CN110781638B - Optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics - Google Patents
Optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics Download PDFInfo
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Abstract
The invention discloses an optimal Thevenin equivalent parameter calculation method based on volt-ampere characteristics of a node port, which is an identification method for calculating optimal Thevenin equivalent parameters based on minimum volt-ampere characteristic deviation of the node port; the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum; calculating the state quantity of a sample point on the volt-ampere characteristic of the node port by a sensitivity method, optimizing a parameter calculation process according to the characteristics of a sample point generation process, and accelerating the calculation speed; and obtaining the optimal Thevenin equivalent parameters of the node port according to the objective function optimization solution. The voltage-current characteristic presented by the port of the element in the power system is also nonlinear, so that the external voltage-current characteristic described by thevenin equivalent parameters solved through network linearization has certain deviation from the real characteristic of the port.
Description
Technical Field
The invention belongs to the field of electricity, and particularly relates to an optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics.
Background
The statements in this section merely provide background information related to the present disclosure and may not necessarily constitute prior art.
With the expansion of the scale of the interconnected power grid and the implementation of the power market, the operation point of the power grid is closer to the static voltage stability limit point. Due to the large-scale access of clean energy, the operation modes of the power system are more complex and diversified, and therefore the requirement for online static voltage stability monitoring is increasingly urgent. The method based on Thevenin equivalent parameter identification in the existing online static voltage stability assessment method is widely applied because of clear concept, simple structure and high calculation speed.
Static voltage stability on-line evaluation requires rapid and accurate acquisition of thevenin equivalent parameters of a node. The current equivalent parameter calculation methods mainly have two types. One is a local measurement-based method, network frame parameters are not needed, pure data drive is achieved, and calculation is simple and fast.
Documents "K.Vu, M.M.Begovic, D.Novosel and M.M.Saha", "Use of local measurements to estimate voltage-stability flag", "Proceedings of the 20th International Conference on Power Industry computers Applications, Columbus, OH, USA,1997, pp.318-323.doi: 10.1109/PICA.1997.599420" first proposed a method for estimating thevenin equivalence parameters based on local measurements and applied the Thevenin equivalence method to static voltage stability analysis. On the basis of the documents, a large number of methods for identifying thevenin equivalent parameters based on local measurement are proposed, such as a method for estimating thevenin parameters in a z-V space, a method for optimizing and searching thevenin equivalent parameters based on an extended PV curve, and the like. The above algorithms all assume that thevenin equivalent parameters in the time window of the measured data remain unchanged, but this assumption does not conform to the actual operating conditions of the power system. In view of the above, the davinan equivalent parameter identification method based on local area measurement is continuously improved. The literature, "leerifu, successes to, scalding, davinan equivalence tracking parameter drift problem research [ J ]. Chinese electro-mechanical engineering, 2005(20): 1-5" analyzes the essential cause of parameter drift, and proposes a method for screening candidate sampling points to inhibit parameter drift.
The literature, "Zhaojinli, mytilus," analysis of voltage instability index working condition based on local phasor measurement [ J ]. electric power system automation, 2006(24):1-4+10, "analyzes the reason for calculating the parameter drift of equivalent parameters by using a curve fitting method, and provides the correct working condition and validity criterion of the voltage instability prediction index.
The voltage stability online monitoring method is characterized in that a voltage stability online monitoring model based on a PMU and an improved Davinan equivalent model is proposed on the basis of qualitative analysis of circuit theories in documents of Liuming pine, Zenbergine, Yao, Sunwandun, Wuwen, and the power system automation 2009,33(10):6-10.
Further improvement is made on the basis of the documents, and methods such as a Thevenin equivalent parameter tracking algorithm based on full differentiation, an iterative optimization solution based on trajectory sensitivity Thevenin equivalent parameters and the like are provided, so that the error of parameter identification is further reduced. From a review of the above documents, it can be seen that Thevenin equivalent parameters based on local measurement have improved the accuracy of identification much on the basis of the original model. However, when the system has line tripping, capacitor switching, generator reactive power out-of-limit and other situations, the limitation of parameter time variation and drift still remains the problem that the local area measurement method cannot completely overcome. The randomness and the fluctuation of the system operation mode caused by the access of large-scale clean energy sources make the limitation more prominent.
