**METHOD AND SYSTEM FOR DETERMINING CURRENTS IN BUNDLED CONDUCTORS**

GENG, Guangchao (11-203 Donadeo Innovation Centre for Engineering, c/o The University of AlbertaEdmonton, Alberta T6G 1H9, T6G 1H9, CA)

*;*

**G01R35/00***;*

**G01R15/14**

**G01R19/25**US9000752B2 | 2015-04-07 | |||

US20120319674A1 | 2012-12-20 | |||

US20150331079A1 | 2015-11-19 | |||

US20130116955A1 | 2013-05-09 | |||

US8718964B2 | 2014-05-06 |

We claim: 1. A system for estimating currents in at least one conductor in a bundle of conductors, the system comprising: - a plurality of sensors arranged to surround at least a cross-sectional portion of said bundle of conductors when said system is deployed, each of said plurality of sensors being for detecting magnetic fields produced by said currents, said sensors producing signals based on magnetic fields detected; - an A/D converter for converting signals produced by said sensors into digital data; - a data processing unit to receive digital data derived from said magnetic fields produced by said currents, said data processing unit calculating an estimate of said currents within said at least one conductor based on said digital data and parameters for said sensors; wherein - said parameters are predetermined and are retrieved by said data processing unit prior to calculating said estimate . 2. The system according to claim 1, wherein said plurality of sensors are arranged in a U shaped configuration such that said bundle of conductors are held in a crook of said U shaped configuration when said system is deployed. 3. The system according to claim 1, wherein said plurality of sensors are arranged to completely encircle said bundle of conductors when said system is deployed. 4. The system according to claim 3, wherein said plurality of sensors are arranged into a pair of cooperating jaws such that said sensors in said pair of jaws completely surround said bundle of conductors when said system is deployed. 5. The system according to claim 4, wherein each one of said pair of cooperating jaws is configured as a half circle. 6. The system according to claim 1, further comprising an amplifier for amplifying said signals produced by said sensors . 7. The system according to claim 1, wherein said parameters for said sensors comprise at least one of: - sensor coordinates; - sensor orientation angle; and - sensor gain factor. 8. The system according to claim 1, further comprising data storage for storing said parameters. 9. The system according to claim 1, wherein said parameters are determined by a method comprising: a) deploying said sensors around at least one conductor, each of said at least one conductor having known coordinates relative to a predetermined reference point b) passing at least one current through said at least one conductor, said at least one current having known and predetermined characteristics; c) measuring at least one output from at least one of said sensors, said output being caused by magnetic fields caused by said at least one current in said at least one conductor ; d) using data derived from said output in a multiple variable least squares formulation problem, said multiple variables being parameters for said sensors; e) determining a solution for said least squares problem once sufficient data from said at least one sensor and said at least one conductor has been generated. 10. The system according to claim 9, wherein said least- squares formulation problem comprises: 11. The method according to claim 9, wherein said at least one conductor comprises a plurality of conductors, each of said plurality of conductors having known coordinates relative to said reference point. 12. The method according to claim 11, wherein step b) comprises passing multiple currents through multiple ones of said plurality of conductors. 13. The system according to claim 9, wherein said plurality of conductors are in a bundle and said method is executed in turn by passing a current through only one conductor at a time . 14. The system according to claim 1, wherein said parameters include at least one of : - sensor coordinates ; sensor orientation angle; and sensor gain factor. 15. A method for calibrating a system for measuring currents in a bundle of conductors, said system including a plurality of sensors for measuring magnetic fields caused by current passing through at least one conductor, the method comprising: a) deploying said sensors around at least one conductor, each of said at least one conductor having known coordinates relative to a predetermined reference point b) passing at least one current through said at least one conductor, said at least one current having known and predetermined characteristics; c) measuring at least one output from at least one of said sensors, said output being caused by magnetic fields caused by said at least one current in said at least one conductor ; d) using data derived from said output in a multiple variable least squares formulation problem, said multiple variables being parameters for said sensors; e) determining a solution for said least squares problem once sufficient data from said at least one sensor and said at least one conductor has been generated. 16. The method according to claim 15, wherein said at least one conductor comprises a plurality of conductors, each of said plurality of conductors having known coordinates relative to said reference point. 17. The method according to claim 16, wherein step b) comprises passing multiple currents through multiple ones of said plurality of conductors. 18. The method according to claim 17, wherein said multiple currents are simultaneously passed through said multiple ones of said plurality of conductors. 19. The method according to claim 16, wherein said multiple currents are passed through said multiple ones of said plurality of conductors one current at a time. 20. The method according to claim 16, wherein steps b) repeated until sufficient data is gathered from said at one sensor using said plurality of conductors. 21. The method according to claim 20, wherein step b) comprises passing at least one current through a subset of said plurality of conductors. 22. The method according to claim 16, wherein said output comprises voltages from said at least one sensor. 23. The method according to claim 15, wherein said least- squares formulation problem comprises: 24. The method according to claim 15, wherein said parameters include at least one of : - sensor coordinates; - sensor orientation angle; and - sensor gain factor. 25. A method for determining currents in at least one conductor in a bundle of conductors, the method comprising: a) deploying a plurality of sensors adjacent said bundle of conductors, said plurality of sensors being arranged in a frame to surround at least a cross-sectional portion of said bundle of conductors; b) detecting magnetic fields caused by said currents using said sensors; c) measuring said magnetic fields; d) producing voltages based on said magnetic fields; e) converting said voltages into digital signals; f) transmitting said digital signals to a data processing device ; g) solving a least squares formulation based on said magnetic fields detected in step b) to result in currents in said at least one of said conductors; wherein parameters for said sensors are predetermined and are used in step g) to solve said least squares formulation. 26. The method according to claim 25, wherein said parameters for said sensors include at least one of: - sensor coordinates; - sensor orientation angle; and - sensor gain factor. 27. The method according to claim 25, wherein said parameters are calibrated and predetermined for said sensors prior to an execution of said method. 28. The method according to claim 25, wherein said parameters are retrieved from a data storage device. 29. The method according to claim 25, wherein said least- squares formulation comprises: |

CONDUCTORS

TECHNICAL FIELD

[0001] The present invention relates to measurements and, more precisely, to methods and systems for in situ measurement of currents within conductors that are bundled together .

BACKGROUND

[0002] Current sensing is one of the basic measurement

techniques for modern power systems. How to measure currents in a bundle of conductors enclosed in a structure has been a challenging task. For example, Romex cables have two or three conductors enclosed in a plastic cover. It has been difficult to measure the currents of these conductors without isolating the conductors first. A method to estimate currents for such cables using external sensors was patented in US Patent 5, 473, 244 [2] . Contactless measurement of overhead line currents tends to encounter similar challenges. In this case, magnetic fields measured by an array of sensors are used to estimate currents of distribution feeders [3] . Recently, the need to monitor home energy use by home owners has motivated research on measuring currents in conductors enclosed in a plastic conduit [4] as a key enabling technique of non-intrusive load monitoring (NILM) [5] .

