A power system model can be considered as an undirected connected weighted graph. Such a structure consists of nodes
and edges
[
20]. Each node
represents a busbar. Transmission lines connecting busbars in substations are identified as edges
. Mathematically, the graph is defined as a pair
.
5.1. Weight Calculation Algorithm Based on Real Power
The first subalgorithm concept is based on real power, which is one of the basic parameters present in the power system. In [
6], this kind of logic dedicated for DC-grids was described. This part of the paper presents the solution applicable to AC-grids, which is a modification of that algorithm. The subalgorithm requires
and
These two values represent the power system in the initial state for a configuration of connected transmission lines. In
Figure 1 dotted lines mark this structure. The subalgorithm calculates the weight of the
-th transmission line.
is the limit representing all edges that can be connected, for instance
for the topology in
Figure 1. Parameter
always starts from
when the subalgorithm is called. The computation logic operates on adjacency matrices. In this case, the following arrays are necessary:
matrices have static dimensions and contain data about grid topology and current state.
matrices are dependent on the considered structure in the state when the
-th transmission line is connected. The matrix of weights has a specific consistency. The weights of all edges energized in the initial structure are of equal singular value, e.g.,
. Numbers calculated by the subalgorithm are assigned only to transmission lines that can be connected.
Weight computation based on real power has the following logic:
- (1)
Calculate power losses and voltages when the -th transmission line is connected. In the computation process, the Newton–Raphson method based on the -matrix can be used, for example.
- (2)
Is the voltage on all busbars within allowable range for -th line of the considered power system’s structure? The limit is set from 0.95 pu to 1.05 pu.
- (a)
If YES, go to step 3.
- (b)
If NO, go to step 7.
- (3)
Is the current within limits for all transmission lines, for the topology when the -th line is connected? Limits are different depending on the season of the year. Permissible values are higher during winter than summer.
- (a)
If YES, go to step 4.
- (b)
If NO, go to step 7.
- (4)
Calculate power delivered by the source to the power system with the
-th line connected. The necessary values are computed by the following equations:
- (5)
Are and within operational limits for the source energizing loads in the considered topology?
- (a)
If YES, go to step 6.
- (b)
If NO, go to step 7.
- (6)
Calculate
for the topology with the
-th transmission line connected and go to step 8. Remember the commutated value in adjacency matrix
. The weight is expressed by the following formula:
- (7)
Rewrite
and
values and then go to step 8.
and
are as follows:
- (8)
Is ?
- (a)
If YES, go to step 9.
- (b)
If NO, go to step 10.
- (9)
End subalgorithm and continue the main program.
- (10)
Update the
value and go to step 1. The
variable is calculated by:
- (11)
The subalgorithm’s logic structure is presented in graphical form in
Figure 2.
5.4. Weight Calculation Algorithm Based on Normalized Factor Including Combined Influence of Real and Reactive Power in Distribution Grid
The structure of power system is complicated. The calculations of grid parameters in electrical engineering are based on complex numbers, which are graphically presented as phasors. The proper relations between power system parameters and weights require consideration of real power and reactive power simultaneously. This is possible when both values are expressed in pu.
The computation logic uses adjacency matrices: , and . The following arrays have static dimensions: , and , arrays have a consistency dependent on the analyzed power system’s topology when the -th transmission line is connected. Adjacency matrix , for a state of the topology before consideration of energization of the -th edge, is made up of singular values, e.g., negative numbers like . The singular value for and has to be different than the one for In this case, the best solution is to set the number as a singular value for and . All other terms of matrices and , different from singular values, are calculated by the subalgorithm for transmission lines considered as possible to connect.
Weight computation based on normalized factor, including the combined influence of real and reactive power in the grid, has the following logic:
- (1)
Calculate power losses and , currents , and voltages when the -th transmission line is connected. In the computation process, the Newton–Raphson method based on the -matrix can be used, for example.
- (2)
Is the voltage at all busbars within tolerable limits for the -th line in the considered power system’s structure? The limit is set from 0.95 pu to 1.05 pu.
- (a)
If YES, go to step 3.
- (b)
If NO, go to step 7.
- (3)
Is the current within rated limits for all transmission lines, for the topology when the -th line is connected? Limits are different depending on the season of the year. Tolerable values are higher during the winter than the summer.
- (a)
If YES, go to step 4
- (b)
If NO, go to step 7.
- (4)
Calculate power delivered by the source to the power system with the -th line connected. The necessary values are computed by Equations (3) and (4).
- (5)
Are and within operational limits for the source energizing loads in the considered topology?
- (a)
If YES, go to step 6.
- (b)
If NO, go to step 7.
- (6)
Put and in matrices and and go to step 8.
- (7)
Rewrite
by Formula (7) and rewrite
value as:
and then go to step 6.
- (8)
Is ?
- (a)
If YES, go to step 10.
- (b)
If NO, go to step 9.
- (9)
Update the value of by Formula (6) and go to step 1.
- (10)
Update the value of
using the formula:
and go to step 11.
- (11)
Are all terms in singular values?
- (a)
If NO, go to step 12.
- (b)
If YES, go to step 18.
- (12)
Analyze matrices
and
and define factors
and
:
and go to step 13.
and
cannot be singular values.
- (13)
Is the real power of the considered line a positive number?
- (a)
If YES, go to step 14.
- (b)
If NO, go to step 17.
- (14)
Calculate
for the topology with the
-th transmission line connected. Put the commutated value into adjacency matrix
and go to step 15. The weight is expressed by the following Formula [
21]:
is a real positive number in the range from
.
- (15)
- (a)
If YES, go to step 21.
- (b)
If NO, go to step 16.
- (16)
Update the value of using Formula (8) and go to step 13.
- (17)
Weight for topology with the -th transmission line connected. Put the commuted value into adjacency matrix and go to step 15.
- (18)
Weight for topology with the -th transmission line connected. Put the commutated value into adjacency matrix and go to step 19.
- (19)
Is ?
- (a)
If YES, go to step 21.
- (b)
If NO, go to step 20
- (20)
Update the value of using Formula (8) and go to step 18.
- (21)
End subalgorithm and continue the main program.
In [
21], factors (14) and (15) are defined as minimal (min) values of
and
respectively. This paper presents a generalized situation. The word “minimal” from [
21] is replaced by “optimal” (opt), because in some greedy algorithms maximal values are important for weights, and minimal ones in others applications [
22].
Figure 3 presents the described subalgorithm graphically.
It is important to notice that the subalgorithm presented here is a novel solution based on Formula (16). Expression (16) in [
21] is used for sectionizing logic for AC-grids represented as graphs. The solution presented in the article is its modification applicable to restoration strategies based on Prim’s algorithm.
It is crucial for the presented algorithm to determine the value of the p coefficient. It can be defined by using an optimization algorithm such as partical swarm optimization (PSO). A swarm of particles is looking for a solution that meets predefined assumptions. In case of the power system finding of the p-coefficient depends on two conditions:
- (1)
All the loads need to be connected to the power system;
- (2)
The losses of active power in the created topology are as small as possible.
The above mentioned guidelines are hierarchical, i.e., the most important goal is to supply all the loads while maintaining the quality requirements for electricity. It means that the voltage on the busbars within the required scope and not to exceed the load capacity of the long-term power supply lines. Only the second optimization criterion is to minimize the active power losses in the power system. The optimization criteria should be determined by the means of mathematical formulas. In case of this article, (1) and (2) coefficients are used for the PSO algorithm.