The most conventional method for solving the load flow problem is the formulation of the power injected at each bus in its polar form [
7]. Interval analysis is useful for modeling uncertainties in numerical analyzes [
11]; thus, it is used in conjunction with the polar formulation of the Newton-Raphson power flow to create what is called an ILF.
3.1. Algorithm Description
The main steps for obtaining a solution to the power flow considering interval variables are described below.
Step 1: Obtain the bus voltages using a deterministic load flow. In this work, the deterministic load flow corresponds to the solution of the power flow through the Newton-Raphson method with input variables fixed in their deterministic values, i.e., power injections (load and generation) fixed in its predicted maximum values.
Step 2: Define the percentage of variation of demands and calculate their respective interval values according to (
9) and (
10):
where
and
are the values of the deterministic active and reactive loads, respectively, at bus
k,
is the demand percentage variation and
and
are the interval values of the active and reactive loads, respectively. It is worth mentioning that in this work, the interval representation is only considered for loads; nonetheless, the logic of Equations (
9) and (
10) can be extended for an interval representation of the generation, if necessary.
Step 3: Calculate the expected interval power through the expected deterministic power and the interval load calculated in (
9) and (
10), as given by (
11) and (
12).
where
and
are, respectively, the active and reactive expected power at bus
k used in the deterministic load flow, while
and
are, respectively, the interval active and reactive expected power at bus
k.
Then, the interval mismatches for the initialization of the bus voltages magnitude and angles can be calculated. The interval mismatches must be obtained with the calculated deterministic powers and the expected interval powers, using expressions (
13) and (
14) as indicated in [
26].
where
and
are the calculated deterministic active and reactive power at bus
k, while
and
are the interval values of the mismatches at bus
k, respectively.
Thus, the interval increments of the bus voltages magnitudes and their respective angles are calculated from the Jacobian matrix of the last iteration of the deterministic power flow and the interval mismatches (
and
) as given by (
15).
where
and
are the vectors of interval mismatches of active and reactive power, respectively. Also,
is the jacobian matrix of the last iteration of the deterministic load flow and
and
are the interval increments of the angles and magnitudes of bus voltages, respectively. Then, the interval voltage magnitudes and phase angles can be calculated using the deterministic solution and the interval increments indicated in (
16).
where
and
are respectively the vectors of the deterministic solution of voltage phase angles and magnitudes, while
and
are the vectors of interval values of voltage phase angles and magnitudes, respectively.
Step 4: Obtain the interval active and reactive power calculated using the admittance matrix of the system as well as the current interval magnitudes and angles of the buses voltages, as described in (
17) and (
18) and, thus recalculate the power mismatches, as proposed by [
26]. Note that these equations are similar to those in the deterministic load flow, only now the variables of the problem are replaced by their respective interval variables.
where
and
are the interval voltage magnitudes at buses
k and
m, respectively. Also,
and
are the interval voltage angles at buses
k and
m, respectively.
and
are respectively the real and complex components of
of the system admittance matrix and
and
are, respectively, the interval active and reactive calculated powers. Thus, the interval mismatches of active and reactive power are updated according to (
19) and (
20) [
26]:
Step 5: Apply the Krawczyk operator according to (
8), where the variables will assume the following values:
Step 6: Update the interval increments for the voltage magnitudes and angles through the intersections between the interval voltages in the previous iteration and the voltage obtained by the Krawczyk operator as indicated in (
24).
Step 7: Check if the greater radius variation of the interval magnitude and angle of the bus voltages between the iterations is less than the specified tolerance, if yes, the interval power flow has converged; otherwise, return to step 5.
3.2. Illustrative Example
A detailed explanation of the ILF algorithm using a didactic 3-bus system, shown in
Figure 1, is presented. In this figure, the line impedances are in ohms, the loads are in MVA and the bus voltages are in per-unit (p.u). This system has a base power of 1 MVA and a base voltage of 23 kV. In
Figure 1, the state variables obtained trough a deterministic load flow in each bus are shown. These calculations correspond to those described in
Step 1 of
Section 3.1. The Jacobian matrix of the last iteration of the deterministic load flow for this system corresponds to:
The Jacobian matrix was considered in its full form. Therefore, high values in the order of were assumed in the main diagonal corresponding to the slack bus, because in this bus the magnitude and phase angle voltage are fixed in 1.0 p.u and 0 degrees, respectively, and thus iterative updating of these variables is not necessary.
A percentage variation of 5%, i.e.,
= 5%, is assumed for the loads of the system to define the interval values, as presented in
Step 2, for
and
as:
Next, the interval increments of the bus voltages magnitudes and angles must be calculated, as stated in the
Step 3 of the ILF algorithm. For this, the interval mismatches of active and reactive powers at each bus are obtained as:
Using the interval mismatches of the power loads and the Jacobian Matrix, the interval increments of the bus voltages magnitudes and their respective angles are computed as:
Using the interval increments of the bus voltages, the interval voltage magnitudes and phase angles are obtained, using Equation (
16), as:
The interval mismatches of the active and reactive power are updated as presented in
Step 4 of the ILF algorithm. For this illustrative example, the updated mismatches correspond to:
Then, the Krawczyk operator must be applied as shown in
Step 5 of the algorithm. The values for
and f(
) are calculated using Equations (
22)–(
24), obtaining:
Following the calculations presented in
Step 6, the updated values for the voltage magnitudes and angles are obtained as:
The convergence of the ILF algorithm is check as stated in Step 7. As the obtained maximum radius variation is 8.52 × in the of the bus 3, the convergence of the ILF is reached.