[0003] All of the above share a common characteristic,

namely, measuring currents on conductors that cannot be accessed individually. This problem is illustrated using Fig. 1. In this Figure, a set of magnetic field sensors are placed around the conduit. The goal is to determine conductor currents inside the conduit using sensor data and without knowing conductor positions. Magnetic sensors [6], including coil [7, 8] and Hall effect based sensors [9, 10], have been utilized to measure magnetic fields and identify currents. Sensor array based approaches have been developed in order to make use of spatial difference and measuring redundancy of multiple sensors to achieve better performance [11] . Reference [2] formulates the contactless current measurement problem as an optimization problem of determining current values using a high number of sensors. Such a solution is infeasible, especially on a device with limited physical size. References [12] and [13] implement current measurement for two or three conductors based on the pre-assumed conductor geometric information from user selected Romex cable specification.

Unfortunately, this has led to large measuring errors and withdrawn products. Reference [14] proposed an on-site calibration method to establish the

relationship between conductor currents and magnetic fields. In addition to this, a dedicated calibration device that draws currents from the conductors downstream of the sensing point was developed.

Unfortunately, the method developed is limited to only situations where the conductors can be accessed so that the calibrating signals can be injected. The method is not as useful as, for example, the method is unable to measure overhead conductor currents . [0005] Existing methods for current measurement are designed for single conductors. One example is the clamp-on current probe. Such devices are widely available in the market. Unfortunately, these devices have limited utility, especially when multiple conductors are bundled together and individual conductors cannot be easily accessed. Examples of such bundled conductors are Romex or Teck cables which have two to four conductors enclosed in a metallic or plastic cover. Existing methods experience difficulties with such cases. Since measurement situations involving bundled or enclosed conductors are common, industry has a strong need for techniques that are able to measure currents in a group of difficult-to-separate conductors .

[0006] Much research has been conducted to solve the above problem. Developments in this field include those described below.

[0007] An article by G.D. Antona, et al ("Processing

Magnetic Sensor Array Data for AC Current Measurement in Multiconductor Systems", IEEE Transactions on Instrumentation and Measurement, vol. 50, no. 5, pp. 1289-1295, 2001.) proposes the use of a finite element method to determine the relationship between conductor currents and magnetic fields so that the currents can be calculated using magnetic field measurements .

[0008] In their article "A Power Sensor Tag with

Interference Reduction for Electricity Monitoring of Two-Wire Household Appliances" (IEEE Transactions on Industrial Electronics, vol. 61, no. 4, pp. 2062- 2070, 2014.), Y. C. Chen, et al proposed the use of inductive coil sensors fabricated on a chip using μ-m level CMOS photo-lithography. The chip was tagged on to a power cord with known geometric information. This would enable the measurement of magnetic fields and, thereby, allow for the calculation of the currents in the power cord.

[0009] US Patent 5,473,244 suggested multiple schemes for deploying sensors to measure conductor currents and positions. WO Patent 2013 068 360 Al implemented current measurement for multiple conductors based on the user selection of the pre-established conductor geometric information. US Patent 5,438,256 disclosed a method to measure currents on overhead power lines using magnetic field sensor array deployed on the ground .

[0010] All of the above methods are based on an assumption that the geometric positions and orientations of the conductors and sensors are known. For example, US patent 5,438,256, and similar approaches such as US Patent 5,473,244, used known geometric parameters of the conductors and sensors to calculate the conductor currents using the sensed magnetic fields. However, in practice, the position information is not available or is not sufficiently accurate. The resulting measurement methods are either unworkable or experience significant errors.

[0011] US Patent 8,718,964 presented an attempt to solve this problem by using an on-site calibration scheme. The scheme involves introducing an intentional, known change to the conductor currents so that the relationship between the conductor currents and sensor outputs can be established. However, this is an intrusive method that can only work when

intentional changes to the currents can be made. Its applications are therefore quite limited.

[0012] From the above, there is therefore a need for

systems, methods, and devices which mitigate if not overcome the shortcomings of the prior art.

SUMMARY

[0013] The present invention provides systems and methods for measuring currents in bundled conductors . A frame holding a number of sensors is used. The geometric and electrical parameters of the sensors in the frame are known or previously determined. The frame is then deployed to substantially encircle at least part of the bundled conductors. The magnetic fields produced by the current in the bundled conductors are then sensed by the sensors. The sensed magnetic fields are then turned into relevant data. The relevant data, in conjunction with the geometric and electrical properties of the sensors are then used to calculate and estimate the current in each of the bundled conductors. The parameters of the sensors may be determined during the assembly or manufacture of the frame and the sensors.

[0014] In one aspect, the present invention uses an off-site calibration method to determine the electrical and geometric parameters of the sensors. Using these parameters, the currents (and positions) of the conductors are then calculated and determined based on the calibrated parameters and the measured magnetic flux density data. This approach does not rely on any preconceived assumptions about the bundled conductor configuration. The present

invention is therefore applicable to various scenarios for current measurement, including its use in measuring currents in bundled conductors such as Romex cables, conductors enclosed in a conduit, and overhead distribution lines.

[0015] The methodology of the present invention involves two stages: (i) off-site calibration of the measurement device and (ii) an on-site current measurement using the calibrated device.

[0016] In one embodiment of the invention, Stage 1

establishes the parameters of the sensors in the measurement device. These parameters include the positions, orientations, and gain factors of the sensors . The calibration is achieved by using multiple calibrating conductors with precisely known positions and currents. This can be implemented as one of the manufacturing steps for the measurement device. The resulting sensor parameters are then saved into the device.

[0017] In this embodiment, Stage 2 calculates conductor

currents and positions using the magnetic fields sensed by the device's sensors and the sensor parameters stored in the device. Since the sensor parameters are known, the conductor currents and positions can be calculated if there is a sufficient number of sensors.

[0018] In another embodiment, the present invention relates to a scheme and its associated device to measure currents in enclosed multiple conductors using a contactless means. The invention utilizes the magnetic fields generated by the conductors to estimate their currents, without physically accessing the individual conductors. This contactless, noninvasive current measurement is useful for

residential, commercial or industrial facilities where inaccessible conductors are used and where the measurement of the current running through these conductors is desired for various applications.

