CALCULATING LINE-TO-NEUTRAL VOLTAGES WITHOUT A CONNECTION TO A SYSTEM NEUTRAL OR EARTH GROUND

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

1. Field

The subject matter disclosed herein relates to calculating voltage and more particularly relates to calculating line-to-neutral voltage without a connection to a system neutral or earth ground.

2. Background Information

Typical three-phase systems include three alternating current (“AC”) voltages that have a fundamental frequency, such a 50 hertz (“Hz”), 60 Hz, 400 Hz, etc. that are offset in time so that the three phases are spaced 120 degrees within one cycle. A three-phase power system can supply power to three-phase loads as well as single-phase loads. Many three-phase loads, such as motors are balanced loads under normal operating conditions and do not require a neutral connection. Other loads may also not require a neutral connection, such as the primary side of a transformer in a delta-wye or delta-delta configuration. Measurement of voltages within a three-phase power system can be important for a variety of reasons, such as power quality monitoring, overcurrent protection, power monitoring, etc.

Some three-phase power systems have neutral connections available for some loads while other three-phase loads do not have a neutral connection available. Often power systems have a connection between the neutral, which is also called the grounded conductor, and a system ground, such as earth ground or a ground to a structure. For safety reasons, a grounding conductor, which is usually marked green or is bare copper, is run with power conductors to a load. This grounding conductor may be called a safety ground and typically connects to structures and frames of equipment being powered in the three-phase power system to provide a low impedance path for fault current. The safety ground is often nearly the same potential as the neutral, however, when current flows in the neutral, voltage drop from the load to the neutral to ground connection, which is usually at a three-phase power source for a system, can be substantial. In fault conditions, at a load voltage between a neutral and a safety ground can vary significantly so determining line-to-neutral voltage by simply measuring line-to-ground can be inaccurate, especially under fault conditions.

Some three-phase power systems are isolated from a grounded structure. For example, some three-phase power systems in marine vessels are isolated from the grounded structure of the vessel. For loads in such an ungrounded system that do not have a neutral available, line-to-neutral voltage measurements are not available. Having a line-to-neutral voltage available for loads with no available neutral connection and for three-phase power systems that are floating is desirable for a variety of reasons.

BRIEF DESCRIPTION

A method for determining voltage is disclosed. An apparatus and computer program product also perform the functions of the method. The method for determining voltage includes measuring three line-to-line voltages for the phases in a three-phase power system, where each line-to-line voltage includes a voltage magnitude, and constructing, on a two-dimensional coordinate system with an origin, a phasor triangle that includes the three line-to-line voltages represented as phasors. A first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin and a second phasor V_bcextends between the origin and a second point. The second point is in a direction vertically and horizontally from the origin. A third phasor V_caextends between the second point and the first point.

The method, in one embodiment, includes adding a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at a third point. The method, in another embodiment, includes adding a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at a fourth point. In one embodiment, the method includes adding a third line segment from the third point to the first point and adding a fourth line segment from the fourth point to the origin, where the third line segment intersects the fourth line segment at a neutral point. The method, in another embodiment, includes determining a line-to-neutral voltage. The line-to-neutral voltage includes a line from the neutral point to a vertex of the phasor triangle.

In one embodiment, determining a line-to-neutral voltage may include determining a line-to-neutral voltage for a phasor V_anfor phase A which includes determining a line from the neutral point to the first point, determining a line-to-neutral voltage for a phasor V_bnfor phase B which includes determining a line from the neutral point to the origin, and/or determining a line-to-neutral voltage for a phasor V_cnfor phase C which includes determining a line from the neutral point to the second point. In another embodiment, the method includes determining a magnitude of phasor V_anas

V
_an=√{square root over ((x_a−x_n)²+(y_a−y_n)²)}{square root over ((x_a−x_n)²+(y_a−y_n)²)};

determining a magnitude of phasor V_bnas

V
_bn=√{square root over ((x_b−x_n)²+(y_b−y_n)²)}{square root over ((x_b−x_n)²+(y_b−y_n)²)}; and

determining a magnitude of phasor V_cnas

V
_cn=√{square root over ((x_c−x_n)²+(y_c−y_n)²)}{square root over ((x_c−x_n)²+(y_c−y_n)²)}.

In one embodiment, the first point has a coordinate of (x_a, y_a), the second point has a coordinate of (x_c, y_c), and the origin has a coordinate of (x_b, y_b) and x_a=the magnitude of the V_abphasor and y_a=0, x_b=0 and

$y_{b} = 0, x_{c} = \frac{V_{ab}^{2} - V_{ca}^{2} + V_{bc}^{2}}{2 \cdot V_{ab}}$

and

$y_{c} = V_{ca} \cdot \sqrt{1 - (\frac{V_{bc}^{2} - V_{ab}^{2} - V_{ca}^{2}}{4 \cdot V_{ab}^{2} \cdot V_{ca}^{2}})},$

where V_abis a magnitude of the first phasor V_ab, V_bcis a magnitude of the second phasor V_bc, and V_cais a magnitude of the third phasor V_ca. In one embodiment, the length of the first line segment has a magnitude of the third phasor V_cadivided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment and the length of the second line segment has a magnitude of the second phasor V_bcdivided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment.

In another embodiment, the length of the first line segment is:

$\frac{V_{ca}}{\sqrt{3}} \cdot \sqrt{1 + {(\frac{- x_{c}}{y_{c}})}^{2}}$

and the length of the second line segment is

$\frac{V_{bc}}{\sqrt{3}} \cdot \sqrt{1 + {(\frac{- (x_{a} - x_{c})}{(y_{a} - y_{c})})}^{2}} .$

In one embodiment, the third point has a coordinate of (x_pbc, y_pbc) and the fourth point has a coordinate of (x_pca, y_pca), where

$x_{pbc} = \frac{x_{c}}{2} - \frac{V_{ca}}{\sqrt{3}};$

$y_{pbc} = \frac{y_{c}}{2} - \frac{- x_{c}}{y_{c}} \cdot \frac{V_{ca}}{\sqrt{3}};$

$x_{pca} = x_{c} + \frac{x_{a} - x_{c}}{2} + \frac{V_{bc}}{\sqrt{3}}; and$

$y_{pca} = \frac{y_{c}}{2} + \frac{- (x_{a} - x_{c})}{(y_{a} - y_{c})} \cdot \frac{V_{bc}}{\sqrt{3}} .$

In another embodiment, the neutral point has a coordinate of (x_n, y_n), where:

$x_{n} = \frac{\frac{y_{pbc}}{x_{a} - x_{pbc}} \cdot x_{a}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}}; and$

$y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} .$

In another embodiment, where y_nis less than zero then y_n=0, and where y_nis greater than or equal to zero and greater than y_cthen y_n=y_c. In another embodiment, where x_nis less than zero then x_n=0, and where x_nis greater than or equal to zero and greater than x_athen x_n=x_a, and where x_nis greater than or equal to zero and less than or equal to x_aand y_c=y_nthen x_n=x_c.

In one embodiment, the length of the first line segment has a magnitude of the second phasor V_bcmultiplied by the square root of three and divided by two, which represents a height of a first equilateral triangle constructed on the second phasor V_bcand extending away from the phasor triangle and the length of the second line segment has a magnitude of the third phasor V_camultiplied by the square root of three and divided by two, which represents a height of a second equilateral triangle constructed on the third phasor V_caand extending away from the phasor triangle. In another embodiment, the third point has a coordinate of (x_pbc, y_pbc) and the fourth point has a coordinate of (x_pca, y_pca), where:

$x_{pbc} = \frac{x_{c} - y_{c} \cdot \sqrt{3}}{2};$

$y_{pbc} = \frac{y_{c} + x_{c} \cdot \sqrt{3}}{2};$

$x_{pca} = x_{a} + \frac{x_{c} - x_{a} + \sqrt{3} \cdot y_{c}}{2}; and$

$y_{pca} = \frac{(y_{c} - y_{a}) + \sqrt{3} \cdot (x_{a} - x_{c})}{2} .$

In another embodiment, the neutral point has a coordinate of (x_n, y_n) where:

$x_{n} = \frac{\frac{x_{a} \cdot y_{pbc}}{x_{a} - x_{pbc}}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}}; and$

$y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} .$

In one embodiment, the line-to-line voltages are measured at a location in the three-phase power system where a neutral connection is unavailable for measurement. In another embodiment, the three-phase power system includes an ungrounded power system. In another embodiment, the three-phase power system includes unbalanced voltages. In another embodiment, the first phasor V_ab, the second phasor V_bc, and the third phasor V_caare separated by 120 degrees.

An apparatus for determining voltages, in one embodiment, includes a measurement module that measures three line-to-line voltages for the phases in a three-phase power system, where each line-to-line voltage includes a voltage magnitude. The apparatus, in one embodiment, includes a triangle module that constructs, on a two-dimensional coordinate system with an origin, a phasor triangle with the three line-to-line voltages represented as phasors, where a first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor V_bcextends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor V_caextends between the second point and the first point.

