Method for fast calculation of electrical network impedances at many frequencies and many system conditions
The calculation of impedances at different frequencies and under a variety of system conditions is computationally burdensome. Accordingly, disclosed embodiments generate a set of reduced bus matrices, retaining nodes of interest from a square bus matrix that models an electrical network, for a plurality of frequencies. This set of reduced bus matrices retains the computational fidelity of the square bus matrix by capturing the influences from removed buses. To evaluate variations in system conditions, change matrices may be generated and applied to the set of reduced bus matrices using matrix algebra.
1 . A method comprising using at least one hardware processor to:
for each of a plurality of frequencies,
acquire a square bus matrix for the frequency, wherein the square bus matrix comprises a row and a column corresponding to each of a plurality of buses within an electrical network of a power system, wherein each location on a diagonal of the square bus matrix represents a sum of parameter values of all electrical components of the electrical network that are connected to one of the plurality buses corresponding to both the row and the column, and wherein each location off the diagonal of the square bus matrix represents a parameter value between one of the plurality of buses corresponding to the row and one of the plurality of buses corresponding to the column,
generate a reduced bus matrix for the frequency from the square bus matrix for the frequency, and
store the reduced bus matrix in a set of reduced bus matrices for the plurality of frequencies, wherein each reduced bus matrix has a same size;
generate a plurality of sets of variation bus matrices by, for each of a plurality of scenarios that each represents a change in system conditions to the electrical network,
generating a change matrix that is the same size as each reduced bus matrix, wherein the change matrix comprises a non-zero value, representing a change in the parameter value, at all locations being changed for that scenario, and a zero value at all locations not being changed for that scenario, and
applying the change matrix to each reduced bus matrix in the set of reduced bus matrices to produce one of the one or more sets of variation bus matrices, wherein the one set of variation bus matrices comprises a variation bus matrix for the scenario for each of the plurality of frequencies;
perform circuit analysis based on the one or more sets of variation bus matrices; and
initiate control of at least one of the electrical components of the electrical network based on the circuit analysis, wherein the control comprises one or more of opening or closing switches, adjusting setpoints, or changing operating modes.
2 . The method of claim 1 , wherein each parameter value is an admittance value.
3 . The method of claim 1 , wherein each parameter value is an impedance value.
4 . The method of claim 1 , wherein applying the change matrix comprises adding the change matrix.
5 . The method of claim 1 , wherein applying the change matrix comprises subtracting the change matrix.
6 . The method of claim 1 , wherein generating a reduced bus matrix for the frequency from the square bus matrix comprises removing one or more of the plurality of buses, while updating the parameter values for retained ones of the plurality of buses based on an influence from the removed one or more buses.
7 . The method of claim 6 , wherein generating a reduced bus matrix for the frequency from the square bus matrix comprises:
partitioning the square bus matrix into
[
K
L
L
T
M
]
wherein K comprises the parameter values for the retained ones of the plurality of buses, M comprises the parameter values for the removed one or more buses, L comprises the parameter values between one of the retained one of the plurality of buses and one of the removed one or more buses, and L T is a transpose of L; and
computing the reduced bus matrix as K−LM −1 L T .
8 . The method of claim 1 , wherein the circuit analysis comprises computing a voltage for each of the plurality of buses in each variation bus matrix in the set of variation bus matrices according to:
V
=
Y
variation
-
1
I
wherein V is a vector of voltages,
Y
variation
-
1
is an invoice of un variation bus matrix, and I is a vector of injected currents, and wherein each parameter value is an admittance value.
9 . The method of claim 1 , wherein each of the plurality of scenarios represents a different topology in the electrical network than at least one other one of the plurality of scenarios.