These applications may include operation monitoring, energy metering and troubleshooting. In a first aspect, the present invention provides a system for estimating currents in at least one conductor in a bundle of conductors, the system comprising :

- a plurality of sensors arranged to surround at least a cross-sectional portion of said bundle of conductors when said system is deployed, each of said plurality of sensors being for detecting magnetic fields produced by said currents, said sensors producing signals based on magnetic fields detected;

- an A/D converter for converting signals produced by said sensors into digital data;

- a data processing unit to receive digital data derived from said magnetic fields produced by said currents, said data processing unit calculating an estimate of said currents within said at least one conductor based on said digital data and parameters for said sensors; wherein

- said parameters are predetermined and are retrieved by said data processing unit prior to calculating said estimate. ^0020] In a second aspect, the present invention provides a method for calibrating a system for measuring currents in a bundle of conductors, said system including a plurality of sensors for measuring magnetic fields caused by current passing through at least one conductor, the method comprising: a) deploying said sensors around at least one conductor, each of said at least one conductor having known coordinates relative to a predetermined reference point b) passing at least one current through said at least one conductor, said at least one current having known and predetermined characteristics; c) measuring at least one output from at least one of said sensors, said output being caused by magnetic fields caused by said at least one current in said at least one conductor; d) using data derived from said output in a multiple variable least squares formulation problem, said multiple variables being parameters for said sensors; e) determining a solution for said least squares problem once sufficient data from said at least one sensor and said at least one conductor has been generated .

^0021] In a third aspect, the present invention provides a method for determining currents in at least one conductor in a bundle of conductors, the method comprising : a) deploying a plurality of sensors adjacent said bundle of conductors, said plurality of sensors being arranged in a frame to surround at least a cross- sectional portion of said bundle of conductors; b) detecting magnetic fields caused by said currents using said sensors; c) measuring said magnetic fields; d) producing voltages based on said magnetic fields; e) converting said voltages into digital signals; f) transmitting said digital signals to a data processing device; g) solving a least squares formulation based on said magnetic fields detected in step b) to result in currents in said at least one of said conductors; wherein parameters for said sensors are predetermined and are used in step g) to solve said least squares formulation .

DESCRIPTION OF THE DRAWINGS ] The embodiments of the present invention will now be described by reference to the following figures, in which identical reference numerals in different figures indicate identical elements and in which:

FIGURE 1 illustrates the configuration of a

measurement device showing the sensors and the bundled conductors;

FIGURE 2 illustrates the geometry modeling of a sensor and a conductor; FIGURE 3 shows the household electrical panel in North American homes;

FIGURE 4 illustrates a flowchart of a two-stage current sensing method according to one aspect of the invention;

FIGURE 5 illustrates a test bed for contactless current measurement;

FIGURE 6 is a schematic diagram of the test bed illustrated in Figure 5;

FIGURE 7 is a bar chart illustrating the

determination of nc by evaluating local and global calculation errors;

FIGURE 8 illustrates absolute and relative mismatches between measured and calculated calibration voltage data;

FIGURE 9 shows the calibration of conductor positions and calibrated sensor positions;

FIGURE 10 schematically illustrates a system

according to one embodiment of the present invention;

FIGURE 11 schematically illustrates a system which may be used to implement one aspect of the present invention;

FIGURE 12 is a flowchart detailing the steps in a method according to another aspect of the invention;

FIGURE 13 is another flowchart detailing the steps in a method according to yet another aspect of the invention; and FIGURE 14 illustrates a variant of measurement device/ sensor frame for one aspect the present invention .

DETAILED DESCRIPTION

[0023] How to measure AC currents in a bundle of

inaccessible, enclosed conductors has been a challenging problem with many potential applications. One aspect of the present invention presents a method to solve the problem using an array of magnetic field sensors. The method consists of two novel ideas. The first idea is to use an off-site calibration method to establish sensor parameters including sensing position and angle. This provides more accurate sensor parameter estimation than simply using nominal values. The second idea is to "measure" (i.e. calculate) conductor currents and positions based on the sensed magnetic fields and the predetermined sensor parameters. This second process is simplified as the sensor information has already been obtained using the first idea. Both calibration and

measurement tasks are formulated as nonlinear least square (NLLS) problems and solved efficiently. The method has the potential for contactless current measurement of Romex, Teck and other enclosed cables as well as for overhead distribution lines.

2.1 Sensing Principle

[0024] An ideal straight conductor with AC current produces an alternating magnetic field around it. This magnetic field can be captured by magnetic sensors such as coils. Fourier transformation is performed to convert time-domain sensor signals to base frequency phasors . A phasor based representation of magnetic flux density B at a point with position and angular distance r and δ to the conductor with current I is obtained as

^{■ }2rr _{( }i _{) }

A voltage is induced on the coil sensor by the projection of the magnetic flux density onto the sensing direction, shown in Fig. 2. This voltage signal is amplified by an analog circuit with output voltage V, then digitized by an analog-to-digital converter (ADC) and finally recorded for analytics.

The effect of coil sensor induction and amplifier circuit can be modeled using a gain factor p and a phase shift φ as

7

(2) From all the parameters involved, one can derive the current f using the measured voltage V.

2.2. Enclosed Multi-Conductor System and Sensor Array For an enclosed multi-conductor system shown in Fig.

1, in order to completely recover all the currents, a measurement device that holds an array of sensors may be used. The measurement device may have a frame that holds the array of sensors on a plane that is perpendicular to the conductors. With the currents flowing through the conductors, the output voltage signals from the sensors sensing the magnetic fields from the currents are recorded simultaneously. These output voltage signals can then be used to solve for the current values.

[0028] One of the unique challenges in contactless current measurement is that the distance r and angle δ between the studied conductors and sensors are practically unknown. Distance r and angle δ can be expressed using Cartesian coordinates, as shown in Fig. 2. The output voltage phasor is a nonlinear function of all present currents and parameters as shown in Eqn. (3), where

i ^{x }s,> ≤} ^{anci } *1 ' -Vj _{" }. ^{are the } coordinates of k-t sensor and i-th conductor, respectively.

)

2.3. Application Example: Home Energy Use Monitoring

[0029] One of the important application scenarios for

contactless current measurement is home energy use monitoring, shown in Fig. 3. The measurement or sensing device can be installed around the plastic conduit outside incoming wires of household

electrical panels. Usually, three conductors, including two hot wires (110V AC) and one neutral wire, are enclosed within the plastic conduit. Both the positions of and currents within the three conductors are to be determined. The measured currents can be used in many applications such as energy use estimation or NILM. According to the existing investigations on magnetic sensor array based approach [14] and clamp-on current probe based approach [15] for home energy use monitoring, a current measurement accuracy of within 5% is required to provide sufficiently accurate estimation for home energy consumption.

