Y∆Vn( )k + f V( * +∆Vn( )k ) f V– ( *)]
µ AZAT[Y∆Vn( )k +f V( * +∆Vn( )k ) f V– ( *)]
∆Vn( )k
---≤κ<1
=
. (5.36)
A simple bound on the ratio can be found by taking the product of the norms of the individual matrices A, Z, , and Y. For each of the p-norms, where p is 1, 2, or infinity, this product is equal to
. (5.37) This bound, however, is too loose and is not always satisfied in a typical distribution system.
A better bound can be found by considering the norms of the matrices and , since
. (5.38)
The matrix is simply the matrix A with each identity block in block column j replaced by . Likewise, the matrix is just the matrix with each identity block in block row j replaced by . It was found that using the infinity-norm yields a tighter bound than the 1-norm.
(5.39)
It is easy to see that, in a typical system, this is much smaller than the bound given in (5.37), since not all load magnitudes are equal to the maximum value and not all of the branch impedances along the highest impedance path are equal to the maximum branch impedance. In fact, in all of the available cases based on data from real systems, this quantity
µ AZATY∆Vn( )k
was significantly smaller than one, guaranteeing the linear convergence of V-I-PARS on these systems.
If constant PQ loads and cogenerators are considered in the network, then the ratio , as given in (5.35), can be bounded as follows:
, (5.40)
where the first term has just been dealt with. In the second term, is the nodal current injected by constant power devices at the solution, and is the injection at iteration k. At a particular node i, the magnitude of the difference between the two injections is
(5.41)
For real systems, it is reasonable to assume that voltage magnitudes at the solution and at each iteration4 are larger than 0.7 per unit, implying that
(5.42) and therefore
. (5.43)
4 Assuming a flat start, the assumption holds for the voltages at the first iteration. The arguments following show a decrease in the magnitude of the error, indicating that volt-ages during subsequent iterations lie closer to the solution than the initial voltvolt-ages did.
µ
Let S be a diagonal matrix whose ith diagonal element is equal to .5 Since the elements of A, and therefore of its transpose, are all posi-tive, the following can be said about the second term of (5.40):
(5.44)
The bound given in (5.39) then becomes
(5.45)
In words, this says that V-I-PARS will converge linearly if the product of the following two terms is smaller than one. Roughly speaking, the first term is the total impedance of the highest impedance path and the second term is the total admittance to ground plus twice the total power injection from all constant power devices.6 In a typical distribution system, all , , and are much smaller than one, yielding a value of which is also smaller than one. When this condition holds, linear convergence from a flat start is guaranteed.
5 The matrix S actually has units of admittance since the factor of 2 comes from an inverse squared voltage quantity.
6 The seeming inconsistency in units is due to the hidden voltage units in the factor of 2.
2 si
5.5 Comments
The backward/forward sweep methods presented in this chapter are applicable to most radial distribution networks. The one modeling limita-tion is that BFS-PARS cannot handle type 4 ungrounded wye to grounded wye connected transformers.
Some general observations with regard to the many variations indi-cate two things. First, the methods based on current generally require less computation per iteration than their power flow based counterparts. Sec-ond, the methods which do not update extra variables such as voltage dur-ing backward sweep and current/power flow durdur-ing forward sweep, require less computation per iteration.
Based on these two observations, V-I-PARS appears to be the most attractive of the BFS-PARS class of algorithms presented, assuming that the number of iterations required for convergence is comparable among the various methods. The comparisons of the algorithms are investigated in more detail in Chapter 8, “Simulation Results”.
It should also be noted that the amount of work per iteration is pro-portional to the number of buses. Therefore, if the number of iterations remains constant, the computational complexity increases linearly with the size of the network, making BFS-PARS suitable for very large radial distribution systems.
84
Fast Decoupled Power Flow A lgorithms for R adial S ystems (DePARS)
One of the most widely used power flow algorithms throughout the power industry is the fast decoupled Newton method proposed in 1974 in [26]. This method exploits some of the numerical properties of the standard power flow formulation to make simplifying assumptions which allow sig-nificant savings in computation over the standard Newton method. Unfor-tunately, this approach is not typically suitable for radial distribution networks. There are often ill-conditioning problems due to the formulation and, in addition, the assumptions necessary for the simplifications used in the standard fast decoupled Newton method are often not valid for these types of systems. Some work, however, has been done to address these problems [16; 31; 21].
This chapter explores a class of algorithms which exploits the radial topological structure to reduce the number of equations and unknowns in the formulation. These algorithms also take advantage of the special
numerical structure of the new formulation to further reduce the computa-tion required for each iteracomputa-tion, in the spirit of the standard fast decoupled method for meshed transmission systems. This class of algorithms will be referred to as fast Decoupled Power flow Algorithms for Radial Systems or DePARS.
This chapter presents four main variants of DePARS. The first of the four methods, VI-DePARS, is a generalization of the method proposed in [32] and will be presented in detail. The other variations, one of which is an extension of the methods proposed in [12], will then be discussed with respect to VI-DePARS.
6.1 Detailed Solution Algorithm
The standard fast decoupled methods used in transmission systems are based on the well-known Newton’s method [28] for solving a non-linear set of equations. In this case, the non-linear equations being solved are the power balance equations which specify that, at each bus, the complex power generated, minus the power absorbed by load, must equal the power injected into the rest of the network. In a distribution system with one source bus and many load buses, the traditional power flow formulation would have six equations for each load bus, balancing the real and reactive part of the power at each of the three phases.
The traditional fast decoupled method for transmission systems improves on the standard Newton method, shown in Table 6.1, by making simplifying approximations which reduce the computational burden for step 5 and step 6.
The DePARS approach is also based on the Newton method. Like the traditional fast decoupled method, it exploits the numerical structure of the Jacobian to greatly reduce the computation required by step 5 and step 6 in Table 6.1. However, DePARS also uses a different formulation of the power flow equations which exploit the radial topological structure of the network resulting in a reduced number of equations and unknowns.
Table 6.2 gives a high level view of DePARS and its basic steps.
Steps 5, 6, and 7 from the original Newton method have been simplified and grouped together into step 2 of DePARS. At the right side of Table 6.2 the details have been kept general enough to cover all four variations of the method. The first option in each step, however, is the one used by VI-DePARS, which will be the focus of the remainder of this section.
Table 6.1 Newton’s Method
Solution of by Newton’s Method 1 Choose an initial guess for the solution, .
2 Set .
3 Evaluate .
4 Stop if .
5 Evaluate the Jacobian, .
6 Solve .
7 Let .
8 Let and go to step 3.
F x( ) = 0
x( )0 i = 0
F( )i = F x( ( )i )
F( )i ≤some tolerance J( )i
∂x
∂F
x( )i
=
J( )i s( )i = –F( )i x(i 1+ ) = x( )i +s( )i i = i 1+
For VI-DePARS, the independent variables are the voltages at the end of each lateral. The power flow equations state that the voltage mismatch at the beginning of each lateral, calculated as a function of the end volt-ages, must be zero at the solution. If this mismatch is not zero it can be used to update the end voltages for the next iteration.