A method for estimating power system operating state based on quantum computing

Abstract

The present invention proposes a method for estimating the operating state of a power system based on quantum computing, which relates to the field of power system operation analysis and state perception. By constructing a power system operating state estimation model based on an unconstrained quadratic binary optimization model (QUBO model), and solving it through an optical quantum computer, the power system operating state estimation result is obtained. The present invention transforms the power system operating state estimation problem into a QUBO model and solves it, converting the traditional computing algorithm for the state estimation problem into a quantum computing algorithm, and providing a model paradigm and feasible algorithm for constructing quantum computing for the power system operating state estimation problem. This algorithm introduces quantum computing into power system state estimation, and quantum computing will be able to meet higher computing power requirements and shorter response time requirements in the future.

Description

Technical Field

[0001] The present invention relates to the field of power system operation analysis and state perception, and specifically to a power system operation state estimation method based on quantum computing. Background Art

[0002] New power systems will incorporate a high proportion of renewable energy and power electronic equipment, resulting in high uncertainty and posing challenges to the safe and stable operation and dispatch of power systems. Power system state estimation is fundamental to power system operation and control. New power systems require shorter response times for power system state estimation, resulting in a surge in demand for computing power. Quantum computing is a new form of computing that uses quantum mechanical principles to control quantum mechanical units to perform calculations. Compared to classical computers, quantum computers use qubits as computational units. Due to the properties of quantum superposition, they can achieve exponential computing power with the same number of bits. Existing research on quantum computing for power system state estimation is based on quantum circuits, using the HHL algorithm to solve linear equations. This algorithm relies on the hardware foundation of general-purpose quantum computers. Currently, the number of available qubits is growing slowly, making it difficult to overcome technical barriers in a short period of time. Coherent Ising machines, specialized optical quantum computers for solving optimization problems, are expected to achieve even higher qubit growth and commercialization in a relatively short period of time. Summary of the Invention

[0003] In view of the fact that general-purpose quantum computers are difficult to realize in the short term, while dedicated optical quantum computers can be commercialized in a relatively short period of time, this invention proposes a method for estimating the operating state of a power system based on quantum computing, which uses a dedicated optical quantum computer (coherent Ising machine) as hardware support to meet the computing power requirements for state estimation of future new power systems.

[0004] To achieve the above object, the present invention provides the following technical solutions:

[0005] The present invention is a method for estimating the operating state of an electric power system based on quantum computing. The hardware-based coherent Ising machine of the present invention can realize the efficient solution of the Ising model. The Ising model and the QUBO model (unconstrained quadratic binary optimization model) are equivalent and can be converted into each other. The least squares algorithm (WLS) is the most widely used state estimation algorithm at present. The Newton iteration method is traditionally used to solve the WLS problem. The present invention converts the solution equation in the Newton iteration method into the solution equation of the QUBO model, realizes the conversion of the traditional algorithm into a quantum computing algorithm, and provides a construction paradigm for the quantum algorithm of the electric power system state estimation.

[0006] The coherent Ising machine is a hybrid quantum computing system consisting of two subsystems: an optical system and an electrical system. The optical system is responsible for the preparation and storage of optical qubits. First, a femtosecond fiber laser with a frequency of 100MHz generates laser pulses. Due to the low output power (100mW) of these laser pulses, they require power amplification by an erbium-doped fiber amplifier (EDFA). The amplified laser is then passed through a periodically programmed lithium niobate (PPLN1) crystal to double the frequency of the optical pulses to 780nm. This frequency-doubled light then serves as the pump light source for a synchronously pumped phase-sensitive amplifier. This light is then converted into 1560nm signal light by a PPLN2 crystal, ultimately forming degenerate optical parametric oscillators (DOPOs). These pulses generate optical pulses with specific phases and amplitudes. These pulses serve as optical qubits, which are stored in a fiber cavity and used for subsequent computations. The electrical system is responsible for the control and calculation of the optical quantum bit. First, the Ising matrix of the problem to be solved is downloaded to the field-programmable gate array (FPGA) through the host computer. After the FPGA obtains the phase and amplitude information of the optical particle bit through a balanced homodyne detector, it calculates the feedback signal and modulates the feedback light through the intensity modulator (IM) and phase modulator (PM). The feedback light and the light pulses in the fiber loop interfere with each other, causing the optical quantum bit to evolve in the direction of the lowest Hamiltonian of the Ising problem. After reaching the lowest Hamiltonian, the phase information of the optical quantum bit is the final solution to the Ising problem.

