# US 2014/0156094 A1 · Sigma algebraic approximants as a diagnostic tool in power networks

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- Number: US 2014/0156094 A1
- Document ID: US20140156094A1
- Status: Published application
- Kind: A1
- Application number: 14/077,856
- Filed: 2013-11-12
- Published: 2014-06-05
- Earliest claimed: 2012-11-13
- Assignees: Gridqunat, Inc.
- Applicants: Gridquant, Inc.
- Inventors: Antonio Trias
- Web page: https://patentgrep.com/patent/US20140156094A1

## CPC

- H02J 13/13: characterised by the transmission of data to equipment in the power network
- G06F 30/00: Computer-aided design [CAD]
- G06F 30/367: Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
- H02J 13/10: characterised by displaying of information or by user interaction, e.g. supervisory control and data acquisition [SCADA] systems
- H02J 13/12: Monitoring network conditions, e.g. electrical magnitudes or operational status
- H02J 13/1323: {using optical fibres}
- H02J 13/333: the equipment forming part of substations
- G05B 23/0243: {model based detection method, e.g. first-principles knowledge model}
- G06F 2119/06: Power analysis or power optimisation
- H02J 2103/30: Simulating, planning, modelling, reliability check or computer assisted design [CAD] of electric power networks
- Y02E 60/00: Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
- Y04S 10/30: State monitoring, e.g. fault, temperature monitoring, insulator monitoring, corona discharge
- Y04S 10/40: Display of information, e.g. of data or controls
- Y04S 40/124: using wired telecommunication networks or data transmission busses
- Y04S 40/20: Information technology specific aspects, e.g. CAD, simulation, modelling, system security
- H02J: ELECTRIC POWER NETWORKS; CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G)
- Y02E: REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
- Y04S: SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS

## Abstract

A system and method for constructively providing a set of complex-valued indices that measures the proximity of an electrical power distribution system to voltage collapse, and providing insights on general static stability issues in power networks. These indices, termed “sigmas”, are designed through a mathematical osculation of the actual load flow solution with an equivalent two-bus system at every node. The procedure for obtaining these sigmas is based on the Holomorphic Embedded Load flow Method (HELM), and the indices are, in fact, proper analytical functions of the embedding parameter, which endows them with powerful diagnostic capabilities. These indices are then used graphically on a two-dimensional chart, in order to appreciate visually the overall distance of the system to voltage collapse, while at the same time spotting the weak nodes out of the whole grid, providing new situational awareness and diagnosis tools.

## Claims

### Claim 1

A computer-implemented method comprising:
generating a first mathematical model of load flow equations (L) representing an electric power system having an electrical grid in which all voltages are equal to a normal or designed voltage level and there is no energy flow in links of said electrical grid, using a computerized device,
said first mathematical model comprising a holomorphic embedding of said load flow equations as (L(s)), s comprising a variable in a complex domain that includes a value s=0 corresponding to a no load case (L(0)) in which all voltages are equal to a normal or designed voltage level and there is no energy flow in links of said electrical grid, and each variable of said load flow equations (L) is contained in L(s) as a function of said variable s by said holomorphic embedding;
transforming said first mathematical model of said load flow equations (L), with said holomorphic embedding, to a second mathematical model comprising a dimensionless equivalent grid model for each node of said electrical grid, using said computerized device,
said second mathematical model comprising said holomorphic embedding of s in said load flow equations (L(s)) according to a sigma index function, said sigma index function comprising a σ function of s in a complex domain;
transcribing said first mathematical model and said second mathematical model into software for use in a computer processor adapted to execute said software; and
using analytical continuation in said computer processor to perform mathematical osculation of said load flow equations at each said node of said electrical grid to calculate voltage at each said node of said electrical grid.

### Claim 2 (depends on 1)

The computer-implemented method of claim 1, further comprising:
displaying a representation of said sigma index for a node of said electrical grid as a measure of proximity to voltage collapse in said electric power system, using said computerized device.

### Claim 3 (depends on 2)

The computer-implemented method according to claim 2, said measure of proximity to voltage collapse being quantified as a distance of said sigma index to a parabola
σ
I
2
-
σ
R
-
1
4
=
0
on a complex sigma-plane.

### Claim 4 (depends on 2)

The computer-implemented method of claim 2, further comprising:
highlighting a representation of said sigma index for a node of said electrical grid within a predetermined distance to a parabola
σ
I
2
-
σ
R
-
1
4
=
0
on a complex sigma-plane, using said computerized device.

### Claim 5 (depends on 4)

The computer-implemented method of claim 4, further comprising:
determining proximity to voltage collapse of said electric power system based on said representation of said sigma index on said complex sigma-plane, using said computerized device.

### Claim 6 (depends on 1)

The computer-implemented method of claim 1, further comprising:
receiving data from a supervisory and data acquisition system representative of conditions of said electrical grid, using said computerized device; and
calculating voltage at each said node of said electrical grid from said data, using said computerized device.