The apparatus, in one embodiment, includes a first line module that adds a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at a third point. In another embodiment, the apparatus includes a second line module that adds a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at a fourth point. The apparatus, in one embodiment, includes a third line module that adds a third line segment from the third point to the first point and a fourth line module that adds a fourth line segment from the fourth point to the origin, where the third line segment intersects the fourth line segment at a neutral point. The apparatus, in another embodiment, includes a line-to-neutral module that determines a line-to-neutral voltage. The line-to-neutral voltage includes a line from the neutral point to a vertex of the phasor triangle.

In one embodiment, the line-to-neutral module determines a line-to-neutral voltage by determining a line-to-neutral voltage for phasor V_anfor phase A which includes determining a line from the neutral point to the first point. In another embodiment, the line-to-neutral module determines a line-to-neutral voltage by determining a line-to-neutral voltage for phasor V_bnfor phase B which includes determining a line from the neutral point to the origin. In another embodiment, the line-to-neutral module determines a line-to-neutral voltage by determining a line-to-neutral voltage for phasor V_cnfor phase C which comprises determining a line from the neutral point to the second point. For example, the apparatus may include a line-to-neutral magnitude module that determines a magnitude of phasor V_anas

V
_an=√{square root over ((x_a−x_n)²+(y_a−y_n)²)}{square root over ((x_a−x_n)²+(y_a−y_n)²)},

determines a magnitude of phasor V_bnas

V
_bn=√{square root over ((x_b−x_n)²+(y_b−y_n)²)}{square root over ((x_b−x_n)²+(y_b−y_n)²)}, and/or

determines a magnitude of phasor V_cnas

V
_cn=√{square root over ((x_c−x_n)²+(y_c−y_n)²)}{square root over ((x_c−x_n)²+(y_c−y_n)²)}.

In one embodiment, first point has a coordinate of (x_a, y_a), the second point has a coordinate of (x_c, y_c), and the origin has a coordinate of (x_b, y_b) where x_a=the magnitude of the V_abphasor and y_a=0, x_b=0 and y_b=0, and

$x_{c} = \frac{V_{ab}^{2} - V_{ca}^{2} + V_{bc}^{2}}{2 \cdot V_{ab}}$

$and$

$y_{c} = V_{ca} \cdot \sqrt{1 - (\frac{V_{bc}^{2} - V_{ab}^{2} - V_{ca}^{2}}{4 \cdot V_{ab}^{2} \cdot V_{ca}^{2}})} .$

V_abis a magnitude of the first phasor V_ab, V_bcis a magnitude of the second phasor V_bc, and V_cais a magnitude of the third phasor V_ca.

In one embodiment, the length of the first line segment has a magnitude of the third phasor V_cadivided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment and the length of the second line segment has a magnitude of the second phasor V_bcdivided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment.

In another embodiment, the third point has a coordinate of (x_pbc, y_pbc) and the fourth point has a coordinate of (x_pca, y_pca), where

In one embodiment, the neutral point has a coordinate of (x_n, y_n), where:

$x_{n} = \frac{\frac{y_{pbc}}{x_{a} - x_{pbc}} \cdot x_{a}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}}; and$

$y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} .$

In one embodiment, where y_nis less than zero then y_n=0, and where y_nis greater than or equal to zero and greater than y_cthen y_n=y_c. In another embodiment, where x_nis less than zero then x_n=0, and where x_nis greater than or equal to zero and greater than x_athen x_n=x_a, and where x_nis greater than or equal to zero and less than or equal to x_aand y_c=y_nthen x_n=x_c.

In one embodiment, the length of the first line segment has a magnitude of the second phasor V_bcmultiplied by the square root of three and divided by two, which represents a height of a first equilateral triangle constructed on the second phasor V_bcand extending away from the phasor triangle and the length of the second line segment has a magnitude of the third phasor V_camultiplied by the square root of three and divided by two, which represents a height of a second equilateral triangle constructed on the third phasor V_caand extending away from the phasor triangle. For example, the third point may have a coordinate of (x_pbc, y_pbc) and the fourth point may have a coordinate of (x_pca, y_pca) where:

In one embodiment, the neutral point has a coordinate of (x_n, y_n) and where:

$x_{n} = \frac{\frac{x_{a} \cdot y_{pbc}}{x_{a} - x_{pbc}}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}}; and$

$y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} .$

In another embodiment, the apparatus includes a meter where the meter includes the measurement module. In another embodiment, the apparatus includes a processor that executes executable code of the measurement module, the triangle module, the first line module, the second line module, the third line module, the fourth line module, and/or the line-to-neutral module.

A computer program product for determining a voltage is included. The computer program product includes a computer readable storage medium having program code embodied therein. The program code is readable/executable by a processor for measuring three line-to-line voltages for the phases in a three-phase power system, where each line-to-line voltage includes a voltage magnitude, and constructing, on a two-dimensional coordinate system with an origin, a phasor triangle with the three line-to-line voltages represented as phasors. A first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin. A second phasor V_bcextends between the origin and a second point, where the second point is in a direction vertically and horizontally from the origin. A third phasor V_caextends between the second point and the first point.

In one embodiment, the program code is readable/executable by a processor for determining a third point by adding a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at the third point. In another embodiment, the program code is readable/executable by a processor for determining a fourth point by adding a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at the fourth point.

The program code, in one embodiment, is readable/executable by a processor for determining a neutral point by adding a third line segment from the third point to the first point and adding a fourth line segment from the fourth point to the origin, where the third line segment intersects the fourth line segment at the neutral point, and determining a line-to-neutral voltage. The line-to-neutral voltage includes a line from the neutral point to a vertex of the phasor triangle.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the advantages of the embodiments of the invention will be readily understood, a more particular description of the embodiments briefly described above will be rendered by reference to specific embodiments that are illustrated in the appended drawings. Understanding that these drawings depict only some embodiments and are not therefore to be considered to be limiting of scope, the embodiments will be described and explained with additional specificity and detail through the use of the accompanying drawings, in which:

FIG. 1 is a schematic block diagram illustrating one embodiment of a system for determining voltages;

FIG. 2 is a schematic block diagram illustrating one embodiment of an apparatus for determining voltages;

FIG. 3 is a diagram of voltages of a three-phase power system with unbalanced voltages;

FIG. 4 is a diagram of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle;

FIG. 5 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides using a first geometric method;

FIG. 6 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides and lines intersecting a neutral point using the first geometric method;

FIG. 7 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with line-to-neutral phasors shown from the neutral point using the first geometric method;

FIG. 8 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides where the length is based on an equilateral triangle using a second geometric method;

FIG. 9 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides and lines intersecting a neutral point using the second geometric method;

FIG. 10 is a diagram of voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle with line-to-neutral phasors shown from the neutral point using the second geometric method;

FIG. 11 is a schematic flow chart diagram illustrating one embodiment of a method for determining voltages;

FIG. 12A is a first part of a schematic flow chart diagram illustrating another embodiment of a method for determining voltages;

FIG. 12B is second part of a schematic flow chart diagram illustrating another embodiment of a method for determining voltages;

FIG. 13 is a plot of laboratory test results showing percent voltage imbalance versus reported voltage error using the first geometric method; and

FIG. 14 is a plot of laboratory test results showing reported voltage error under balanced conditions using the first geometric method.

DETAILED DESCRIPTION

Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, appearances of the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment, but mean “one or more but not all embodiments” unless expressly specified otherwise. The terms “including,” “comprising,” “having,” and variations thereof mean “including but not limited to” unless expressly specified otherwise. An enumerated listing of items does not imply that any or all of the items are mutually exclusive and/or mutually inclusive, unless expressly specified otherwise. The terms “a,” “an,” and “the” also refer to “one or more” unless expressly specified otherwise.

Furthermore, the described features, advantages, and characteristics of the embodiments may be combined in any suitable manner. One skilled in the relevant art will recognize that the embodiments may be practiced without one or more of the specific features or advantages of a particular embodiment. In other instances, additional features and advantages may be recognized in certain embodiments that may not be present in all embodiments.

These features and advantages of the embodiments will become more fully apparent from the following description and appended claims, or may be learned by the practice of embodiments as set forth hereinafter. As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method, and/or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module,” or “system.” Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having program code embodied thereon.

Many of the functional units described in this specification have been labeled as modules, in order to more particularly emphasize their implementation independence. For example, a module may be implemented as a hardware circuit comprising custom VLSI circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. A module may also be implemented in programmable hardware devices such as field programmable gate arrays, programmable array logic, programmable logic devices or the like.

Modules may also be implemented in software for execution by various types of processors. An identified module of program code may, for instance, comprise one or more physical or logical blocks of computer instructions which may, for instance, be organized as an object, procedure, or function. Nevertheless, the executables of an identified module need not be physically located together, but may comprise disparate instructions stored in different locations which, when joined logically together, comprise the module and achieve the stated purpose for the module.

Indeed, a module of program code may be a single instruction, or many instructions, and may even be distributed over several different code segments, among different programs, and across several memory devices. Similarly, operational data may be identified and illustrated herein within modules, and may be embodied in any suitable form and organized within any suitable type of data structure. The operational data may be collected as a single data set, or may be distributed over different locations including over different storage devices, and may exist, at least partially, merely as electronic signals on a system or network. Where a module or portions of a module are implemented in software, the program code may be stored and/or propagated on in one or more computer readable medium(s).