[0007] The Ising model is a stochastic model that describes physical phase transitions. The mathematical form of the Ising model is as follows: Where H represents the Hamiltonian, σ is the spin variable to be determined, which takes a value of ±1, and J and h are the coefficients of the quadratic term and the linear term, respectively.

[0008] The QUBO model is an unconstrained quadratic binary optimization model. The mathematical modeling of the QUBO model is as follows: Where Γ = {1, 2, 3, ..., k} is the set of subscripts of the variable x, representing a total of k 0-1 variables, β is the coefficient of the quadratic term, and α is the coefficient of the linear term.

[0009] Power system state estimation involves establishing a relationship between distribution network state variables x (e.g., node voltage amplitude and phase angle, n×1) and the measurement function h (essentially a mapping of power flow equations, typically a nonlinear system of equations) given a measurement vector z (e.g., node voltage, node injection power, and line power flow, all m×1). Taking into account the inherent noise inherent in the measurements, the relationship between the distribution network state variables x (e.g., node voltage amplitude and phase angle, n×1) and the measurement vector z is: z = h(x) + e. Redundant measurements of the operating state are obtained through measurement devices as input, and the voltage amplitude and phase angle are output as the desired state variables. The resulting estimate is then determined to best represent the system's current operating state.

[0010] The weighted least squares (WLS) algorithm is: This method assumes that the measurement error obeys The normal distribution of the measurement and the estimated value of the measurement is minimized. Based on the state estimation model, given the measurement vector z and the measurement function h, the objective function constructed by the weighted least squares criterion is as follows: min J(x) = [zh(x)] T W[zh(x)]

[0011] Where: W is the weighted coefficient matrix, which represents the confidence of the measurement. Its specific construction form is a diagonal matrix as shown below:

[0012] The Newton iteration method is as follows: Since the measurement h(x) in the model is a nonlinear function and x cannot be expressed analytically, h(x) needs to be Taylor expanded, the higher-order terms are omitted to linearize it, and the WLS problem is solved iteratively using the Newton method. At this time, for the kth iteration, the objective function can be written as follows:

[0013] J(Δx)=[Δz k -H(x k )Δx k ] T W[Δz k -H(x k )Δx k ] (1)

[0014] in,

[0015]

[0016] Δz k =z k -h(x k ) (3)

[0017] By expanding formula (3), we can get its iteration direction

[0018] x k+1 =x k +Dxk (4)

[0019] G(x k )Dx k =H(x k ) T WΔz k (5)

[0020] in,

[0021] G(x k )=H(x k ) T WH(x k ) (6)

[0022] Therefore, the k-th iteration result is:

[0023] Δx k =G(k) -1 H(x k ) T W[zh(x k )] (7)

[0024] Where: H(x k ) is the value of h and x k The Jacobian matrix of .

[0025] The method for estimating the operating state of a power system based on quantum computing of the present invention comprises the following steps:

[0026] Step S1: Obtain the measured data of the voltage and power of each node, preset the measurement standard deviation, and construct the weighted coefficient matrix W. Initialize the node voltage and phase angle, and preset the step value for skipping the iteration.

[0027] Step S2: Calculate the measurement function h(x) and the Jacobian matrix model H(x) according to the initial voltage phase angle.

[0028] Step S3: Determine the binary expression matrix by the preset model accuracy (number of bits) Find the Q matrix of the QUBO model.

[0029] Step S4: Input the Q matrix into CIM (simulated annealing / GUROBI will also be used for verification later), set the solution parameters, obtain the solution vector, and binary expression matrix The product of this vector is the state quantity correction vector, and the estimated values of the node voltage and phase angle are corrected according to formula (25).