### Claim 7

A system comprising:
a supervisory control and data acquisition system collecting data from an electrical grid of an electric power system, said electrical grid having a plurality of nodes, said data being indicative of electrical conditions in said electrical grid, and
a processor-controlled energy management system in communication with said supervisory control and data acquisition system,
said processor-controlled energy management system further comprising a program of instructions executable by said energy management system to:
process said data received from said supervisory control and data acquisition system into load flow equations (L) representing said electrical grid;
generate a first mathematical model of said load flow equations (L),
said first mathematical model comprising a holomorphic embedding of said load flow equations as (L(s)), s comprising a variable in a complex domain that includes a value s=0 corresponding to a no load case (L(0)) in which all voltages are equal to a normal or designed voltage level and there is no energy flow in links of said electrical grid, and each variable of said load flow equations (L) is contained in L(s) as a function of said variable s by said holomorphic embedding; and
transform said first mathematical model of said load flow equations (L), with said holomorphic embedding, to a second mathematical model comprising a dimensionless equivalent grid model for each node of said electrical grid,
said second mathematical model comprising a holomorphic embedding of s in said load flow equations (L(s)) according to a sigma index, said sigma index comprising a variable σ in a complex domain.

### Claim 8 (depends on 7)

The system according to claim 7, said processor-controlled energy management system further comprising executable computer instructions to:
use analytical continuation to perform mathematical osculation of said load flow equations at each node of said electrical grid to calculate a at each said node of said electrical grid.

### Claim 9 (depends on 7)

The system according to claim 7, said processor-controlled energy management system further comprising executable computer instructions to:
display a representation of said sigma index for a node of said electrical grid as a measure of proximity to voltage collapse in said electric power system.

### Claim 10 (depends on 9)

The system according to claim 9, said measure of proximity to voltage collapse being quantified as a distance of said sigma index to a parabola
σ
I
2
-
σ
R
-
1
4
=
0
on a complex sigma-plane.

### Claim 11 (depends on 7)

The system according to claim 7, said processor-controlled energy management system further comprising executable computer instructions to:
highlight a representation of said sigma index for a node of said electrical grid within a predetermined distance to a parabola
σ
I
2
-
σ
R
-
1
4
=
0
on a complex sigma-plane.

### Claim 12

A computer-implemented method comprising:
generating a first mathematical model of an electrical power network having an electrical grid with a plurality of nodes using a topology comprising a dimensionless, two-bus equivalent model at each node of said electrical grid, using a computerized device, said model containing only a complex voltage of each said node and a complex parameter, sigma, at each said node;
embedding load flow equations (L) representing said first mathematical model in a holomorphic embedding as (L(s)), using said computerized device, s being a variable in a complex domain that includes a value s=0 corresponding to a no load case (L(0)) in which said complex voltage is equal to a normal or designed voltage level and there is no energy flow in links of said electrical grid and a value s=1 corresponding to an objective case (L(1)) representative of said electrical grid in a condition for which stability is to be determined, each variable of said load flow equations (L) being contained in L(s) as a function of said variable s by said holomorphic embedding;
transforming said first mathematical model of said load flow equations (L) with said holomorphic embedding, to a second mathematical model comprising a dimensionless equivalent grid model for each node of said electrical grid, using said computerized device,
said second mathematical model comprising a holomorphic embedding of s in said load flow equations (L(s)) according to said complex parameter, sigma, at each said node;
transcribing said first mathematical model and said second mathematical model into software for use in a computer processor adapted to execute said software;
developing in power series, values of unknowns in parameters of said holomorphic embedding (L(s)), using said computer processor, said values being in a neighborhood of said value for said no load case of each parameter of said load flow equations;
using said computer processor to compute an n-order algebraic approximant to said power series;
receiving data from connectivity points of said electrical grid, using said computerized device;
using analytical continuation in said computer processor to perform mathematical osculation of said load flow equations at each said node of said electrical grid to calculate said complex voltage at each said node and said complex parameter, sigma, at each said node; and
displaying a representation of said complex parameter, sigma, for a node of said electrical grid as a measure of proximity to voltage collapse in said electrical power network, using said computerized device.

### Claim 13 (depends on 12)

The computer-implemented method of claim 12, further comprising:
after said receiving data from said connectivity points of said electrical grid, testing said data in order to make inferences on any missing information and quality of available measurements, using said computerized device.

### Claim 14 (depends on 13)

The computer-implemented method according to claim 13, said testing of said data including logical considerations concerning said connectivity points and said data received from said connectivity points of said electrical grid.

### Claim 15 (depends on 12)

The computer-implemented method according to claim 12, said receiving data from said connectivity points of said electrical grid comprising receiving data from a supervisory and data acquisition system representative of conditions of said electrical grid, and forming said load flow equations (L) from said data.

### Claim 16 (depends on 12)

The computer-implemented method according to claim 12, said data comprising loads, generations, voltages, flows, and states of circuit breakers.

### Claim 17 (depends on 12)

The computer-implemented method according to claim 12, said measure of proximity to voltage collapse being quantified as a distance of said complex parameter, sigma, to a parabola σI2−σR−¼=0 on a complex sigma-plane.

### Claim 18 (depends on 12)

The computer-implemented method of claim 12, further comprising:
highlighting a representation of said complex parameter, sigma, within a predetermined distance to a parabola σI2−σR−¼=0 on a complex sigma-plane, using said computerized device.

### Claim 19 (depends on 12)

The computer-implemented method of claim 12, further comprising:
diagnosing an electrical state of said electrical power network by analyzing evolution of said complex parameter, sigma, on a two-dimensional sigma plane using functional dependency of said complex parameter, sigma, on said holomorphic embedding parameter s, using said computerized device.

### Claim 20 (depends on 12)

The computer-implemented method according to claim 12, using said representation of said complex parameter, sigma, on a two-dimensional chart to provide assessment of proximity to voltage collapse in said electrical power network.