The computer readable medium may be a tangible computer readable storage medium storing the program code. The computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, holographic, micromechanical, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing.

More specific examples of the computer readable storage medium may include but are not limited to a portable computer diskette, a hard disk, a random access memory (“RAM”), a read-only memory (“ROM”), an erasable programmable read-only memory (“EPROM” or Flash memory), a portable compact disc read-only memory (“CD-ROM”), a digital versatile disc (“DVD”), an optical storage device, a magnetic storage device, a holographic storage medium, a micromechanical storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain, and/or store program code for use by and/or in connection with an instruction execution system, apparatus, or device.

The computer readable medium may also be a computer readable signal medium. A computer readable signal medium may include a propagated data signal with program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electrical, electro-magnetic, magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport program code for use by or in connection with an instruction execution system, apparatus, or device. Program code embodied on a computer readable signal medium may be transmitted using any appropriate medium, including but not limited to wire-line, optical fiber, Radio Frequency (“RF”), or the like, or any suitable combination of the foregoing

In one embodiment, the computer readable medium may comprise a combination of one or more computer readable storage mediums and one or more computer readable signal mediums. For example, program code may be both propagated as an electro-magnetic signal through a fiber optic cable for execution by a processor and stored on RAM storage device for execution by the processor.

Program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Smalltalk, C++, PHP or the like and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The program code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (“LAN”) or a wide area network (“WAN”), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). The computer program product may be shared, simultaneously serving multiple customers in a flexible, automated fashion.

The computer program product may be integrated into a client, server and network environment by providing for the computer program product to coexist with applications, operating systems and network operating systems software and then installing the computer program product on the clients and servers in the environment where the computer program product will function. In one embodiment software is identified on the clients and servers including the network operating system where the computer program product will be deployed that are required by the computer program product or that work in conjunction with the computer program product. This includes the network operating system that is software that enhances a basic operating system by adding networking features.

Furthermore, the described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In the following description, numerous specific details are provided, such as examples of programming, software modules, user selections, network transactions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments. One skilled in the relevant art will recognize, however, that embodiments may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment.

Aspects of the embodiments are described below with reference to schematic flowchart diagrams and/or schematic block diagrams of methods, apparatuses, systems, and computer program products according to embodiments of the invention. It will be understood that each block of the schematic flowchart diagrams and/or schematic block diagrams, and combinations of blocks in the schematic flowchart diagrams and/or schematic block diagrams, can be implemented by program code. The program code may be provided to a processor of a general purpose computer, special purpose computer, sequencer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions/acts specified in the schematic flowchart diagrams and/or schematic block diagrams block or blocks.

The program code may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function/act specified in the schematic flowchart diagrams and/or schematic block diagrams block or blocks.

The program code may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the program code which executed on the computer or other programmable apparatus provide processes for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.

The schematic flowchart diagrams and/or schematic block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of apparatuses, systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the schematic flowchart diagrams and/or schematic block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions of the program code for implementing the specified logical function(s).

It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the Figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more blocks, or portions thereof, of the illustrated Figures.

Although various arrow types and line types may be employed in the flowchart and/or block diagrams, they are understood not to limit the scope of the corresponding embodiments. Indeed, some arrows or other connectors may be used to indicate only the logical flow of the depicted embodiment. For instance, an arrow may indicate a waiting or monitoring period of unspecified duration between enumerated steps of the depicted embodiment. It will also be noted that each block of the block diagrams and/or flowchart diagrams, and combinations of blocks in the block diagrams and/or flowchart diagrams, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and program code.

FIG. 1 is a schematic block diagram illustrating one embodiment of a system 100 for determining voltages. The system 100 includes a voltage measurement apparatus 102, which in some embodiments may include a meter 104, a processor 106 and/or memory 108, a three-phase voltage source 110, a first load 112 and a second load 114 connected to the voltage source 110 with a phase A, a phase B, and a phase C, and may include a grounded structure 116, which are described below.

The system 100 includes a voltage measurement apparatus 102 that determines line-to-neutral voltages. In one embodiment, the voltage measurement apparatus 102 determines line-to-neutral voltages from line-to-line voltage of a three-phase power system, such as the system 100 depicted in FIG. 1. The voltage measurement apparatus 102, in one embodiment, calculates line-to-neutral voltage where a neutral is not available. For instance, the three-phase voltage source 110 may be isolated from a grounded structure 116 and loads of the system 100 (e.g. the first and second loads 112, 114) may be three-phase loads without a neutral connection.

In one embodiment, the voltage measurement apparatus 102 includes a meter 104. The meter 104, in one example, is located with the voltage measurement apparatus 102. For example, functionality of the voltage measurement apparatus 102 may be included in the meter 104. For example, the meter 104 may be a PowerMonitor 5000™ by Allen-Bradley® and functionality of the voltage measurement apparatus 102 may be incorporated into the PowerMonitor 5000. In another embodiment, the meter 104 may be apart from the voltage measurement apparatus 102. For example, the meter 104 may measure voltages and may send voltage measurement information to the voltage measurement apparatus 102. In another embodiment, a portion of the meter 104 may be located external to the voltage measurement apparatus 102 while a portion is within the voltage measurement apparatus 102.

In another embodiment, the voltage measurement apparatus 102 includes a processor 106 and/or memory 108. The processor 106 may execute code associated with the voltage measurement apparatus 102. For example, the voltage measurement apparatus 102 may be embodied by computer program product for determining a voltage. The computer program product may be stored on a computer readable storage medium, such as the memory 108. The memory 108 may include RAM, ROM, a hard disk drive, etc. that has the program code embodied therein and the program code may be readable/executable by the processor 106. In another embodiment, the voltage measurement apparatus 102 is fully or partially embodied by logic hardware and may be embodied in part, in some embodiments, by executable code. Functionality of the voltage measurement apparatus 102 is explained further in the apparatus 200 of FIG. 2.

The system 100, in one embodiment, includes a three-phase voltage source 110. The three-phase voltage source 110 may be three-phase power from an electric utility, may be a generator, may be an uninterruptable power supply, may be a switching power supply, or other power source that provides three-phase power. Three-phase power, in one embodiment, includes three sinusoidal voltage waveforms offset by 120 degrees. The waveforms may include a sinusoidal fundamental frequency waveform and may include harmonic waveforms. The sinusoidal fundamental frequency waveform of the three waveforms is each offset by 120 degrees.

In one embodiment, the system 100 includes one or more loads, for example the first load 112 and the second load 114. The first load 112 may be a motor, such as a three-phase motor. Often three-phase motors do not include a neutral connection and a neutral wire is not run between the three-phase voltage source 110 and the motor load 112. The second load 114 may be a branch panel. While branch panels typically include a neutral connection, in some examples a transformer (not shown) is located between the branch panel 114 and the three-phase voltage source 110 and the transformer may be a delta-wye transformer where no neutral is run upstream of the transformer. Other loads and power system configurations may also not have a neutral connection or ground connection available. In some embodiments, a grounded structure 116 or earth ground may not be available, for example for floating power systems that require isolation from ground.

Typically, the three waveforms from the three-phase voltage source 110 are balanced so that line-to-line voltages are equal and line-to-neutral voltages are equal. During this condition, determining line-to-neutral voltage from line-to-line voltage may reasonably be determined by dividing the line-to-line voltage by the square root of three. When the three voltage waveforms from the three-phase voltage source 110 or as measured at some point within the system 100 become unbalanced, determination of line-to-neutral voltage from the line-to-line voltages is not trivial. The voltage measurement apparatus 102 may be used to determine line-to-neutral voltages from line-to-line voltages.

FIG. 2 is a schematic block diagram illustrating one embodiment of an apparatus 200 for determining voltages. The apparatus 200 includes a measurement module 202, a triangle module 204, a first line module 206, a second line module 208, a third line module 210, a fourth line module 212, a line-to-neutral module 214, and in some embodiments, a line-to-neutral magnitude module 216, which are described below.

The apparatus 200, in one embodiment, includes a measurement module 202 that measures three line-to-line voltages for the phases in a three-phase power system, such as the system 100 depicted in FIG. 1. Each line-to-line voltage includes a voltage magnitude. For example, the line-to-line voltage magnitude may be a root-mean-square (“RMS”) voltage measured between two phases. For instance the measurement module 202 may measure RMS voltages. The measurement module 202 may connect to each of the three phases A, B, and C, and a line-to-line measurement may be between Phase A and Phase B, between Phase B and Phase C, or between Phase C and Phase A. In another embodiment, the voltage magnitude may be a peak voltage. For example, the peak voltage may be a highest voltage within a cycle of a fundamental sinusoidal frequency within the three line-to-line voltages and the measurement module 202 may measure a peak voltage of the three phases, A, B, and C. The peak voltages, in another embodiment, are converted RMS voltage measurements. One example of the measurement module 202 is the meter 104 depicted in the system 100 of FIG. 1.