2.4. Measurement Concept

[0030] The problem as described above in Section 2.2 is

restated as follows: solve for conductor currents with measured magnetic fields. Such a problem is unsolvable since both sensor and conductor positions are unknown. The issue is that there are more unknowns than available equations. Increasing the number of sensors can establish more equations but it also introduces more unknown sensor positions. It is recognized that sensor parameters are the property of the sensing device and are independent of conductor configurations. It is therefore sensible to first determine sensor positions and orientations with respect to a reference point. Once the sensor parameters have been obtained, the problem of "measuring" conductor currents then becomes solvable. In addition, adding more sensors leads to redundant equations for improved accuracy.

[0031] Determining sensor parameters is, unfortunately, not a simple task. Firstly, it can be quite difficult to install sensors at precise positions and with desired orientations. It is even more difficult to measure sensor positions and angles using mechanical means. Secondly, each sensor has an amplifier circuit, whose gain factor and phase shift must be established as well. Otherwise, additional unknown variables are introduced. In view of these challenges, the present invention includes an electrical method for

determining sensor parameters. This method, called sensor calibration, lets each sensor measure the magnetic fields of a set of conductors with positions and currents in the conductors being precisely known and predetermined. The sensor parameters can then be solved from the resultant equations. This off-site calibration process may be done as part of the manufacturing process for the measurement device. Once a device is assembled, it can be mounted onto a calibrator, the outputs of all sensors are collected, and the parameters for the sensors are solved for. The resulting sensor parameters can then be saved so that they can be used when the measurement device is deployed. The sensor parameters can be stored in an EPROM device embedded with the measurement device. This allows for each device to have its own unique sensor parameters to be predetermined regardless of the precision of sensor installation when the measurement device is deployed. Once the sensor parameters have been determined, the device can determine currents for various conductor configurations. This can be accomplished by solving conductor currents and positions based on known sensor parameters and outputs. A nonlinear least square (NLLS) analysis is utilized to solve this problem. NLLS performs nonlinear regression with m observations and n unknown parameters (m ≥ n) . It is a special form of nonlinear programming (NLP) . In fact, the sensor calibration is also formulated as an

NLLS problem, resulting in consistent solutions for both calibration and measurement problems . 3.1. Principle and Method 033] The goal of sensor calibration is to find the sensor- specific parameters, including pt, k, χ ,γ , and 6k for k-t sensor. The idea is to use a set of measured currents with known positions to determine sensor parameters by measuring the calibrating current and the sensor output voltages simultaneously. In order to simplify the process and avoid inter-conductor

interference, only one conductor is energized and used at any one time. Each conductor is then, in turn, energized with a current and the resulting sensor outputs are measured.

[0034] The following nc times of single conductor

calibrating experiments are performed, which constitute the set of calibration data set Sc. A single conductor with an AC current measured by current probe is placed at right angles to the plane of the sensors at a specified location inside the conduit with a known and predetermined position.

3.2. Calibration Data Processing

[0035] Calibration is done sensor by sensor. For the k-t

sensor, phase shift fyk can be determined as an average angular difference between the measured current and voltage phasor as in Eqn. (4) . Note that we use the subscript (-)q to refer to current and voltage obtained in the calibration stage. For the j- th calibrating experiment, the output voltages of all sensors {V ^{* }'* _{r } k ε Sc) are recorded simultaneously with the calibrating current V _{a } on the conductor at position { ^, ) . (4) After obtaining fyk , the remaining parameters can be calibrated using the magnitudes of the measured phasors, i.e., l _{a }' and based on the NLLS formulation shown as Eqn. (5). Auxiliary variables are used to reduce the nonlinearity in Eqn.

which is representing output voltage on Je-th sensor caused by the j-th calibrating experiment with conductor position at ( x ^{J } _{c }, y _{c } ^{J } ) . Also, multiple inequality constraints are considered for better optimum searching. Physical sensor position ( xji ; jf ) is used as a reference, the calibrated position is constrained in a circle with tolerance radius Rs . Note that, if calibrating conductors are distributed on the two sides of the line along the sensing direction, one has to firstly correct the opposite directions of voltage phasors before applying Eqn.

(4). Negative values of Υ^' ^{Λ } are expected in this case. Equation (5) is given as:

[0037] Because of the above, parameters for the k-t sensor pt, Qk, jcjf and f, are found by solving NLLS problem

Eqn. (5) . Since there are essentially four

independent unknown parameters (i.e., n = 4), one has to obtain at least four sets of observations (i.e., m ≥ n) to fully determine the unknowns. Using redundant calibration data samples allows one to improve the accuracy of the calibrating parameters. The optimal number of calibrating experiments (i.e., nc) is investigated in Section 6.2.

4. On-Site Current Measurement

4.1. Pre-processing

[0038] In the measurement stage, the sensing device is

deployed or placed outside the enclosed conduit as shown Fig. 1. The aim of the measurement stage is to find the accurate currents and positions of m conductors in the set of Si inside the enclosed conduit, based on the information of ns sensor output voltages in the set of Ss. Necessary pre-processing procedures related to sensor-circuit response have to be performed as follows. Note that we use the subscripts (·)ρ and (-)c to denote measured and pre- processed voltage phasors in the on-site measurement stage .

[0039] Parameter k describes the phase delay effect of the output voltage V with respect to its inducing magnetic field shown as Eqn. (2) . Such a phase delay can be compensated for before one determines the currents in the conductors. Given the calibrated fyk obtained by Eqn. (4) in the calibration stage, phase- compensated voltage phasor l' ^{ft } can be obtained using .,- + 1 ^{1 } =— ι· ^{> } j>

where _{c }*. and V _{t } refer to real and imaginary parts of T? ^{fc }, which will be the NLLS formulation shown as Eqn. (7) instead of the measured V . Equation (7) is given

4.2. Measurement Formulation

[0040] The current measurement formulation is based on NLLS

Eqn. (7), which utilizes the following data: sensor geometrical parameters and (x , Ok) and electrical parameters (pk) from the calibration result in Section 3, as well as phase-compensated voltages (¾.

+ il' _{c } ^{¾ } _{j }) from pre-processing in Section 4.1. 041] In this formulation, position (x , y ) and the current i _{-i } ^{J } ) in phasor form of all conductors are to be determined. This essentially follows the nonlinear equation given as Eqn. (3) above. The objective function is to minimize absolute mismatches between measured and calculated sensor output voltages. By using absolute mismatches, measurements of small magnitude are with proportional small weight, since they are commonly with large relative error that has to be eliminated.

[0042] Boundary constraints are enforced for each variable, where V^, _{BX } is set as the voltage measurement range of

ADC module and lj„ _{ax } is set as the rated current capacity of the conductor. Conductors are possibly anywhere within the conduit with radius Ri, so a corresponding geometry constraint is added.

[0043] Phasor based auxiliary variables Υ/' ^{Λ } + IV/' are inserted into Eqn. (7), representing the voltage components from k-t sensor caused by the j-th conductor, in order to reduce the nonlinearity after re-arrangement of the NLLS formulation.