[0030] Step S5: Determine whether the maximum absolute value of the iteration step is less than a preset value. If it is greater than the preset value, return to step 2 and continue the next iteration. If it is less than the preset value, exit the loop to obtain the final state estimation node voltage value and phase angle value.

[0031] In step S1, the initialized voltage per unit value is generally 1, the initialized phase angle per unit value is generally 0, and W is a weighting coefficient matrix that represents the confidence of the measurement. Its construction method is consistent with that in WLS:

[0032]

[0033] Where σ is the standard deviation of a single measurement, representing the confidence level of that measurement.

[0034] In step S2, the measurement function of the distribution network is as follows:

[0035] 1) Node voltage measurement model

[0036]

[0037] 2) Node Injection Active Power Measurement Model

[0038]

[0039] 3) Node injection reactive power measurement model

[0040]

[0041] 4) Line transmission active power measurement model

[0042]

[0043] 5) Line transmission reactive power measurement model

[0044]

[0045] The measurement function h(x) in step S2 is derived, and its corresponding Jacobian matrix is as follows:

[0046]

[0047] Among them, the partial derivative results of each measurement function are as follows:

[0048] 6) Node voltage measurement model

[0049]

[0050] 7) Node Injection Active Power Measurement Model

[0051]

[0052] 8) Node Injection Reactive Power Measurement Model

[0053]

[0054] 9) Line transmission active power measurement model

[0055]

[0056] 10) Line transmission reactive power measurement model

[0057]

[0058]

[0059] The symbols in the above formula are explained as follows:

[0060] 1) Subscript i: node number;

[0061] 2) Superscript z: indicates that the value is a measured value;

[0062] 3) V i : voltage amplitude of node i (pu);

[0063] 4)θ i : voltage phase angle value of node i (degree);

[0064] 5)θ ij : the phase angle difference between node i and node j (degree);

[0065] 6)P i : injected active power of node i (pu);

[0066] 7)Q i : injected reactive power of node i (pu);

[0067] 8)P ij : The active power transmitted from node i to node j (pu);

[0068] 9)Q ij : The reactive power transmitted from node i to node j (pu);

[0069] 10)G ij : admittance matrix conductance;

[0070] 11)B ij : admittance matrix susceptance;

[0071] 12) Self-susceptance of node i.

[0072] In step S3, the binary expression matrix Convert the continuous variable into a 0-1 binary variable. For the continuous variable w, the continuous variable w is expressed as follows through binary expression:

[0073]

[0074] Where: is an n×nK matrix; is an nK×1 vector of 0-1 binary variables, where n is the number of independent variables and the value of K determines the accuracy of the binary representation.

[0075] In step S3, the transformation of the QUBO model is derived as follows:

[0076] The form of each iterative solution in Newton iteration is:

[0077] Δx k =[H(x k ) T WH(x k )] -1 H(x k ) T W[zh(x k )] (26)

[0078] For the following least squares regression problem:

[0079]

[0080] Its analytical solution is:

[0081] w=(X T X) -1 X T Y (28)

[0082] Therefore, let have to,

[0083]

[0084] make:

[0085] X1=W1H(x k ),Y1=W1[zh(x k )] (30)

[0086] The solution in Newton iteration can be written in the form of the following least quadratic solution:

[0087] Δx k =(X1 T X1) -1 X1 T Y1 (31)

[0088] Combining the binary expansion of Equation (25) and Equation (27), each iterative solution can be regarded as solving the following QUBO model:

[0089]

[0090] Then the corresponding Q matrix is:

[0091]

[0092] Step S4 uploads the Q matrix to the quantum cloud platform, and the optical quantum computer can be called to solve the QUBO model, return the solution vector, and obtain the iteration vector according to formula (25) to update the state vector, including the state estimation values of voltage and phase angle.

[0093] Step S5 determines whether the maximum iteration step of the iteration vector in S4 is less than a preset value. If it is less than the preset value, it is considered that the model has converged, otherwise the iteration continues.