The second type is based on a wide area measurement method, and node Thevenin equivalent parameters are obtained by single-state section data. The method can overcome the limitation of time variation and drift of the parameters, and the identification of the parameters is more accurate than that of a local area measurement method. Documents "Y.Wang et al", "Voltage Stability Monitoring Based on the Concept of Coupled Single-Port Circuit", "in IEEE Transactions on Power Systems, vol.26, No.4, pp.2154-2163, and Nov.2011.doi: 10.1109/TPWRS.2011.2154366" propose the Concept of Coupled Single Port, realize the identification of Thevenin equivalent parameters Based on Single-state section data, and the parameter identification precision is greatly improved compared with the local measurement method. The document ' Davinan, Yijun, Houjunxian, Sun China east, Shaoyao, Linweifang ' Thevenin equivalent parameter tracking calculation method [ J ] based on time domain simulation, China electro-mechanical engineering, 2010,30(34):63-68 ' proposes a Thevenin equivalent parameter calculation method based on time domain simulation, wherein the load in a network is equivalent to impedance under the current section to correct a node admittance matrix of the system, and an open-circuit voltage is calculated by adopting a compensation method. The document shows that the method takes account of the wide-area Thevenin equivalent parameter online calculation method for the reactive power out-of-limit of the generator [ J ] the power system automation, 2016,40(11):53-60+67 ], the load in the network is equivalent to impedance to correct the node admittance matrix of the system, and then the Thevenin equivalent potential of each node is solved according to a node voltage equation. The problem of quantitative calculation of load node Thevenin equivalent parameters when the reactive power of the generator exceeds the limit is solved. The two documents differ in that a method of compensating the current source and a method of disconnecting the branch impedance are respectively adopted when the open-circuit voltage is obtained. However, when solving thevenin equivalent parameters of different nodes, the two documents need to solve a linear equation for many times, and the calculation time is slow, so that the online identification of the parameters cannot be met.
Aiming at the problem document,' Liangchen, Liudawei, coking army, Mashiying, Octopus, Thevenin equivalent parameters added based on an accelerating branch are calculated on line [ J ]. the power grid technology, 2017,41(09):2972 and 2978.
For a linear network, the two-node system corresponding to the Thevenin equivalent parameter and the original system show completely the same linear volt-ampere characteristics, and theoretically, the two-node system and the original system can be equivalently replaced. For an actual power system, a node port externally shows nonlinear volt-ampere characteristics, and the nonlinear volt-ampere characteristics of the node port are approximated by the Thevenin equivalent which is essentially the linear volt-ampere characteristics formed by combining the equivalent potential and the equivalent impedance of a node with the voltage and current under the current section. In the existing literature, nonlinear part linearization processing in a power network is performed by observing a node port to a system side, so that the network is changed into a linear network under a current state section, and the volt-ampere characteristic of the node port is expressed as linearity. The documents "Y.Wang et al", "Voltage Stability Monitoring Based on the Concept of Coupled Single-Port Circuit", "in IEEE Transactions on Power Systems, vol.26, No.4, pp.2154-2163, and Nov.2011.doi: 10.1109/TPWRS.2011.2154366" derive the volt-ampere characteristic expression of the node Port, and under the assumption that the load of each node of the system increases in proportion and the Voltage amplitude proportion is approximately unchanged, the volt-ampere characteristic relation of the node Port can be obtained to be a linear relation. But in actual operation the power of the individual nodes cannot fluctuate exactly to scale. Therefore, the node port voltage-current characteristic obtained by the existing document has a certain deviation from the actual node port voltage-current characteristic.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides an optimal Thevenin equivalent parameter calculation method based on the node port volt-ampere characteristic.
In order to achieve the above object, one or more embodiments of the present invention provide the following technical solutions:
an optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics comprises the following steps:
rapidly obtaining node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method;
the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
The further technical scheme is a method for measuring the precision of thevenin equivalent parameters, which comprises the following steps: and measuring the accuracy of the Thevenin equivalent parameters by keeping the loads of other nodes unchanged, adding fluctuation quantity to the loads at the calculation nodes, and calculating the modulus value of the difference between the node voltage phasor obtained by calculating the Thevenin equivalent parameters and the node voltage phasor obtained by calculating the actual load flow.
According to the further technical scheme, the precision of thevenin equivalent parameters is measured, and the method specifically comprises the following steps: adding random disturbance to a load node i, and setting the load after the load node i is disturbed as SikK is 1,2 …, m, and the total number of perturbations is m;
with SikThe power of the original node i is replaced to carry out load flow calculation to obtain the standard value of the node voltage phasor
Then the S is mixedikEstimation value of node voltage phasor calculated by two-node system with Thevenin equivalent parameter correspondence
And measuring the accuracy of the Thevenin equivalent parameters by the proportion of the phasor difference modulus of the estimated value of the node voltage and the standard value to the modulus of the node voltage phasor standard value.
The further technical scheme is used for calculating the sample points on the node port volt-ampere characteristic curve and has two characteristics: firstly, the variation of other load nodes in the active power delta P of the load node is zero; secondly, only the voltage amplitude and the phase angle variation corresponding to the load node need to be solved.
An optimal Thevenin equivalent parameter calculation system based on node port volt-ampere characteristics comprises:
a sample point acquisition module for rapidly acquiring node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method;
an optimization target module, wherein the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and the solving module is used for solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, so that the real part and the imaginary part of the optimal Thevenin equivalent potential are obtained.