The apparatus 200, in one embodiment, includes a triangle module 204 that constructs, on a two-dimensional coordinate system with an origin, a phasor triangle that includes the three line-to-line voltages, measured by the measurement module 202, represented as phasors. A first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin of the coordinate system. A second phasor V_bcextends between the origin and a second point. The second point is in a direction vertically and horizontally from the origin. A third phasor V_caextends between the second point and the first point. An example of a phasor triangle constructed from three line-to-line voltage phasors of FIG. 3 is shown in FIG. 4. In FIG. 3, the third phasor V_cais two-thirds the voltage of the first and second phasors, V_aband V_bc.

FIG. 3 is a diagram of voltages of a three-phase power system with unbalanced voltages. FIG. 4 is a diagram of the voltages of the three-phase power system with unbalanced voltages repositioned as a phasor triangle. FIG. 3 is a typical phasor diagram with phasor V_aboriented at zero degrees, phasor V_bcat 120 degrees, and phasor V_caat −120 degrees (240 degrees) where angles are measured clockwise from the positive X axis. Note that with a voltage imbalance, angles between the voltage phasors V_ab, V_bc, and V_cachange from the 120 degree separation in FIG. 3 and angles of the phasor triangle are determined by magnitudes of the phasors. The first point is at the termination of phasor V_aband is (x_a, y_a) so x_a=V_aband y_b=0. The origin is (x_b, y_b) so x_b=0 and y_b=0. The second point is (x_c, y_c) and may be determined using the law of cosines, where

$\begin{matrix} x_{c} = \frac{V_{ab}^{2} - V_{ca}^{2} + V_{bc}^{2}}{2 \cdot V_{ab}} & Equation (1) \\ y_{c} = V_{ca} \cdot \sqrt{1 - (\frac{V_{bc}^{2} - V_{ab}^{2} - V_{ca}^{2}}{4 \cdot V_{ab}^{2} \cdot V_{ca}^{2}})} & Equation (2) \end{matrix}$

V_ab, V_bc, and V_caare magnitudes of the three line-to-line voltages measured by the measurement module 202.

First Geometric Method

The apparatus 200, in one embodiment, includes a first line module 206 that adds a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at a third point. FIG. 5 is a diagram of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides using a first geometric method. FIG. 5 depicts the first line segment for the first geometric method. In the embodiment, the first line segment terminates at a third point, (x_pbc, y_pbc). In the depicted first geometric method, the length of the first line segment has a magnitude of the third phasor V_cadivided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment. A second geometric method for is depicted in FIGS. 8-10. The slope of the first line segment may be expressed as:

$\begin{matrix} {slope}_{pbc} = \frac{- x_{c}}{y_{c}} & Equation (3) \end{matrix}$

The length of the first line segment, in one embodiment, then may be expressed as:

$\begin{matrix} \frac{V_{ca}}{\sqrt{3}} \cdot \sqrt{1 + {(\frac{- x_{c}}{y_{c}})}^{2}} & Equation (4) \end{matrix}$

From the position and length of the first line segment, in one example, the third point may be determined as:

$\begin{matrix} x_{pbc} = \frac{x_{c}}{2} - \frac{V_{ca}}{\sqrt{3}} & Equation (5) \\ y_{pbc} = \frac{y_{c}}{2} - \frac{- x_{c}}{y_{c}} \cdot \frac{V_{ca}}{\sqrt{3}} & Equation (6) \end{matrix}$

In one embodiment, the apparatus 200 includes a second line module 208 that adds a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at a fourth point. FIG. 5 also depicts the second line segment for the first geometric method. In the embodiment, the second line segment terminates at a fourth point, (x_pca, y_pca). In the first geometric method, the length of the second line segment has a magnitude of the second phasor V_bcdivided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment. In one embodiment, the slope of the second line segment may be expressed as:

$\begin{matrix} {slope}_{pca} = \frac{- (x_{a} - x_{c})}{(y_{a} - y_{c})} & Equation (7) \end{matrix}$

In another embodiment, the length of the second line segment then may be expressed as:

$\begin{matrix} \frac{V_{bc}}{\sqrt{3}} \cdot \sqrt{1 + {(\frac{- (x_{a} - x_{c})}{(y_{a} - y_{c})})}^{2}} & Equation (8) \end{matrix}$

From the position and length of the second line segment, the fourth point may be determined as:

$\begin{matrix} x_{pca} = x_{c} + \frac{x_{a} - x_{c}}{2} + \frac{V_{bc}}{\sqrt{3}} & Equation (9) \\ x_{pca} = \frac{y_{c}}{2} + \frac{- (x_{a} - x_{c})}{(y_{a} - y_{c})} \cdot \frac{V_{bc}}{\sqrt{3}} & Equation (10) \end{matrix}$

The apparatus 200, in one embodiment, includes a third line module 210 that adds a third line segment from the third point to the first point and a fourth line module 212 that adds a fourth line segment from the fourth point to the origin. The third line segment intersects the fourth line segment at a neutral point, (x_n, y_n). FIG. 6 is a diagram, consistent with the first geometric method, of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides and lines intersecting a neutral point. FIG. 6 depicts the neutral point (x_n, y_n). The slope of the third line segment may be expressed as:

$\begin{matrix} {slope}_{3 rd} = \frac{- y_{pbc}}{x_{a} - x_{pbc}} & Equation (11) \end{matrix}$

An equation for the third line segment is y=slope_3rd·x+Z where Z=−slope_3rd·x_a. The slope of the fourth line segment may be expressed as:

$\begin{matrix} {slope}_{4 th} = \frac{y_{pca}}{x_{pca}} & Equation (12) \end{matrix}$

An equation for the fourth line segment is y=slope_4th·x. The x-coordinate of the intersection of the third and fourth lines segments may be expressed as:

$\begin{matrix} x_{n} = \frac{z}{{slope}_{4 th} - {slope}_{3 rd}} & Equation (13) \end{matrix}$

The y-coordinate of the intersection of the third and fourth line segments may be expressed as:

y
_n=slope_4th·x_n Equation (14)

The neutral point (x_n, y_n), in one embodiment, may then be expressed as:

$\begin{matrix} x_{n} = \frac{\frac{y_{pbc}}{x_{a} - x_{pbc}} \times x_{a}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}} & Equation (15) \\ y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} & Equation (16) \end{matrix}$

The apparatus 200, in one embodiment, includes a line-to-neutral module 214 that determines a line-to-neutral voltage where the line-to-neutral voltage is a line from the neutral point to a vertex of the phasor triangle. In one embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_anfor phase A, which includes determining a line from the neutral point to the first point. In another embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_bnfor phase B, which includes determining a line from the neutral point to the origin. In another embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_cnfor phase C, which includes determining a line from the neutral point to the second point.

FIG. 7 is a diagram of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle with line-to-neutral phasors shown from the neutral point. FIG. 7 depicts the phasor V_anfor phase A, the phasor V_bnfor phase B, and the phasor V_cnfor phase C. In one embodiment, the apparatus 200 includes a line-to-neutral magnitude module 216 that determines a magnitude of the line-to-neutral voltage phasors. In one embodiment, the line-to-neutral magnitude module 216 determines magnitudes of the line-to-neutral phasors as:

V
_an=√{square root over ((x_a−x_n)²+(y_a−y_n)²)}{square root over ((x_a−x_n)²+(y_a−y_n)²)} Equation (17)

V
_bn=√{square root over ((x_b−x_n)²+(y_b−y_n)²)}{square root over ((x_b−x_n)²+(y_b−y_n)²)} Equation (18)

V
_cn=√{square root over ((x_c−x_n)²+(y_c−y_n)²)}{square root over ((x_c−x_n)²+(y_c−y_n)²)} Equation (19)

The equations listed above may be less computationally intensive than other methods of calculating line-to-neutral voltages from line-to-line voltages. For example, the equations do not use sine or cosine functions.

In some embodiments, certain extreme voltage imbalance may cause the neutral point to be outside the phasor triangle. For example, in the case of a phase loss, the neutral point may be slightly outside the phasor triangle. For instance, as certain equations have a denominator that approaches zero, the equations may calculate a neutral point outside the phasor triangle. To correct for these situations, in one embodiment the following equations may be used to correct the neutral point location:

$\begin{matrix} y_{n} := if y_{n} < 0 then y_{n} = 0; else if y_{n} > y_{c}, then y_{n} = y_{c}; else y_{n} . & Equation (20) \\ x_{n} := if x_{n} < 0 then x_{n} = 0; else if x_{n} > x_{a} then x_{n} = x_{a}; else if y_{c} = y_{a} then x_{n} = x_{c}; else x_{n} = x_{n} . & Equation (21) \end{matrix}$

Second Geometric Method

For the second geometric method, the first line module 206 again adds a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle and the first line segment terminates at a third point. However, the third point, (x_pbc, y_pbc) terminates at a point that is a vertex of a first equilateral triangle formed along the second phasor V_bcand away from the phasor triangle where the three sides of the equilateral triangle all have a length of the magnitude of the second phasor V_bc. The length of the first line segment is the magnitude of the second phasor V_bcmultiplied by the square root of three and divided by two.