4.3. Auxiliary Constraints

4.3.1. Net Current Constraint

[0044] In some circumstance, the net current of the studied conductor set can be known beforehand, or easily measured as (In,r +iln,±) , by adding a clamp-on AC current probe on the conduit. The following constraint can be added into Eqn. (7) to assist

Optimum se^ r r-hi n rr

(8) 4.3.2. Cable Geometry Constraints

0045] For the cases that the geometry distances of

conductors are known a priori, especially for cables with marked model numbers and, hence, an available specification, one is able to integrate the relevant constraints into the NLLS formulation Eqn. (7) . Such constraints can be added to specify the relative position of the conductors, where D and d are the external diameter of the cable and the diameter of internal con _{r }. _{t } ■ .-, ., ,. .-, .,

d ^{1 } < f.rf - .if}- - f j _{j }f - ιή ^{1 })- < D-

( < O - .r ^{J } ) ^{2 } + (if - yff < D ^{2 }

(9)

Procedure Flowchart 046] As a brief summary of the two-stage contactless

current measurement approach detailed above, a procedure flowchart is provided in Figure 4. From Figure 4, it can be seen that the off-site calibration stage is based on calibrating data set Sc to determine sensor parameters. The resultant sensor parameters are used in the current measurement stage along with sensor output voltages V to find conductor positions

(x , y ) and current values ( _{f }, _{E }") in the enclosed conduit or bundle of conductors .

[0047] Note that once the calibration procedures have been completed, the calibrated sensor parameters that have been calculated or measured can then be stored or otherwise made available to the current measuring system. The parameters can be stored in computer memory (e.g. in an EEPROM, an accessible magnetic medium, or any suitable medium) and this memory can be integrated into the complete current measurement system. Alternatively, the parameters can be suitably hardwired into the current measurement system such that the parameters are available whenever the system is used for measuring the current in bundled conductors.

6. Case Studies

6.1. Test Bed and Implementation

[0048] Laboratory experiments have been performed to

validate the effectiveness of the current measurement method on a test bed as shown in Figure 5. The schematic diagram is visualized in Figure 6. The overall dimensions of the U-shaped sensing device are 10 x 4 x 7 cm (1 x w x h) . The inner diameter is 44 mm to accommodate the conductor holder and the conduit .

[0049] The reference currents are obtained using a three- phase power source and a set of adjustable loads, connected by a testing conductor set inside a polyvinyl chloride (PVC) conduit, on to which the U- shaped sensing device is installed. The reference current values are measured using AEMC MN252 current probes [16] on each individual conductor, for both calibration and measurement stages . The current magnitude and phase accuracy of the current probe is 2.5% of the reading ±5mV and ≤5.0°, respectively (its output ratio is 10 mV/A) . Balanced and unbalanced currents with different magnitudes are obtained by using different load combinations for each phase. This is done in order to emulate practical operation currents . [0050] Six pairs of two-axis coil sensors are installed around the conduit as a semi-circle perpendicular to the conductors, shown in Figures 1 and 6. Each cylinder-shaped coil sensor is 5 mm in diameter and 8 mm in length. To save the space of printed circuit board (PCB) , a pair of sensors is embedded in a sensor holder or frame and positioned in different sensing directions perpendicular to the conductors. The sensor holders are installed on the PCB. The geometrical layout of the sensors is demonstrated in the calibration result in Section 6.2. Small position mismatches caused by manual sensor installation were observed in the sensor pairs with the mismatches being identified in the off-site calibration. Such mismatches will not affect the accuracy performance of on-site current measurement.

[0051] As can be seen from Figure 6 and from the above

description, the magnetic fields sensed by the various sensors are converted into suitable voltage signals . These signals are then received by an amplifier or amplifier circuit. The amplifier amplifies these signals and passes them on to an A/D (analog to digital) converter (ADC) for conversion into a suitable digital signal. The digital signal is then received by a data processing unit

(represented by a laptop in Figure 6) . The data processing unit then calculates the relevant parameters for the various sensors. As noted above, the process may need to be iterated a number of times as each sensor's parameters may need to be

calculated. This may involve isolating the readings from each sensor for every one of the conductors in the conduit and calculating the parameters for each sensor accordingly. [0052] All the measurements, including currents and sensor output voltages, were acquired using a National Instrument CompactDAQ with four 9215 simultaneous analog input modules and recorded by a laptop with LabVIEW data acquisition software. The ADC modules used in the test bed are within ±10 V range and 16- bit resolution. The minimum recognizable signal step is (2 x 10V) / 2 ^{16 } = 0.31 mV. The developed algorithm processes sensor signals to calculate currents and compare them with the reference values for the currents .

[0053] Two types of enclosed multi-conductor systems were tested as shown in Fig. 5. The first type tested uses straight aluminum round rods with a diameter 3.618 mm installed on a conductor holder with 21 known positions. This type of multi-conductor system was used to obtain calibration data and to evaluate sensor error. The second type is a three-conductor residential service cable with individual conductor diameters of around 7 mm. For this type of multi- conductor system, the conductors are not strictly straight and perpendicular to the sensing device. This type of multi-conductor bundle was used to emulate practical situations that go beyond ideal as sumptions .

[0054] The NLLS formulation is implemented in AMPL [17] and solved using KNITRO [18] as the optimizer. Interior point method (IPM) is selected to solve the

formulated NLP problems . The computing performance data is obtained using a laptop with an Intel Core i7-2860QM 2.50 GHz CPU. While the above description mainly discusses the three-conductor problem and a semi-circular

configuration for the coil sensors, the calculations, analysis, methods, and processes are also applicable to other conductor and/or sensor arrangements. As an example, the above described systems and methods can be used in a two-wire problem [19] . The reason for the applicability of the above to other situations and circumstances is the flexible nature of NLLS as a constrained nonlinear optimization problem. Based on the NLLS formulation, reinforcing constraints can be easily customized such as in (8)-(9), as well as scale up the formulation for additional sensors and conductors .

In the discussion below, currents and voltages are shown in peak values instead of root mean square (RMS) values in order to be consistent with the phasor representations used above.

.2. Calibration of Sensor Parameters

.2.1. Determining nc

One has to determine nc before calibration, that is, how many calibrating samples are sufficient to obtain sensor parameters that are accurate enough.