[0094] Compared to existing technologies, this invention offers the following advantages: It proposes a quantum computing-based method for estimating the operating state of power systems, providing a model for the future evolution of quantum algorithms for state estimation. Compared to traditional algorithms, this algorithm introduces quantum computing into power system state estimation, potentially enabling higher computing power and shorter response times. Compared to existing research based on quantum circuits, this invention, implemented using a dedicated optical quantum computer, is expected to be applied and commercialized much sooner. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 Flowchart of the present invention;

[0096] Figure 2 This is a comparison chart of voltage state estimation results;

[0097] Figure 3 This is a comparison chart of the phase angle state estimation results; DETAILED DESCRIPTION

[0098] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0099] Step S1: Obtain the measured data of the voltage and power of each node, preset the measurement standard deviation, and construct the weighted coefficient matrix W. Initialize the node voltage and phase angle, and preset the step value for skipping the iteration.

[0100] Step S2: Calculate the measurement function h(x) and the Jacobian matrix model H(x) according to the initial voltage phase angle.

[0101] Step S3: Determine the binary expression matrix by the preset model accuracy (number of bits) Find the Q matrix of the QUBO model.

[0102] Step S4: Input the Q matrix into CIM (simulated annealing / GUROBI will also be used for verification later), set the solution parameters, obtain the solution vector, and binary expression matrix The product of this vector is the state quantity correction vector, and the estimated values of the node voltage and phase angle are corrected according to formula (25).

[0103] Step S5: Determine whether the maximum absolute value of the iteration step is less than a preset value. If it is greater than the preset value, return to step 2 and continue the next iteration. If it is less than the preset value, exit the loop to obtain the final state estimation node voltage value and phase angle value.

[0104] In step S1, the initialized voltage per unit value is generally 1, the initialized phase angle per unit value is generally 0, and W is a weighting coefficient matrix that represents the confidence of the measurement. Its construction method is consistent with that in WLS:

[0105]

[0106] Where σ is the standard deviation of a single measurement, representing the confidence level of that measurement.

[0107] In step S2, the measurement function of the distribution network is as follows:

[0108] 1) Node voltage measurement model

[0109] V i z =V i (35)

[0110] 2) Node Injection Active Power Measurement Model

[0111]

[0112] 3) Node injection reactive power measurement model

[0113]

[0114] 4) Line transmission active power measurement model

[0115]

[0116] 5) Line transmission reactive power measurement model

[0117]

[0118] The measurement function h(x) in step S2 is derived, and its corresponding Jacobian matrix is as follows:

[0119]

[0120] Among them, the partial derivative results of each measurement function are as follows:

[0121] 11) Node voltage measurement model

[0122]

[0123] 12) Node Injection Active Power Measurement Model

[0124]

[0125] 13) Node Injection Reactive Power Measurement Model

[0126]

[0127]

[0128] 14) Line transmission active power measurement model

[0129]

[0130] 15) Line transmission reactive power measurement model

[0131]

[0132] The symbols in the above formula are explained as follows:

[0133] 1) Subscript i: node number;

[0134] 2) Superscript z: indicates that the value is a measured value;

[0135] 3) V i : voltage amplitude of node i (pu);

[0136] 4)θ i : voltage phase angle value of node i (degree);

[0137] 5)θ ij : the phase angle difference between node i and node j (degree);

[0138] 6)P i : injected active power of node i (pu);

[0139] 7)Q i : injected reactive power of node i (pu);

[0140] 8)P ij : The active power transmitted from node i to node j (pu);

[0141] 9)Q ij : The reactive power transmitted from node i to node j (pu);

[0142] 10)G ij : admittance matrix conductance;

[0143] 11)B ij : admittance matrix susceptance;

[0144] 12) Self-susceptance of node i.

[0145] In step S3, the binary expression matrix Convert the continuous variable into a 0-1 binary variable. For the continuous variable w, the continuous variable w is expressed as follows through binary expression:

[0146]

[0147] Where: is an n×nK matrix; is an nK×1 vector of 0-1 binary variables, where n is the number of independent variables and the value of K determines the accuracy of the binary representation.