Also for the second geometric method, the second line module 208 again adds a second line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the third phasor V_caand away from the phasor triangle and the second line segment terminates at a fourth point. However, the fourth point, (x_pca, y_pca) terminates at a point that is a vertex of a second equilateral triangle formed along the third phasor V_caand away from the phasor triangle where the three sides of the equilateral triangle all have a length of the magnitude of the third phasor V_ca. The length of the second line segment is the magnitude of the third phasor V_camultiplied by the square root of three and divided by two. FIG. 8 depicts the first and second equilateral triangles. The first equilateral triangle has sides equal to the second phasor V_bc. The second equilateral triangle has sides equal to the third phasor V_ca. The first line segment terminates at the third point at a vertex of the first equilateral triangle and the second line segment terminates at the fourth point at a vertex of the second equilateral triangle.

The slope of the first line segment is the same as stated above in Equation 3. Also, the slope of the second line segment is as stated above in Equation 7. From the position and length of the first line segment, in one example, the third point and the fourth point may be determined as:

$\begin{matrix} x_{pbc} = \frac{x_{c} - y_{c} \cdot \sqrt{3}}{2} & Equation (22) \\ y_{pbc} = \frac{y_{c} + x_{c} \cdot \sqrt{3}}{2} & Equation (23) \\ x_{pca} = x_{a} + \frac{x_{c} - x_{a} + \sqrt{3} \cdot y_{c}}{2} & Equation (24) \\ y_{pca} = \frac{(y_{c} - y_{a}) + \sqrt{3} \cdot (x_{a} - x_{c})}{2} & Equation (25) \end{matrix}$

Again, the third line module 210 adds a third line segment from the third point to the first point and the fourth line module 212 adds a fourth line segment from the fourth point to the origin. The third line segment again intersects the fourth line segment at a neutral point, (x_n, y_n). Consistent with the second geometric method, FIG. 9 is a diagram of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle with perpendicular lines bisecting two sides and lines intersecting a neutral point. FIG. 9 again depicts the neutral point (x_n, y_n). The slope of the third line segment may be expressed as stated above in Equation 11. An equation for the third line segment is y=slope_3rd·x+Z where Z=−slope_3rd·x_a. The slope of the fourth line segment may be expressed as stated above in Equation 12. An equation for the fourth line segment is y=slope_4th·x. The x-coordinate of the intersection of the third and fourth lines segments may be expressed as stated above in Equation 13. The y-coordinate of the intersection of the third and fourth line segments may be expressed as stated above in Equation 14.

The neutral point (x_n, y_n), in one embodiment, may then be expressed as:

$\begin{matrix} x_{n} = \frac{\frac{x_{a} \cdot y_{pbc}}{x_{a} - x_{pbc}}}{\frac{y_{pca}}{x_{pca}} + \frac{y_{pbc}}{x_{a} - x_{pbc}}} & Equation (26) \\ y_{n} = \frac{y_{pca}}{x_{pca}} \cdot x_{n} & Equation (27) \end{matrix}$

The neutral point may be called the isogonic center, which is a point inside of a triangle where no interior angle is greater than 120 degrees and where an observer would see the vertices at equal degree distances, e.g. 120 degrees apart. Typically the neutral point for the second geometric method remains within the phasor triangle and the line-to-neutral voltages are at a constant 120 degrees apart typically due to machine geometry. However, machine error may cause variations, for example when denominators are small. The line-to-neutral module 214 again determines a line-to-neutral voltage where the line-to-neutral voltage is a line from the neutral point to a vertex of the phasor triangle. In one embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_anfor phase A, which includes determining a line from the neutral point to the first point. In another embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_bnfor phase B, which includes determining a line from the neutral point to the origin. In another embodiment, the line-to-neutral module 214 determines the line-to-neutral voltage for a phasor V_cnfor phase C, which includes determining a line from the neutral point to the second point.

Consistent with the second geometric method, FIG. 10 is a diagram of voltages of a three-phase power system with unbalanced voltages repositioned as a phasor triangle with line-to-neutral phasors shown from the neutral point. FIG. 10 depicts the phasor V_anfor phase A, the phasor V_bnfor phase B, and the phasor V_cnfor phase C. The line-to-neutral magnitude module 216 determines a magnitude of the line-to-neutral voltage phasors. The line-to-neutral magnitude module 216 determines magnitudes of the line-to-neutral phasors as stated above in Equations 17, 18 and 19.

FIG. 11 is a schematic flow chart diagram illustrating one embodiment of a method 1100 for determining voltages. The method 1100 begins and measures 1102 three line-to-line voltages for the phases in a three-phase power system, such as the system 100 depicted in FIG. 1. Each line-to-line voltage includes a voltage magnitude. In one embodiment, the measurement module 202 measures the three line-to-line voltages.

The method 1100 constructs 1104, on a two-dimensional coordinate system with an origin, a phasor triangle with the three line-to-line voltages represented as phasors, where a first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor V_bcextends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor V_caextends between the second point and the first point. In one embodiment, the triangle module 204 constructs the phasor triangle.

The method 1100 adds 1106 a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at a third point. For the first geometric method, the length of the first line segment has a magnitude of the third phasor V_cadivided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment. For the second geometric method, the length of the first line segment is the magnitude of the second phasor V_bcmultiplied by the square root of three and divided by two. The first line module 206, in one embodiment, adds 1106 the first line segment.

The method 1100 adds 1108 a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at a fourth point. For the first geometric method, the length of the second line segment has a magnitude of the second phasor V_bcdivided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment. For the second geometric method, the length of the second line segment is the magnitude of the third phasor V_camultiplied by the square root of three and divided by two. The second line module 208, in one example, adds 1108 the second line segment.

The method 1100 adds 1110 a third line segment from the third point to the first point and adds 1112 a fourth line segment from the fourth point to the origin. The third line segment intersects the fourth line segment at a neutral point. In one embodiment, the third line module 210 adds 1110 the third line segment and the fourth line module 212 adds 1112 the fourth line segment. The method 1100 determines 1114 a line-to-neutral voltage where the line-to-neutral voltage is a line from the neutral point to a vertex of the phasor triangle, and the method 1100 ends. The line-to-neutral module 214, in one embodiment, determines 1114 the line-to-neutral voltages.

FIG. 12A is a first part and FIG. 12B is a second part of a schematic flow chart diagram illustrating another embodiment of a method 1200 for determining voltages. The method 1200 begins and measures 1202 three line-to-line voltages for the phases in a three-phase power system, such as the system 100 depicted in FIG. 1. Each line-to-line voltage includes a voltage magnitude. The method 1200 constructs 1204, on a two-dimensional coordinate system with an origin, a phasor triangle with the three line-to-line voltages represented as phasors, where a first phasor V_aboriginates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor V_bcextends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor V_caextends between the second point and the first point.

The method 1200 determines 1206 coordinates of the vertices of the phasor triangle. For example, the method 1200 may determine 1206 that the first point is at the termination of phasor V_aband is (x_a, y_a) so x_a=V_aband y_b=0. The origin is (x_b, y_b) so the method 1200 may determine 1206 that x_b=0 and y_b=0. The method 1200 may determine 1206 the second point (x_c, y_c) using equations 1 and 2.

The method 1200 adds 1208 a first line segment that extends from a point that bisects the second phasor V_bcin a direction perpendicular to the second phasor V_bcand away from the phasor triangle. The first line segment terminates at a third point. For the first geometric method, the length of the first line segment has a magnitude of the third phasor V_cadivided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment. For the second geometric method, the length of the first line segment is the magnitude of the second phasor V_bcmultiplied by the square root of three and divided by two. The method 1200 adds 1210 a second line segment that extends from a point that bisects the third phasor V_cain a direction perpendicular to the third phasor V_caand away from the phasor triangle. The second line segment terminates at a fourth point. For the first geometric method, the length of the second line segment has a magnitude of the second phasor V_bcdivided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment. For the second geometric method, the length of the second line segment is the magnitude of the third phasor V_camultiplied by the square root of three and divided by two.

The method 1200 adds 1212 a third line segment from the third point to the first point and adds 1214 a fourth line segment from the fourth point to the origin. The third line segment intersects the fourth line segment at a neutral point. The method 1200 determines 1216 coordinates of a neutral point (x_n, y_n). Theoretically, steps 1218-1240 are be more applicable to the first geometric method than the second geometric method for voltages greater than zero. However, calculation error and possible negative voltage readings are possible so steps 1218-1240 may be used for both the first and the second geometric methods. The method 1200 determines 1218 if y_n<0 (follow “A” on FIG. 12A to “A” on FIG. 12B). If the method 1200 determines that y_nis less than zero, the method 1200 then assigns 1220 y_nto be equal to zero. If the method 1200 determines 1218 that y_nis not less than zero, then the method 1200 determines 1222 if y_nis greater than y_c. If the method 1200 determines 1222 that y_nis greater than y_c, then the method 1200 assigns 1224 y_nto be equal to y_c. If the method 1200 determines 1222 that y_nis not greater than y_c, the method 1200 determines 1226 that y_nis not to be modified. For example, the method 1200 may use the results of equation 17 to determine y_nwithout further modification.