10 . A system comprising:
at least one hardware processor; and
software configured to, when executed by the at least one hardware processor,
for each of a plurality of frequencies,
acquire a square bus matrix for the frequency, wherein the square bus matrix comprises a row and a column corresponding to each of a plurality of buses within an electrical network of a power system, wherein each location on a diagonal of the square bus matrix represents a sum of parameter values of all electrical components of the electrical network that are connected to one of the plurality buses corresponding to both the row and the column, and wherein each location off the diagonal of the square bus matrix represents a parameter value between one of the plurality of buses corresponding to the row and one of the plurality of buses corresponding to the column,
generate a reduced bus matrix for the frequency from the square bus matrix for the frequency, and
store the reduced bus matrix in a set of reduced bus matrices for the plurality of frequencies, wherein each reduced bus matrix has a same size,
generate a plurality of sets of variation bus matrices by, for each of a plurality of scenarios that each represents a change in system conditions to the electrical network,
generating a change matrix that is the same size as each reduced bus matrix, wherein the change matrix comprises a non-zero value, representing a change in the parameter value, at all locations being changed for that scenario, and a zero value at all locations not being changed for that scenario, and
applying the change matrix to each reduced bus matrix in the set of reduced bus matrices to produce one of the one or more sets of variation bus matrices, wherein the one set of variation bus matrices comprises a variation bus matrix for the scenario for each of the plurality of frequencies,
perform circuit analysis based on the one or more sets of variation bus matrices, and
initiate control of at least one of the electrical components of the electrical network based on the circuit analysis, wherein the control comprises one or more of opening or closing switches, adjusting setpoints, or changing operating modes.
11 . The system of claim 10 , wherein each parameter value is an admittance value.
12 . The system of claim 10 , wherein each parameter value is an impedance value.
13 . The system of claim 10 , wherein applying the change matrix comprises adding or subtracting the change matrix.
14 . The system of claim 10 , wherein generating a reduced bus matrix for the frequency from the square bus matrix comprises removing one or more of the plurality of buses, while updating the parameter values for retained ones of the plurality of buses based on an influence from the removed one or more buses.
15 . The system of claim 10 , wherein the circuit analysis comprises computing a voltage for each of the plurality of buses in each variation bus matrix in the set of variation bus matrices based on one or more injected currents.
16 . The system of claim 10 , wherein the one or more scenarios are a plurality of scenarios, and wherein the one or more sets of variation bus matrices are a plurality of sets of variation matrices that comprise a variation bus matrix for each of the plurality of scenarios for each of the plurality of frequencies.
17 . A non-transitory computer-readable medium having instructions stored thereon, wherein the instructions, when executed by a processor, cause the processor to:
for each of a plurality of frequencies,
acquire a square bus matrix for the frequency, wherein the square bus matrix comprises a row and a column corresponding to each of a plurality of buses within an electrical network of a power system, wherein each location on a diagonal of the square bus matrix represents a sum of parameter values of all electrical components of the electrical network that are connected to one of the plurality buses corresponding to both the row and the column, and wherein each location off the diagonal of the square bus matrix represents a parameter value between one of the plurality of buses corresponding to the row and one of the plurality of buses corresponding to the column,
generate a reduced bus matrix for the frequency from the square bus matrix for the frequency, and
store the reduced bus matrix to a set of reduced bus matrices for the plurality of frequencies, wherein each reduced bus matrix has a same size;
generate a plurality of sets of variation bus matrices by, for each of a plurality of scenarios that each represents a change in system conditions to the electrical network,
generating a change matrix that is the same size as each reduced bus matrix, wherein the change matrix comprises a non-zero value, representing a change in the parameter value, at all locations being changed for that scenario, and a zero value at all locations not being changed for that scenario, and
applying the change matrix to each reduced bus matrix in the set of reduced bus matrices to produce one of the one or more sets of variation bus matrices, wherein the one set of variation bus matrices comprises a variation bus matrix for the scenario for each of the plurality of frequencies;
perform circuit analysis based on the one or more sets of variation bus matrices; and
initiate control of at least one of the electrical components of the electrical network based on the circuit analysis, wherein the control comprises one or more of opening or closing switches, adjusting setpoints, or changing operating modes.