[0148] In step S3, the transformation of the QUBO model is derived as follows:

[0149] The form of each iterative solution in Newton iteration is:

[0150]

[0151] For the following least squares regression problem:

[0152]

[0153] Its analytical solution is:

[0154]

[0155] Therefore, let have to,

[0156]

[0157] make:

[0158] X1=W1H(x k ),Y1=W1[zh(x k )] (63)

[0159] The solution in Newton iteration can be written in the form of the following least quadratic solution:

[0160] Δx k =(X1 T X1) -1 X1 T Y1 (64)

[0161] Combining the binary expansion of Equation (25) and Equation (27), each iterative solution can be regarded as solving the following QUBO model:

[0162]

[0163] Then the corresponding Q matrix is:

[0164]

[0165] Step S4 uploads the Q matrix to the quantum cloud platform, and the optical quantum computer can be called to solve the QUBO model, return the solution vector, and obtain the iteration vector according to formula (25) to update the state vector, including the state estimation values of voltage and phase angle.

[0166] Step S5 determines whether the maximum iteration step of the iteration vector in S4 is less than a preset value. If it is less than the preset value, it is considered that the model has converged, otherwise the iteration continues.

[0167] After specific actual experiments, the system used for simulation testing in the present invention is an adapted 4-node distribution network system, which is adapted from the case4_dist.m system provided in MATPOWER. The power supply at the original bus numbered 400 is removed to make it conform to the power flow equation of the distribution network.

[0168] Measurement data description: The busbar number indicates the measurement location of the variable; the true value is the theoretical value obtained through power flow calculation; the standard deviation is used to describe the error between the measured value and the true value due to factors such as sensor delay and accuracy error. At the same time, the standard deviation is used to construct the weighting coefficient matrix in the WLS model; the measured value is the simulated measurement data obtained by random sampling from the normal distribution based on the true value and standard deviation; the error is the numerical difference between the measured value and the true value.

[0169] Calculation result data description: The residual is the absolute maximum value of the voltage and phase angle updates during each iteration. When the residual is very small, it means that the iteration result has approached the optimal value. When the residual is lower than the preset value, the iteration can be exited to obtain the final state estimate.

[0170] Table 1 shows the true value, standard deviation setting and measured value of the adapted 4-bus system.

[0171] Table 2 compares the final state estimation results. The true system value is obtained through power flow calculations, and three different state estimation methods are used: the traditional least squares algorithm (WLS), the QUBO model using the GUROBI solver, and the QUBO model using an optical quantum computer.

[0172] From Table 2 and Figure 2 and Figure 3 The correctness of the power system state estimation method based on the QUBO model of this invention has been verified, and the final results are close to the traditional algorithm and the system true value. However, due to the limitations of current experimental conditions, the maximum available bits of the optical quantum real machine (i.e., the CIM real machine) are 100, and the accuracy of the Q matrix data is within 8 bits. Therefore, the binary expansion accuracy in the experiment is low, and the Q matrix has been scaled, so the error is slightly greater than the result of the traditional WLS algorithm.

[0173] Table 1 Adapted 4-node measurement data

[0174] Serial number name Standard deviation Busbar number Measurements True value error 0 Voltage Magnitude 0.001 1 1.0006449 1 0.000645 1 Active Power Injection 0.001 1 -0.0003996 0.0004 0.0008 2 Active Power Injection 0.001 2 -0.0003989 0.0004 0.000799 3 Active Power Injection 0.001 3 -0.0004000 0.0004 0.0008 4 Reactive Power Injection 0.001 1 -0.0002003 0.0002 0.0004 5 Reactive Power Injection 0.001 2 -0.0002000 0.0002 0.0004 6 Reactive Power Injection 0.001 3 -0.0002001 0.0002 0.0004

[0175] Table 2 System true value and final state estimation results

[0176]

[0177] Compared to existing technologies, this invention introduces quantum computing into power system state estimation, representing a forward-looking exploration of quantum computing in this field, potentially addressing future demands for higher computing power. Compared to quantum circuit implementations, this invention, based on a dedicated optical quantum computer, is more likely to achieve higher quantum bit counts in a shorter timeframe, enabling commercialization.