Theoretically, an NLLS problem can be solved if the number of observations (m) is no less than the number of unknown parameters (n) , i.e., m ≥ n or as long as m is at least as large as n. In this case, four sets of calibrating measurements are sufficient to determine the four sensor parameters (i.e., pt, k, Χρ,Ύς) . However, in practice, redundant measurements are preferred since uncertainties always exist. Regression using an exact number of observations leads to poor calibration results. In order to find a proper nc setting, Fig. 7 illustrates the norm of mismatches between calculated and measured sensor outputs, i.e., objective values in NLLS formulation in Eqn. (5) . Calibrated and uncalibrated conductors refer to the conductor positions that are used and not used in calibration, respectively. The process is actually one of fitting local error when only four conductors are used for calibration and leads to a significant global error. Meanwhile, the algorithm is able to reduce global error when 10 or more

calibrating conductors are used. However, increasing nc to a number greater than 10 does not reduce calibration error level. Hence, nc is selected as 10 for the studied sensing system as a proper setting value. As noted, 10 is a preferred number but is by no means the only value which can be used. .2.2. Evaluating Errors The voltage values in calibration range from 0.01 to 6 V. Absolute and relative errors between calculated and measured sensor outputs have to be evaluated and checked after calibration, which are visualized using color maps shown in Fig. 8 and can be used for diagnosing the sensing device. If apparent large values are found in a row, this indicates an

erroneous calibrating data sample. If apparent large values are found in a column, this indicates a flawed sensor. If the errors are averagely distributed and with small values, just as shown in Fig. 8, then this indicates that an accurate calibration result has been obtained. .2.3. Calibration Results The calibration result of the U-shaped sensing device noted above is shown in Figure 9. In Figure 9, the circles and squares refer to the calibrating

conductor positions and the calibrated sensor positions, respectively. The short lines across the sensors indicate their sensing directions. Since small spacing between the positions of sensor pairs was observed, this indicates mismatches in manual sensor installation in the two-axis sensor holder. These mismatches were identified in the calibration stage and these will not affect the accuracy of on- site current measurement. Once the calibration procedures are done, the sensor parameters are stored and re-used during on-site current measurement by solving the NLLS formulation in Eqn . ( 7 ) . The above therefore details a flexible and efficient method to contactlessly measure currents for enclosed multi-conductor systems using a sensor array. The system and method includes a two-stage scheme of off- site sensor calibration and on-site current

measurement. This approach simplifies measuring procedures and increases accuracy. The off-site calibration identifies sensor parameters that are independent from conductor configurations and that are valid for various measuring scenarios. Conductor currents and positions can, using these parameters, be determined simultaneously in on-site measurement. The effectiveness of this approach was verified using ideal conductor assumptions as well as practical cable limitations with satisfactory accuracy. [0062] In one implementation, the present invention takes the form of a system as shown in Figure 10. A conductor with AC current produces an alternating magnetic field around it. The magnetic field can be measured with magnetic sensors. As shown in Fig. 10, an array of magnetic field sensors are installed on the measurement device to detect the magnetic fields generated by the enclosed multiple conductors. Each sensor detects magnetic fields and generates a voltage signal. This signal is amplified by an amplifier circuit and the signal is converted into a digital signal by an analog-to-digital converter (ADC) . A processing unit takes sensor output signals as input and calculates values of the currents in the enclosed conductors. To accomplish this, the processing unit retrieves stored sensor parameters and uses these parameters in calculating the currents in the conductors. The calculated result can then be shown in a user interface .

[0063] For a single conductor, the output of a magnetic

field sensor can be in the form of an output voltage. It is related to the current as follows : v = f{ _{V },q ) (10) where v is the sensor output voltage; p represents the sensor parameters which consist of (i) sensor position and

orientation in space and (ii) the gain factor which transforms magnetic flux density into output voltage; q represents the conductor position; and i is the conductor current.

[0064] The variables v and i can be modeled as phasors or waveforms. In the discussion below, the phasor form is used.

[0065] For multiple conductors, it is assumed that the

conductors are parallel to each other and

perpendicular to the plane where the sensors are installed as shown in Fig. 10. The magnetic fields generated by the conducts add to each other at the sensor positions. Assuming n conductors and a sensor array with m sensors, the output voltage of k-t sensor is represented as v _{k } =∑}- 1 f{p _{k }, q t) kE[1 ,,,ΐΐϊ] (11)

[0066] As noted above, the magnetic field sensors are used to measure magnetic fields and, from these

measurements, conductor currents are calculated from the above equations. If there are m sensors, there is a total of 2m equations (i.e. real and imaginary part of the sensor output voltage phasor) . The variables of the equations include 4m sensor parameters pk

(i.e., sensor position in X-Y coordinates, sensor orientation angle, and gain factor), 2n conductor positions qj (i.e., conductor position in X-Y coordinates) and 2n the current values ij (i.e., real and imaginary part of current phasors) . It can be seen that there are more variables (4m+4n in total) than the equations (2m in total) . The problem cannot be solved without additional information. [0067] US patents 5,438,256 and 5,473,244 assume known p and g parameters to solve for the currents. As explained before, such parameters especially conductor positions g cannot be easily obtained on-site.

[0068] As noted above, it is recognized that sensor

parameters p are the property of the measurement device. They are independent of the conductors to be measured. Therefore, it is possible to first establish the sensor parameters independently. Once the sensor parameters become known, the number of unknown variables becomes 4n (g and i) . The problem of current measurement can thus be solved as long as

2m≥ 4rt (12)

[0069] For example, six sensors are able to estimate the currents in three conductors while eight sensors are able to measure the currents in a four-conductor system. In addition, adding more sensors leads to redundant equations for improved accuracy.

[0070] The sensor parameters are determined using an off- site calibration method. In this method, a newly constructed measurement device is set to measure the magnetic fields produced by a set of conductors with known positions and currents. The sensor parameters are solved by utilizing equation (11) as well. In this case, the unknown variables are p and the known variables are v, g and i. As long as there are sufficient calibrating conductors, p can be solved accurately. This off-site calibration process can be performed as a part of the manufacturing process of the measurement device: once a device is assembled, it can be mounted onto a calibrator, and the outputs of all the sensors are collected. The sensor

parameters can then be solved and the resulting parameters can be saved into a memory embedded in the device .

[0071] Once sensor parameters become available, the system can be used to measure currents of various conductor configurations. This second stage involves on-site current measurement using the calibrated device.

Stage 1: Off-site Calibration of the Measurement

Device

The goal of the calibration step is to find the sensor parameters p. The method uses a set of reference conductors with known positions g and known current values i to determine sensor parameters p. The current and sensor output voltages are measured simultaneously. During the calibration step, the conductor is placed at various positions so as to provide sufficient data to determine sensor

parameters .

[0072] Each sensor has four parameters, its X-Y coordinates, sensing orientation angle, and gain factor.

Therefore, four conductor positions are theoretically sufficient to determine the four sensor parameters. More conductor positions will further increase accuracy. For the k-t sensor, the sensor parameters can be solved using the following least-squares formulation . mln _{pft } - f(p _{k }, q _{}l } ί | ^{" } (13) where 2 is the number of conductor positions (1 ≥ 4) . The sensor output voltage v _{Jr }k, the reference position of conductor q and the reference current value ij are known, while sensor parameters pk are to be determined. Calibration is performed by solving Eqn. (13) one by one for each sensor.

