Application of KCL

The current or power flow component of the update formulas (2.2)-(2.5) is based on the current or power lost in bus k’s incoming branch and the application of Kirchhoff’s Current Law (KCL) at bus k. The application of KCL at bus k requires the currents injected by cogenerators, shunt

Y_k^BR

I_k I_k′

Y_k¹¹ Y_k¹² Y_k²¹ Y_k²²

V_{k 1–} V_k

=

V_k V_{k 1–} I_k V_{k 1–}

V_k I_k′

I_k I_k′ I_k = (S_k. /V_{k 1–} ) ∗

I_k′ = (S_k′. /V_k) ∗ V_k V_{k 1–}

S_k = V_{k 1–} .* I_k∗ S_k′ = V_k.*I_k′∗

capacitors, and loads, represented by , , and , respectively. Each of these quantities is a function of and is hence designated with a tilde.

It also requires the currents injected into sub-laterals branching off from bus k. Here and is the set of buses adjacent to bus k on sub-laterals.

KCL at bus k can then be written as

. (2.20)

In the case where power flow is used there is an analogous equation expressing the conservation of complex power at bus k.

(2.21) Table 2.6 General Branch Update Formulas

Based

21

Detailed Component Models

In any problem where mathematics and numerical algorithms are used to analyze a physical system, the results are only as accurate as the mathematical models used. In power systems analysis, the solutions found by any power flow algorithm are only useful to the user if they provide results which are meaningful with respect to some real system. It is there-fore important to model each component of the system as accurately as possible. On the other hand, care must be taken to avoid using models which are overly detailed and therefore either computationally impractical or unusable due to unavailability of parameter data. The algorithms pre-sented in this dissertation are based on models which attempt to meet these two requirements. Most are based on standard three-phase models as presented in [2; 8; 10].

This chapter describes in detail the models used for loads, shunt capacitors, cogenerators, distribution lines, switches, and transformers.

These models provide relationships between the relevant voltages, cur-rents, and power flows. By convention, injected currents and power flows

are always used for loads, shunt capacitors, and cogenerators, as shown in Figure 2.2.

Bus voltages are typically the phase voltages , , and refer-enced to ground. However, it is possible to have floating sections of the net-work in which there is no reference to ground. For example, there might be a feeder connected to the secondary side of a grounded wye to delta trans-former which has only ungrounded, i.e. delta connected, loads. The terms grounded and ungrounded, respectively, will be used to distinguish between parts of the network which have a reference to ground and those floating sections which do not. Which buses are in grounded sections and which are in ungrounded sections is determined according to the ground-ing of the transformer connections durground-ing the initial traversal of the net-work described in Section 2.2.2.2, “Breadth-First Search”. It is assumed that any part of a network supplied through an ungrounded transformer connection will be entirely ungrounded.¹

In the ungrounded sections, to avoid arbitrarily picking a particular phase as the voltage reference, the line-to-line voltages and are used. In this case, the third line voltage is redundant since it is always equal to , and the dimension of the equations is reduced by one. Similarly, the current in one of the phases is redundant since , so typically only phase a and phase b currents are used for calculation.

1 A sub-network, supplied through an ungrounded transformer connection, which does have some grounded elements could be handled by the network reduction methods of Chapter 4, though the details of such a case are not discussed. In their current forms, however, the methods of Chapter 5 and Chapter 6 are unable to handle this case.

V^a V^b V^c

V^ab V^bc V^ca

V^ab+V^bc

( )

–

I^c = –(I^a+I^b)

When computing power flows as opposed to currents, ground is used as a reference in grounded sections. For example, . However, in ungrounded sections, the power used is that defined by the line-to-line voltages and , and the currents and .

(3.1) (3.2) It is important to note that, although it is possible to calculate total power flows in grounded sections of the network by the sum , the total power flow in an ungrounded section is not equal to . Total power flows must be calculated using a common voltage base. For example, using phase c as a voltage reference, the total power can be computed as

. (3.3)

3.1 Load Model

The model used for loads is a flexible one. It includes constant com-plex power, constant current, and constant impedance types.² Three-phase loads can be balanced or unbalanced and can be connected in a grounded wye configuration or an ungrounded delta configuration. It is also possible to have single-phase or two-phase grounded loads. Typically, the load val-ues are given as nominal power delivered to the load and must be con-verted into the appropriate constant model parameters. Depending on the type of power flow algorithm being utilized, it may be necessary to compute

2 Each load could actually be a linear combination of these three types. In fact, it is straightforward to generalize the model presented here to a current injection expressed as an arbitrary function of voltage.

S^a V^aI^a∗

=

V^ab V^bc I^a I^b

S^ab V^abI^a∗

=

S^bc V^bcI^b∗

=

S^a+S^b+S^c S^ab+S^bc

S^total = S^ac+S^bc

the following quantities from the bus voltage and the constant model parameters:

• admittance matrix

• injected current

• injected power

With a grounded wye connected load as shown in Figure 3.1, for each phase p, the parameter is given. This is the nominal complex power absorbed by the element connected between phase p and ground. In other words, for a three-phase load, the nominal load is

. (3.4)

These values are converted to the appropriate constant model parameters , , or , according to the type of load and the nominal voltage , using the equations in Table 3.1. , , and are n x 1

com-Figure 3.1 Grounded Wye Connected Load V_k

Y_Lk

I_Lk = –Y_LkV_k S_Lk = V_k.*I_Lk∗

S_{Lk nom}^p _,

S_{Lk nom,}

S_{Lk nom}^a _, S_{Lk nom}^b _, S_{Lk nom}^c _,

=

y_Lk^b

y_Lk^a y_Lk^c

V_k^c I_Lk^a

S_Lk^b

S_Lk^c S_Lk^a

I_Lk^b I_Lk^c V_k^a

V_k^b

I_Lk S_Lk y_Lk

V_{k nom,} I_Lk S_Lk y_Lk

plex vectors of current, power, and admittance, respectively, where n is the number of phases present. Note that and are injected quantities, hence the negative sign in (3.5) and (3.6).

Figure 3.2 shows an ungrounded delta connected load for which the nominal power given is the power absorbed by the elements between each phase. In this case, the nominal load is

. (3.14)

†where .

Table 3.1 Load Parameters from Nominal Loads

Connection Load Type Parameter Calculation

grounded

constant Z^† (3.10)

constant S (3.11)

In the conversion from the nominal load to the appropriate constant model parameters shown in (3.8)-(3.13) in Table 3.1, the voltages used are phase-to-ground or line-to-line voltages, respectively, depending on whether the load is in a grounded or ungrounded section of the network.