The method 1200 determines 1228 if x_nis less than zero. If the method 1200 determines 1228 if x_nis less than zero, the method assigns 1230 x_nto be equal to zero. If the method 1200 determines 1228 that x_nis not less than zero, the method 1200 determines 1232 if x_nis greater than x_a. If the method 1200 determines 1232 that x_nis greater than x_a, the method 1200 assigns 1234 x_nto be equal to x_a. If the method 1200 determines 1232 that x_nis not greater than x_a, the method 1200 determines 1236 if y_cis greater than y_n. If the method 1200 determines 1236 that y_cis greater than y_n, the method 1200 assigns 1238 x_nto be equal to x_c. If the method 1200 determines 1236 that y_cis not greater than y_n, the method 1200 determines 1240 that x_nis not to be modified. For example, the method 1200 may use the results of equation 16 to determine x_nwithout further modification. The method 1200 determines 1242 a line-to-neutral voltage where the line-to-neutral voltage is a line from the neutral point to a vertex of the phasor triangle, and the method 1200 ends.

FIG. 13 is a plot of laboratory test results showing percent voltage imbalance versus reported voltage error for the first geometric method. Testing was completed using an apparatus similar to the apparatus 200 of FIG. 2 tested on a simple three-phase power system, which may be similar to the system 100 of FIG. 1. Data for FIG. 13 is shown in Table 1 and Table 2. Table 1 includes in the first three columns line-to-neutral voltage settings on the test equipment. The voltage settings vary on each phase between zero and 400 volts and represent various under and over-voltage conditions for three phases, 1, 2 and 3. The next three columns are expected line-to-line voltage conditions based on the test equipment settings. The next three columns are reported line-to-line voltages, which are reported from the metering device. The reported line-to-neutral voltages are the voltages calculated by the metering device using one embodiment of the apparatus 200 and methods 1100, 1200 described above in relation to FIGS. 2, 11, and 12. The calculated percent voltage imbalance of the second to last column is based on the test equipment set voltages and the reported percent voltage imbalance of the last column is the voltage imbalance reported by the metering device. The first three columns of Table 2 show reported line-to-line voltage error from expected line-to-line voltages as reported from the metering device. The last three columns of Table 2 show reported line-to-neutral voltage error comparing reported line-to-neutral voltages from the metering to the set line-to-neutral voltages.

FIG. 13 is a plot of the results in Table 2. FIG. 13 has percent voltage imbalance is shown on the horizontal axis and reported voltage error is shown on the vertical axis. VL1-L2 is the line-to-line voltage between a first and a second phase, VL2-L3 is the line-to-line voltage between the second and a third phase, and VL3-L1 is the line-to-line voltage between the third and the first phase. VL1N, VL2N and VL3N are line-to-neutral voltages for the first, second, and third phases, respectively. The “One Phase Lossed” line indicates a percent voltage imbalance corresponding to one phase of the 3-phase power system being shorted to ground and the “Two Phase Lossed” line indicated a percent voltage imbalance corresponding to when two phases are shorted to ground. Note that reported voltage error is greatest for when voltage approaches zero for one phase, but the error in most situations remains low. Two single phase loss conditions are represented in the graph. In one instance, the first phase (e.g. Phase A in the system 100 of FIG. 1) is reduced to zero, and in the second instance the second phase (e.g. Phase B in the system 100 of FIG. 1) is reduced to zero.