[0073] Figure 11 illustrates the equipment for practicing one embodiment of this method. The calibrating device has :

(i) a conductor holder embedded with a set of aluminum conductors at precisely known positions,

(ii) a switching device that can connect any one of the conductors to the source,

(iii) a power source that supplies current to the switching device,

(iv) a current probe that can precisely measure the current entering into the conductor,

(v) a data acquisition system that obtains values of Vj,k and ij from the measurement device and the current probe, respectively,

(vi) a computer that controls the switch and calculates the sensor parameters by solving Eqn.

( 13 ) , and

(vii) a memory writing device that saves the calculated sensor parameters into the memory embedded in the measurement device .

[0074] The process of calibration is shown in the flowchart of Figure 12 and is explained below. The method starts with initializing the j counter (step 10) .

The j-th calibrating conductor is then selected (step 20) and energized using a reference current (step 30) . Reference current value ij and sensor output voltages v _{Jr }k are recorded by the data acquisition system (step 40) . Step 50 then checks if there are any other calibrating conductors left (there are only 2 conductors and if j is equal to 2 then no

conductors are left) . If there are other conductors left, then the counter is incremented (step 60) and the logic loops to step 20. Once all the calibrating conductors have been energized and the current and sensor voltages have been measured for each of the conductors, the data is then used to solve Eqn. (13) . Once the data has been gathered, step 70 is to initialize the counter to be used for the second half of the process. For each sensor, the relevant reference current, sensor voltage, and conductor position are used to solve the least-squares problem Eqn. (13) for that sensors parameter (step 80) . In one embodiment, the least-squares problem is solved using the interior-point method by the computer to result in the sensor parameters. For each sensor, once the sensor parameters have been found, the parameters are then written to data storage (step 90) . Step 100 then checks if other sensors still need their parameters to be solved for. It should be noted that, as noted above, there are m sensors and, as such, the process loops until k = m. If there are still sensor parameters to be calculated, the variable k is incremented (step 110) and the logic loops back to step 80. It should be clear that the process of solving Eqn. (13) and writing the

parameters to memory is accomplished for each sensor in turn . [0076] Note that once the measurement device has been calibrated, the sensor parameters are stored in an embedded data storage device such as an EEPROM. These parameters are valid for multiple current measurement operations and for various types of enclosed multiple conductors or bundled conductors. The calibration is thus only performed once, preferably as a part of the manufacturing process.

Stage 2: On-site Current Measurement using the

Calibrated Device

[0077] In the on-site current measurement stage using the calibrated device, a set of conductors with currents enclosed in a structure is placed close to the magnetic field sensors as shown in Figure 10. The aim is to find the accurate currents i and conductor positions q, based on the measurement of sensor output voltages v and calibrated sensor parameters p. As can be seen in Figure 10, the sensors are arranged on a plane to encircle or surround at least a cross- sectional portion of the bundle of conductors.

[0078] Consider a case with n conductors and m sensors, a least-squares based formulation Eqn. (14) is established to find the conductor currents and positions that best fit the measured sensor output voltages. Therefore, the mismatches between the measured voltages v and the calculated values are minimized .

(14)

[0079] By solving Eqn. (14), the current values i and

conductor positions q are found simultaneously. It does not depend on any prior knowledge regarding conductor configurations and, as such, it is adaptable to many different conductor configurations. This means that this formulation can be used for any number of conductors and for any distances among conductors .

[0080] Fig. 13 shows the flowchart of on-site current

measurement. This process starts (step 200) with reading sensor parameter values from the data storage (which could be embedded memory) . As the sensors capture the composed magnetic field generated by the conductors inside the enclosed conduit or jacket, the sensor output voltages are obtained from ADC (step 210) . This data is used as input for solving the least-squares problem shown in Eqn. (14) . The solution gives the optimal combination of current values and conductor positions . The measurement results for the currents in the bundled conductors are then shown on the user interface (step 230) . If further measurements are to be taken, then decision 240 loops the logic back to step 210, otherwise, the process ends, usually due to a user request.

[0081] As can be imagined, there are many variations and

extensions for the above process and method. Some of them may be as detailed below.

[0082] The method above also works for time-domain signals as the established least-squares formulation in Eqn. (15) can be directly used calculate time-domain current signals. The method can thus be used in applications such as detecting characterized operating signatures of home appliances or detecting the fault pattern of a motor. [0083] It should be noted that the method does not impose any limitation on the number of sensors or on the number of conductors . The least-squares formulation in Eqn. (15) is solvable as long as the number of equations is not less than the number of unknowns.

[0084] It should be clear that the method is not limited to a U-shaped sensor array configuration shown in Figure 10. While the configuration in the figure has the bundled conductors in the crook of the U-shaped frame or structure so that the sensors encircle the bundle of conductors, other configurations are possible. Other types of configurations, as long as the sensors are sufficiently diverse in spatial distribution, may also be used. One example of such a configuration is a two half-ring sensor array assembly that can open and close, shown as Figure 14. As can be seen in Figure 14, the sensors are divided into two half circle or half ring jaws. When deployed, the bundle of conductors is encircled by the half ring jaws.

[0085] The above discussion assumes the measurement device is portable. Portability would be useful for troubleshooting purposes. The device can also be used for permanent installation. For such permanent installation instances, the method of current measurement could be further simplified: After the conductor positions q are obtained, the relationship between the sensor voltage vector V and the conductor current vector X is reduced to a linear form. This linear form is described by a constant trans- impedance matrix Z as shown in Eqn. (15) . Matrix Z can be evaluated using the obtained sensor parameters and conductor positions (i.e., p and q) . Therefore, for the scenario of permanent installation, the current measurement problem can be simplified as the solution of linear equations Eqn. (15) .

V = ZI (15)

[0086] The method does not assume any a prior knowledge of conductor information. However, the availability of limited conductor information can be used to improve the current estimation accuracy. This is done by adding auxiliary constraints to the least-squares problem. For example, the following constraint in Eqn. (16) can be considered for a multi-conductor cable where the external cable diameter D and the internal individual conductor diameter d are known and where χ ( · ) and y(-) are the conductor coordinates on the X-Y axis.

(16 )

[0087] The method does not assume the geometry of individual conductors. It can be modeled as a dimensionless point described by its coordinate of X-Y axis, or other patterns of conductor geometry which can be described by more conductor parameters. For example, for an ellipse distributed conductor used in the cables with concentric neutral wire, one is able to use five parameters to describe this conductor:

central point coordinate of X-Y axis, rotating angle, major/minor axis.