[0178] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

[0179] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A method for estimating the operating state of a power system based on quantum computing, characterized in that: The steps include: Step S1: Obtain the measured data of the voltage and power of each node, preset the measurement standard deviation, construct the weighted coefficient matrix W, initialize the node voltage and phase angle, and preset the step value for jumping out of the iteration; Step S2: Calculating the measurement function h(x) and the Jacobian matrix model H(x) according to the initial voltage phase angle; Step S3: Determine the binary expression matrix using the preset model accuracy Obtain the Q matrix of the QUBO model; Step S4: Input the Q matrix into CIM, set the solution parameters, obtain the solution vector, and binary expression matrix The product of this vector is the state quantity correction vector, and the estimated values of the node voltage and phase angle are corrected according to formula (25); Binary expression matrix Convert the continuous variable into a 0-1 binary variable. For the continuous variable w, the continuous variable w is expressed as follows through binary expression: Where: is an n×nK matrix; It is an nK×1 vector composed of 0-1 binary variables, where n is the number of independent variables and the value of K determines the accuracy of the binary expression; Step S5: Determine whether the maximum absolute value of the iteration step is less than a preset value. If it is greater than the preset value, return to step 2 and continue the next iteration. If it is less than the preset value, exit the loop and obtain the final state estimation node voltage value and phase angle value; In step S4, the Q matrix is uploaded to the quantum cloud platform, and the optical quantum computer is called to solve the QUBO model, the solution vector is returned, and the iteration vector is obtained according to formula (25), and the state vector is updated, including the state estimation values of the voltage and phase angle.

2. The method for estimating the operating state of a power system based on quantum computing according to claim 1, characterized in that: In step S1, the initialized voltage per unit value is 1, the initialized phase angle per unit value is 0, and W is a weighting coefficient matrix that represents the confidence of the measurement. Its construction method is consistent with that in WLS: Where σ is the standard deviation of a single measurement, representing the confidence level of that measurement.

3. The method for estimating the operating state of a power system based on quantum computing according to claim 2, characterized in that: In step S2, the measurement function of the distribution network is as follows: 1) Node voltage measurement model 2) Node Injection Active Power Measurement Model 3) Node injection reactive power measurement model 4) Line transmission active power measurement model 5) Line transmission reactive power measurement model The measurement function h(x) in step S2 is derived, and its corresponding Jacobian matrix is as follows: Among them, the partial derivative results of each measurement function are as follows: 1) Node voltage measurement model 2) Node Injection Active Power Measurement Model 3) Node injection reactive power measurement model 4) Line transmission active power measurement model 5) Line transmission reactive power measurement model In the above formula: 1) Subscript i: node number; 2) Superscript z: indicates that the value is a measured value; 3) V i : voltage amplitude of node i (pu); 4)θ i : voltage phase angle value of node i (degree); 5)θ ij : the phase angle difference between node i and node j (degree); 6)P i : injected active power of node i (pu); 7)Q i : injected reactive power of node i (pu); 8)P ij : The active power transmitted from node i to node j (pu); 9)Q ij : The reactive power transmitted from node i to node j (pu); 10)G ij : admittance matrix conductance; 11)B ij : admittance matrix susceptance; 12) Self-susceptance of node i.

4. The method for estimating the operating state of a power system based on quantum computing according to claim 3, characterized in that: In step S3, the transformation of the QUBO model is derived as follows: The form of each iterative solution in Newton iteration is: For the following least squares regression problem: Its analytical solution is: w=(X T X) -1 X T Y (28) Therefore, let have to, make: X1=W1H(x k ),Y1=W1[z-h(x k )] (30) The solution in Newton iteration can be written in the form of the following least quadratic solution: Combining the binary expansion of Equation (25) and Equation (27), each iterative solution can be regarded as solving the following QUBO model: Then the corresponding Q matrix is:

5. The method for estimating the operating state of a power system based on quantum computing according to claim 4, characterized in that: The step S5 determines whether the maximum iteration step of the iteration vector in S4 is less than a preset value. If it is less than the preset value, the model is considered to have converged. Otherwise, the iteration continues. The final iteration value obtained by the algorithm is compared with the true value and the traditional WLS solution value to obtain the effectiveness of the method.

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