TABLE 1

Calc.
Reported

Set L-N V
Expected L-L V
Reported L-L V
Reported L-N V
% V
% V

L1N
L2N
L3N
L12
L23
L31
L12
L23
L31
L1N
L2N
L3N
Imbal.
Imbal

400
400
0
693
400
400
692
399
400
400
398
0
39%
39%

400
400
20
693
410
410
692
409
410
410
409
0
37%
37%

400
400
50
693
427
427
692
426
427
406
405
37
34%
34%

400
400
400
693
693
693
692
692
691
399
400
398
0%
0%

400
400
375
693
671
671
692
670
670
399
399
375
2%
2%

400
400
350
693
650
650
692
649
649
399
399
351
4%
4%

400
400
325
693
629
629
692
628
628
398
398
327
7%
7%

400
400
300
693
608
608
692
607
607
398
398
302
9%
9%

400
400
240
693
560
560
692
559
559
397
397
244
15%
15%

400
400
200
693
529
529
692
528
528
397
397
203
19%
19%

400
400
180
693
514
514
692
513
513
397
397
184
21%
21%

400
400
150
693
492
492
692
492
492
398
398
152
24%
24%

400
400
130
693
478
478
692
478
478
399
399
130
26%
26%

400
400
120
693
472
472
692
471
471
399
399
120
27%
27%

400
300
120
608
375
472
608
374
471
390
307
122
26%
26%

400
240
120
560
317
472
559
317
471
386
250
122
29%
29%

400
200
120
529
280
472
529
280
471
385
211
124
34%
34%

400
180
120
514
262
472
514
261
471
385
191
123
37%
37%

400
150
120
492
234
472
492
234
471
386
159
124
41%
41%

400
130
120
478
217
472
478
216
471
388
137
125
44%
44%

400
120
120
472
208
472
471
208
471
389
125
125
46%
46%

300
120
120
375
208
375
374
208
374
292
124
123
35%
35%

240
120
120
317
208
317
317
208
317
234
123
122
26%
26%

200
120
120
280
208
280
280
208
280
196
122
121
19%
19%

180
120
120
262
208
262
261
208
261
177
121
121
15%
15%

150
120
120
234
208
234
234
208
234
148
121
120
8%
8%

130
120
120
217
208
217
216
208
216
129
120
120
3%
3%

120
120
120
208
208
208
208
208
208
120
120
120
0%
0%

115
120
120
204
208
204
203
208
203
115
120
119
1%
1%

110
120
120
199
208
199
199
208
199
110
120
119
3%
3%

105
120
120
195
208
195
195
208
195
105
120
119
4%
4%

100
120
120
191
208
191
191
208
190
101
119
119
6%
6%

90
120
120
182
208
182
182
208
182
91
119
119
9%
9%

80
120
120
174
208
174
174
208
174
81
119
119
12%
12%

70
120
120
166
208
166
166
208
166
71
119
119
15%
15%

60
120
120
159
208
159
159
208
159
61
119
119
19%
19%

50
120
120
151
208
151
151
208
151
51
119
119
22%
22%

40
120
120
144
208
144
144
208
144
41
119
119
26%
26%

30
120
120
137
208
137
137
208
137
30
120
120
29%
29%

20
120
120
131
208
131
131
208
131
18
121
121
33%
33%

10
120
120
125
208
125
125
208
125
5.2
122
122
36%
36%

5
120
120
123
208
123
123
208
123
0
123
122
38%
38%

0
120
120
120
208
120
120
208
120
0
120
120
39%
39%

0
110
120
110
199
120
110
199
120
0
110
120
39%
39%

0
100
120
100
191
120
100
191
120
0
100
120
39%
39%

0
80
120
80
174
120
80
174
120
0
80
120
40%
40%

0
60
120
60
159
120
60
159
120
0
60
120
47%
47%

0
40
120
40
144
120
40
144
120
1.2
39
119
61%
61%

0
20
120
20
131
120
20
131
120
1.7
19
119
78%
78%

0
10
120
10
125
120
9.9
125
120
1.2
9.4
119
88%
88%

0
0
120
0
120
120
0.9
120
120
0.8
0.1
119
99%
99%

TABLE 2

Reported Line-to-Line

Voltage Error
Reported Line-to-Neutral

From Expected Value
Voltage Error

VL1-L2
VL2-L3
VL3_L1
VL1N
VL2N
VL3N

−0.10%
−0.32%
−0.05%
−0.13%
0.42%
—

−0.09%
−0.28%
−0.09%
2.43%
2.22%
100.00%

−0.08%
−0.28%
−0.09%
1.43%
1.28%
−26.00%

−0.10%
−0.13%
−0.21%
−0.17%
0.05%
−0.40%

−0.08%
−0.13%
−0.18%
−0.28%
0.22%
−0.13%

−0.08%
−0.15%
−0.20%
−0.38%
0.33%
0.26%

−0.09%
−0.16%
−0.19%
−0.47%
0.42%
0.52%

−0.10%
−0.14%
−0.19%
−0.60%
0.53%
0.67%

−0.09%
−0.13%
−0.18%
−0.75%
0.70%
1.67%

−0.10%
−0.16%
−0.16%
−0.72%
0.72%
1.60%

−0.08%
−0.16%
−0.17%
−0.65%
0.70%
2.06%

−0.12%
−0.19%
−0.19%
−0.53%
0.53%
1.27%

−0.09%
−0.20%
−0.15%
−0.30%
0.35%
0.31%

−0.09%
−0.21%
−0.15%
−0.22%
0.25%
0.25%

−0.08%
−0.19%
−0.15%
−2.45%
2.37%
1.58%

−0.11%
−0.15%
−0.17%
−3.45%
4.33%
1.75%

−0.09%
−0.14%
−0.15%
−3.75%
5.55%
3.17%

−0.12%
−0.17%
−0.15%
−3.68%
5.94%
2.75%

−0.11%
−0.13%
−0.15%
−3.47%
6.20%
3.42%

−0.11%
−0.17%
−0.15%
−3.10%
5.46%
4.25%

−0.10%
−0.12%
−0.10%
−2.80%
4.50%
4.50%

−0.11%
−0.12%
−0.19%
−2.67%
3.08%
2.50%

−0.12%
−0.12%
−0.15%
−2.38%
2.17%
1.67%

−0.11%
−0.17%
−0.18%
−1.95%
1.42%
0.83%

−0.05%
−0.17%
−0.20%
−1.67%
1.08%
0.67%

−0.09%
−0.12%
−0.22%
−1.07%
0.50%
0.25%

−0.08%
−0.07%
−0.17%
−0.62%
0.08%
−0.08%

−0.07%
−0.07%
−0.17%
−0.17%
0.08%
−0.33%

−0.06%
−0.12%
−0.16%
0.00%
0.17%
−0.83%

−0.07%
−0.17%
−0.17%
0.18%
0.33%
−0.67%

−0.10%
−0.17%
−0.15%
0.38%
0.42%
−0.67%

−0.10%
−0.12%
−0.20%
0.60%
0.50%
−0.83%

−0.10%
−0.12%
−0.16%
1.11%
0.58%
−0.83%

−0.09%
−0.12%
−0.20%
1.50%
0.67%
−0.83%

−0.08%
−0.12%
−0.14%
1.86%
0.75%
−0.92%

−0.09%
−0.07%
−0.15%
2.17%
0.67%
−0.92%

−0.15%
−0.12%
−0.15%
2.20%
0.67%
−0.75%

−0.08%
−0.12%
−0.08%
1.50%
0.50%
−0.50%

−0.06%
−0.17%
−0.06%
−1.33%
0.08%
−0.25%

−0.04%
−0.12%
−0.04%
−10.00%
0.67%
0.67%

−0.08%
−0.07%
0.08%
−48.00%
1.83%
1.75%

−0.06%
−0.07%
0.02%
100.00%
2.08%
1.92%

−0.25%
−0.07%
0.00%
—
0.17%
−0.25%

−0.18%
−0.12%
0.00%
—
0.18%
−0.25%

−0.20%
−0.10%
0.00%
—
0.20%
−0.25%

−0.25%
−0.09%
0.00%
—
0.25%
−0.25%

−0.33%
−0.15%
0.00%
—
0.17%
−0.25%

−0.25%
−0.15%
0.00%
—
2.00%
−0.83%

−1.00%
−0.19%
0.00%
—
4.50%
−0.83%

−1.00%
−0.24%
0.00%
—
6.00%
−0.67%

—
−0.33%
0.00%
—
—
−0.83%

FIG. 14 is a plot of laboratory test results showing reported voltage error under balanced conditions using the first geometric method. The apparatus and three-phase power system used for the results shown in FIG. 13 were again used for testing. In this particular test, line-to-neutral voltages were varied from zero to 400 volts and the voltage on each phase was the same as for the other phases so that the voltages remained balanced. The measured voltages were compared to the calculated voltages. Table 3 and Table 4 show the test results used for FIG. 14. In Table 3, the first three columns are the set line-to-neutral voltages of the test equipment. The next three columns show expected line-to-line voltages, and the next three columns show reported line-to-line voltages from the metering device. The next three columns indicate reported line-to-neutral voltages that were calculated by the metering device using and embodiment of the apparatus 200 and methods 800, 1200 described above. The last three columns show measured output voltage of the test equipment using a separate reference metering device. Table 4 includes calculated error for line-to-line voltages and line-to-neutral voltages. The results of Table 4 are included in FIG. 14. Note that the error was greatest near zero, but under most conditions the error was very low. Test results reveal that the voltage measurement apparatus 102 may be a viable method for determining line-to-neutral voltages where a neutral or ground is unavailable.

TABLE 2

Set L-N Volt.
Expected L-L V
Reported L-L V
Reported L-N V
CA L-N V

L1N
L2N
L3N
L12
L23
L31
L12
L23
L31
L1N
L2N
L3N
L1N
L2N
L3N

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

5
5
5
9
9
9
0
0
0
0
0
0
5
5
5

10
10
10
17
17
17
0
0
0
0
0
0
10
10
10

12
12
12
21
21
21
21
21
21
12
12
12
12
12
12

14
14
14
24
24
24
24
24
24
14
14
14
14
14
14

15
15
15
26
26
26
26
26
26
15
15
15
15
15
15

20
20
20
35
35
35
35
35
35
20
20
20
20
20
20

30
30
30
52
52
52
52
52
52
30
30
30
30
30
30

40
40
40
69
69
69
69
69
69
40
40
40
40
40
40

50
50
50
87
87
87
87
86
86
50
50
50
50
50
50

60
60
60
104
104
104
104
104
104
60
60
60
60
60
60

75
75
75
130
130
130
130
130
130
75
75
75
75
75
75

100
100
100
173
173
173
173
173
173
100
100
100
100
100
100

TABLE 3

Set L-N Volt.
Expected L-L V
Reported L-L V
Reported L-N V
CA L-N V

L1N
L2N
L3N
L12
L23
L31
L12
L23
L31
L1N
L2N
L3N
L1N
L2N
L3N

120
120
120
208
208
208
208
208
207
120
120
120
120
120
120

140
140
140
242
242
242
242
242
242
140
140
140
140
140
140

160
160
160
277
277
277
277
277
277
160
160
160
160
160
160

180
180
180
312
312
312
312
311
311
180
180
179
180
180
180

200
200
200
346
346
346
346
346
346
200
200
199
200
200
200

250
250
250
433
433
433
433
433
432
250
250
249
250
250
250

300
300
300
520
520
520
519
519
519
300
300
299
300
300
300

350
350
350
606
606
606
606
606
605
349
350
349
350
350
350

400
400
400
693
693
693
692
692
692
400
400
399
400
400
400

L-to-L Error
L-to-N Error

Set VLN
VL1-L2
VL2-L3
VL3-L1
VL1N
VL2N
VL3N

0
#DIV/0!
#DIV/0!
#DIV/0!
#DIV/0!
#DIV/0!
#DIV/0!

5
100.00%
100.00%
100.00%
100.00%
100.00%
0.00%

10
100.00%
100.00%
100.00%
100.00%
100.00%
0.00%

12
0.07%
−0.41%
−0.89%
−0.83%
−1.67%
0.00%

14
−0.20%
−0.20%
−0.61%
−0.71%
−0.71%
0.00%

15
−0.31%
−0.31%
−0.31%
−1.33%
−1.33%
0.00%

20
−0.12%
−0.41%
−0.41%
−1.00%
−0.50%
0.00%

30
−0.12%
−0.12%
−0.12%
−0.33%
−0.33%
0.00%

40
−0.12%
−0.12%
−0.26%
−0.25%
−0.50%
0.00%

50
−0.12%
−0.23%
−0.23%
−0.20%
−0.60%
0.00%

60
−0.12%
−0.21%
−0.21%
−0.33%
−0.33%
0.00%

75
−0.08%
−0.16%
−0.16%
−0.27%
−0.40%
0.00%

100
−0.06%
−0.18%
−0.23%
−0.10%
−0.40%
−0.01%

120
−0.07%
−0.12%
−0.21%
−0.08%
−0.42%
0.00%

140
−0.04%
−0.12%
−0.20%
−0.14%
−0.36%
−0.01%

160
−0.08%
−0.08%
−0.15%
−0.31%
−0.19%
−0.01%

180
−0.09%
−0.12%
−0.15%
−0.17%
−0.33%
−0.01%

200
−0.09%
−0.12%
−0.15%
−0.10%
−0.35%
0.00%

250
−0.07%
−0.10%
−0.16%
−0.08%
−0.36%
−0.01%

300
−0.08%
−0.12%
−0.18%
−0.07%
−0.33%
−0.01%

350
−0.07%
−0.10%
−0.20%
−0.06%
−0.37%
−0.01%

400
−0.10%
−0.12%
−0.19%
−0.05%
−0.38%
−0.01%

The described examples and embodiments are to be considered in all respects only as illustrative and not restrictive. This written description uses examples and embodiments to disclose the invention, including best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The examples and embodiments may be practiced in other specific forms. The patentable scope of this invention is defined by the claims and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural element with insubstantial differences from the literal languages of the claims.