[0088] The method also does not assume any prior knowledge regarding current patterns, e.g. three-phase balanced or unbalanced currents or if the neutral current is zero. However, the availability of such information is also beneficial to improving current estimation accuracy. If the net current of the multi-conductor system can be measured simultaneously with sensor signals (e.g., using a clamp-on current probe) or is known to be zero beforehand (e.g., cable serving a delta connected load) , similar auxiliary constraints can be applied. For example, the constraint shown as Eqn. (17) represents the net current equalling an external measurement i _{e } .

For a better understanding of the various aspects of the invention, reference may be made to the following documents, all of which are hereby incorporated by reference .

[1] S. Ziegler, R. Woodward, H.-C. Iu, and L.

Borle, "Current sensing techniques: A review," IEEE Sensors J., vol. 9, no. 4, pp. 354-376, April 2009.

[2] J. Libove and J. Singer, Apparatus for

measuring voltages and currents using non-contacting sensors. US Patent 5,473,244, Dec 1995.

[3] X. Sun, Q. Huang, Y. Hou, L. Jiang, and P. W. T. Pong, "Noncontact operation-state monitoring technology based on magnetic-field sensing for overhead high-voltage transmission lines," IEEE Trans. Power Del., vol. 28, no. 4, pp. 2145-2153, Oct 2013.

[4] J. Sharood, G. Bailey, D. Carr, J. Turner, and D. Peachey, Appliance retrofit monitoring device with a memory storing an electronic signature. US Patent 6,934,862, Aug 2005. [5] G. Hart, "Nonintrusive appliance load

monitoring," Proc . IEEE, vol. 80, no. 12, pp. 1870- 1891, Dec 1992.

[6] J. Lenz and A. S. Edelstein, "Magnetic sensors and their applications," IEEE Sensors J., vol. 6, no. 3, pp. 631-649, June 2006.

[7] J. Bull, N. Jaeger, and F. Rahmatian, "A new hybrid current sensor for high-voltage applications, " IEEE Trans. Power Del., vol. 20, no. 1, pp. 32-38, Jan 2005.

[8] K. Draxler, R. Styblikova, J. Hlavacek, and R. Prochazka, "Calibration of rogowski coils with an integrator at high currents," IEEE Trans. Instrum. Meas., vol. 60, no. 7, pp. 2434-2438, July 2011.

[9] K.-L. Chen and N. Chen, "A new method for power current measurement using a coreless hall effect current transformer," IEEE Trans. Instrum. Meas., vol. 60, no. 1, pp. 158-169, Jan 2011.

[10] C.-T. Liang, K.-L. Chen, Y . -P . Tsai, and N. Chen, "New electronic current trans-former with a self-contained power supply," IEEE Trans. Power Del., vol. 30, no. 1, pp. 184-192, Feb 2015.

[11] G. D'Antona, L. Di Rienzo, R. Ottoboni, and A. Manara, "Processing magnetic sensor array data for ac current measurement in multiconductor systems," IEEE Trans. Instrum. Meas., vol. 50, no. 5, pp. 1289-1295, Oct 2001.

[12] M. Bourkeb, C. Joubert, R. Scorretti, 0. Ondel, H. Yahoui, L. Morel, L. Duvillaret, C. Kern, and G. Schmitt, Device for measuring currents in the conductors of a sheathed cable of a polyphase network. WO Patent WO Patent 2013 068 360 Al, May 2013.

[13] AVO International Limited, Megger FlexiClamp 200 User Guide. [Online]. Available:

http : //www . biddlemegger . com/biddle-ug/Flexiclamp 200 UG.pdf

[14] P. Gao, S. Lin, and W. Xu, "A novel current sensor for home energy use moni-toring, " IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 2021-2028, July 2014.

[15] Energy Inc.'s The Energy Detective (TED), Specifications of TED Spyder . [On-line] . Available: http : //www . theenergydetective . com/downloads/spyderspec . pdf

[16] AEMC Instruments, AC Current Probe Model MN252. [Online] . Available:

http : //www . aemc . com/products/pdf/2115. 8. pdf

[17] R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical

Programming. Cengage Learning, 2002.

[18] R. H. Byrd, J. Nocedal, and R. A. Waltz,

\KNITRO: An integrated package for nonlinear

optimization," in Large-scale Nonlinear Optimization. Springer, 2006, pp. 35-59] .

[19] J. Zhang, Y. Wen, and P. Li, \Nonintrusive current sensor for the two-wire power cords, " IEEE Trans. Magn . , vol. 51, no. 11, pp. 1-4, Nov 2015.

[20] L. D. Rienzo and Z. Zhang, \Spatial harmonic expansion for use with magnetic sensor arrays," IEEE Trans. Magn., vol. 46, no. 1, pp. 53-58, Jan 2010. [0090] The embodiments of the invention may be executed by a computer processor or similar device programmed in the manner of method steps, or may be executed by an electronic system which is provided with means for executing these steps. Similarly, an electronic memory means such as computer diskettes, CD-ROMs, Random Access Memory (RAM) , Read Only Memory (ROM) or similar computer software storage media known in the art, may be programmed to execute such method steps. As well, electronic signals representing these method steps may also be transmitted via a communication network .

[0091] Embodiments of the invention may be implemented in any conventional computer programming language. For example, preferred embodiments may be implemented in a procedural programming language (e.g."C") or an object-oriented language (e.g. "C++", "java", "PHP", "PYTHON" or "C#") . Alternative embodiments of the invention may be implemented as pre-programmed hardware elements, other related components, or as a combination of hardware and software components.

[0092] Embodiments can be implemented as a computer program product for use with a computer system. Such implementations may include a series of computer instructions fixed either on a tangible medium, such as a computer readable medium (e.g., a diskette, CD- ROM, ROM, or fixed disk) or transmittable to a computer system, via a modem or other interface device, such as a communications adapter connected to a network over a medium. The medium may be either a tangible medium (e.g., optical or electrical communications lines) or a medium implemented with wireless techniques (e.g., microwave, infrared or other transmission techniques) . The series of computer instructions embodies all or part of the functionality previously described herein. Those skilled in the art should appreciate that such computer instructions can be written in a number of programming languages for use with many computer architectures or operating systems. Furthermore, such instructions may be stored in any memory device, such as semiconductor, magnetic, optical or other memory devices, and may be transmitted using any

communications technology, such as optical, infrared, microwave, or other transmission technologies. It is expected that such a computer program product may be distributed as a removable medium with accompanying printed or electronic documentation (e.g., shrink- wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server over a network (e.g., the Internet or World Wide Web) . Of course, some embodiments of the invention may be implemented as a combination of both software (e.g., a computer program product) and hardware. Still other embodiments of the invention may be implemented as entirely hardware, or entirely software (e.g., a computer program product). A person understanding this invention may now conceive of alternative structures and embodiments or variations of the above all of which are intended to fall within the scope of the invention as defined in the claims that follow.

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