Claims

1. A method for determining voltage, the method comprising: measuring three line-to-line voltages for the phases in a three-phase power system, each line-to-line voltage comprising a voltage magnitude;constructing, on a two-dimensional coordinate system with an origin, a phasor triangle comprising the three line-to-line voltages represented as phasors, wherein a first phasor Vab originates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor Vbc extends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor Vca extends between the second point and the first point;adding a first line segment that extends from a point that bisects the second phasor Vbc in a direction perpendicular to the second phasor Vbc and away from the phasor triangle, the first line segment terminating at a third point;adding a second line segment that extends from a point that bisects the third phasor Vca in a direction perpendicular to the third phasor Vca and away from the phasor triangle, the second line segment terminating at a fourth point;adding a third line segment from the third point to the first point;adding a fourth line segment from the fourth point to the origin, wherein the third line segment intersects the fourth line segment at a neutral point; anddetermining a line-to-neutral voltage, the line-to-neutral voltage comprising a line from the neutral point to a vertex of the phasor triangle.
2. The method of claim 1, wherein determining a line-to-neutral voltage comprises one or more of: determining a line-to-neutral voltage for a phasor Van for phase A which comprises determining a line from the neutral point to the first point;determining a line-to-neutral voltage for a phasor Vbn for phase B which comprises determining a line from the neutral point to the origin; anddetermining a line-to-neutral voltage for a phasor Vcn for phase C which comprises determining a line from the neutral point to the second point.
3. The method of claim 2, further comprising one or more of: determining a magnitude of phasor Van as Van=√{square root over ((xa−xn)2+(ya−yn)2)}{square root over ((xa−xn)2+(ya−yn)2)};determining a magnitude of phasor Vbn as Vbn=√{square root over ((xb−xn)2+(yb−yn)2)}{square root over ((xb−xn)2+(yb−yn)2)}; anddetermining a magnitude of phasor Vcn as Vcn=√{square root over ((xc−xn)2+(yc−yn)2)}{square root over ((xc−xn)2+(yc−yn)2)}.
4. The method of claim 1, wherein the first point comprises a coordinate of (xa, ya), the second point comprises a coordinate of (xc, yc), and the origin comprises a coordinate of (xb, yb) wherein: xa=the magnitude of the Vab phasor and ya=0;xb=0 and yb=0; and
5. The method of claim 1, wherein the length of the first line segment comprises a magnitude of the third phasor Vca divided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment and wherein the length of the second line segment comprises a magnitude of the second phasor Vbc divided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment.
6. The method of claim 5, wherein the length of the first line segment comprises
7. The method of claim 5, wherein the third point comprises a coordinate of (xpbc, ypbc) and the fourth point comprises a coordinate of (xpca, ypca), wherein
8. The method of claim 7, wherein the neutral point comprises a coordinate of (xn, yn) and wherein:
9. The method of claim 8, wherein where yn is less than zero then yn=0, and where yn is greater than or equal to zero and greater than yc then yn=yc.
10. The method of claim 9, wherein where xn is less than zero then xn=0, and where xn is greater than or equal to zero and greater than xa then xn=xa, and where xn is greater than or equal to zero and less than or equal to xa and yc=y, then xn=xc.
11. The method of claim 1, wherein the length of the first line segment has a magnitude of the second phasor Vbc multiplied by the square root of three and divided by two, which represents a height of a first equilateral triangle constructed on the second phasor Vbc and extending away from the phasor triangle and wherein the length of the second line segment has a magnitude of the third phasor Vca multiplied by the square root of three and divided by two, which represents a height of a second equilateral triangle constructed on the third phasor Vca and extending away from the phasor triangle.
12. The method of claim 11, wherein the third point comprises a coordinate of (xpbc, ypbc) and the fourth point comprises a coordinate of (xpca, ypca), wherein
13. The method of claim 12, wherein the neutral point comprises a coordinate of (xn, yn) and wherein:
14. The method of claim 1, wherein the line-to-line voltages are measured at a location in the three-phase power system where a neutral connection is unavailable for measurement.
15. The method of claim 1, wherein the three-phase power system comprises an ungrounded power system.
16. The method of claim 1, wherein the three-phase power system comprises unbalanced voltages.
17. The method of claim 1, wherein the first phasor Vab, the second phasor Vbc, and the third phasor Vca are separated by 120 degrees.
18. An apparatus comprising: a measurement module that measures three line-to-line voltages for the phases in a three-phase power system, each line-to-line voltage comprising a voltage magnitude;a triangle module that constructs, on a two-dimensional coordinate system with an origin, a phasor triangle comprising the three line-to-line voltages represented as phasors, wherein a first phasor Vab originates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor Vbc extends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor Vca extends between the second point and the first point;a first line module that adds a first line segment that extends from a point that bisects the second phasor Vbc in a direction perpendicular to the second phasor Vbc and away from the phasor triangle, the first line segment terminating at a third point;a second line module that adds a second line segment that extends from a point that bisects the third phasor Vca in a direction perpendicular to the third phasor Vca and away from the phasor triangle, the second line segment terminating at a fourth point;a third line module that adds a third line segment from the third point to the first point;a fourth line module that adds a fourth line segment from the fourth point to the origin, wherein the third line segment intersects the fourth line segment at a neutral point; anda line-to-neutral module that determines a line-to-neutral voltage, the line to-neutral voltage comprising a line from the neutral point to a vertex of the phasor triangle,wherein at least a portion of said modules comprise one or more of hardware and executable code, the executable code stored on one or more computer readable storage media.
19. The apparatus of claim 18, wherein the line-to-neutral module determines a line-to-neutral voltage by: determining a line-to-neutral voltage for phasor Van for phase A which comprises determining a line from the neutral point to the first point;determining a line-to-neutral voltage for phasor Vbn for phase B which comprises determining a line from the neutral point to the origin; anddetermining a line-to-neutral voltage for phasor Vcn for phase C which comprises determining a line from the neutral point to the second point.
20. The apparatus of claim 19, further comprising a line-to-neutral magnitude module that one or more of: determines a magnitude of phasor Van as Van=√{square root over ((xa−xn)2+(ya−yn)2)}{square root over ((xa−xn)2+(ya−yn)2)},determines a magnitude of phasor Vbn as Vbn=√{square root over ((xb−xn)2+(yb−yn)2)}{square root over ((xb−xn)2+(yb−yn)2)}, anddetermines a magnitude of phasor Vcn as Vcn=√{square root over ((xc−xn)2+(yc−yn)2)}{square root over ((xc−xn)2+(yc−yn)2)}.
21. The apparatus of claim 18, wherein first point comprises a coordinate of (xa, ya), the second point comprises a coordinate of (xc, yc), and the origin comprises a coordinate of (xb, yb) wherein: xa=the magnitude of the Vab phasor and ya=0;xb=0 and yb=0; and
22. The apparatus of claim 21, wherein the length of the first line segment comprises a magnitude of the third phasor Vca divided by the square root of three and multiplied by the square root of one plus the square of the slope of the first line segment and wherein the length of the second line segment comprises a magnitude of the second phasor Vbc divided by the square root of three and multiplied by the square root of one plus the square of the slope of the second line segment.
23. The apparatus of claim 21, wherein the third point comprises a coordinate of (xpbc, ypbc) and the fourth point comprises a coordinate of (xpca, ypca), wherein
24. The apparatus of claim 23, wherein the neutral point comprises a coordinate of (xn, yn) and wherein:
25. The apparatus of claim 24, wherein where yn is less than zero then yn=0, and where yn is greater than or equal to zero and greater than yc then yn=yc.
26. The apparatus of claim 25, wherein where xn is less than zero then xn=0, and where xn is greater than or equal to zero and greater than xa then xn=xa, and where xn is greater than or equal to zero and less than or equal to xa and yc=yn then xn=xc.
27. The apparatus of claim 18, wherein the length of the first line segment has a magnitude of the second phasor Vbc multiplied by the square root of three and divided by two, which represents a height of a first equilateral triangle constructed on the second phasor Vbc and extending away from the phasor triangle and wherein the length of the second line segment has a magnitude of the third phasor Vca multiplied by the square root of three and divided by two, which represents a height of a second equilateral triangle constructed on the third phasor Vca and extending away from the phasor triangle.
28. The apparatus of claim 27, wherein the third point comprises a coordinate of (xpbc, ypbc) and the fourth point comprises a coordinate of (xpca, ypca) and wherein
29. The apparatus of claim 28, wherein the neutral point comprises a coordinate of (xn, yn) and wherein:
30. The apparatus of claim 18, further comprising a meter, the meter comprising the measurement module.
31. The apparatus of claim 18, further comprising a processor, the processor executing executable code of one or more of the measurement module, the triangle module, the first line module, the second line module, the third line module, the fourth line module, and the line-to-neutral module.
32. A computer program product for determining a voltage, the computer program product comprising a computer readable storage medium having program code embodied therein, the program code readable/executable by a processor for: measuring three line-to-line voltages for the phases in a three-phase power system, each line-to-line voltage comprising a voltage magnitude;constructing, on a two-dimensional coordinate system with an origin, a phasor triangle comprising the three line-to-line voltages represented as phasors, wherein a first phasor Vab originates at a first point and extends in a direction along a horizontal axis of the coordinate system to the origin, a second phasor Vbc extends between the origin and a second point, the second point in a direction vertically and horizontally from the origin, and a third phasor Vca extends between the second point and the first point;determining a third point by adding a first line segment that extends from a point that bisects the second phasor Vbc in a direction perpendicular to the second phasor Vbc and away from the phasor triangle, the first line segment terminating at the third point;determining a fourth point by adding a second line segment that extends from a point that bisects the third phasor Vca in a direction perpendicular to the third phasor Vca and away from the phasor triangle, the second line segment terminating at the fourth point;determining a neutral point by adding a third line segment from the third point to the first point and adding a fourth line segment from the fourth point to the origin, wherein the third line segment intersects the fourth line segment at the neutral point; anddetermining a line-to-neutral voltage, the line-to-neutral voltage comprising a line from the neutral point to a vertex of the phasor triangle.

CALCULATING LINE-TO-NEUTRAL VOLTAGES WITHOUT A CONNECTION TO A SYSTEM NEUTRAL OR EARTH GROUND

Information

Publication Number

Date Filed

Date Published

Inventors

Original Assignees

CPC

International Classifications

Abstract

Description

Claims