The invention also discloses a computer device, which comprises a memory, a processor and a computer program which is stored on the memory and can run on the processor, and is characterized in that the processor executes the program to realize the following steps:
rapidly obtaining node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method; (ii) a
The optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
The invention also discloses a computer readable storage medium having a computer program stored thereon, characterized in that the program, when executed by a processor, implements the steps of:
acquiring a sample point on a current section real volt-ampere characteristic curve of a node port in a power system;
the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
obtaining node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points obtained under the current state section based on an optimization target;
and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the obtained node voltage vector and current vector estimated values, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
The above one or more technical solutions have the following beneficial effects:
the invention provides a method for finding the optimal Thevenin equivalent parameter according to the minimum volt-ampere characteristic expression deviation of a node port in order that a simplified circuit obtained through the node Thevenin equivalent parameter has a port volt-ampere characteristic close to that of an original system circuit. Different from a method for carrying out linear processing on a current power flow section in the prior art, the method for calculating the voltage-current characteristic of the current power flow section of the power flow system rapidly obtains sample points on a real voltage-current characteristic curve of a node port through a sensitivity method, obtains the linear characteristic with the minimum voltage-current characteristic deviation with the actual node port under the current section through a fitting method, obtains the optimal Thevenin equivalent parameter, and finally optimizes a parameter calculation method according to the characteristics of a sample point generation process so as to improve the calculation speed. In order to verify the reasonability of the method, the accuracy of thevenin equivalent parameters is measured by keeping loads of other nodes unchanged, adding fluctuation quantity to the loads at the calculation nodes and calculating the module value of the difference between the node voltage phasor obtained by calculating the thevenin equivalent parameters and the node voltage phasor obtained by calculating the actual load flow, so that the limitation that the voltage amplitude is only compared in the prior art is avoided.
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The accompanying drawings, which are incorporated in and constitute a part of this specification, are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification, illustrate exemplary embodiments of the invention and together with the description serve to explain the invention and not to limit the invention.
FIG. 1 is a schematic diagram illustrating fast calculation of node state quantity variation according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an inversion process of a Jacobian matrix according to an embodiment of the invention;
FIG. 3 is a schematic diagram of errors of Thevenin equivalent parameters of different sampling intervals and sampling quantities of a node system in embodiment 9 of the present invention;
FIG. 4 is a schematic diagram of errors of Thevenin equivalent parameters of different sampling intervals and sampling quantities of a node system in embodiment 39 of the present invention;
FIG. 5 is a schematic diagram of a real voltammetric characteristic curve of a node 29 according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of an imaginary volt-ampere characteristic curve of a node 29 according to an embodiment of the present invention;
FIG. 7 is a diagram illustrating a comparison of the mean values of the relative errors of the nodes according to the embodiment of the present invention.
Detailed Description
It is to be understood that the following detailed description is exemplary and is intended to provide further explanation of the invention as claimed. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the invention. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
The embodiments and features of the embodiments of the present invention may be combined with each other without conflict.
The general idea provided by the invention is as follows:
and (3) providing a method for finding the optimal Thevenin equivalent parameter according to the minimum volt-ampere characteristic expression deviation of the node port. Different from the method for linearly processing the current power flow section in the documents, the method disclosed by the invention can be used for quickly obtaining the sample points on the real volt-ampere characteristic curve of the node port by a sensitivity method, obtaining the linear characteristic with the minimum volt-ampere characteristic deviation with the actual node port under the current section by a fitting method, obtaining the optimal Thevenin equivalent parameter, and optimizing the parameter calculation method according to the characteristics of the sample point generation process so as to improve the calculation speed.
Example one
The embodiment discloses an optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics. Therefore, the method for finding the optimal Thevenin equivalent parameter according to the minimum volt-ampere characteristic expression deviation of the node port is provided according to the disclosure.
To understand the technical idea of the present application, the current thevenin theorem is explained:
thevenin's theorem shows that a port linear network containing an independent power supply and a linear resistor shows linearity to the volt-ampere characteristic of an external circuit, and can be equivalently replaced by a series combination of a voltage source and the resistor. Thevenin equivalent parameters are coefficients of a linear volt-ampere characteristic expression. The Thevenin equivalent theorem of the linear network can simplify a complex linear network into a simple circuit formed by combining a voltage source and a resistor in series, and facilitates analysis.
For the nonlinear network observed from the i port of the node to the system side in the power system, if the equivalent potential passes through thevenin
Equivalent impedance E of TheveninthFor the equivalence of a two-node system for supplying power to the node, a proper Thevenin equivalent parameter needs to be found, so that the volt-ampere characteristic of a node port is simplified into a linear expression shown in formula (1):
in the formula:
and
representing a port voltage current vector;
and Z is the coefficient of the port volt-ampere characteristic equation, namely Thevenin equivalent parameter.
The volt-ampere characteristic relation of the port of the node i can be obtained as follows:
where the subscripts L, T and G represent the load node, the tie node, the generator node,
and
is the voltage current vector of the node, and Y is the node admittance matrix.
The existing idea of linearizing the volt-ampere characteristic of the node port is to directly linearize the nonlinear network observed from the node port to the system side under the current state section, so that the volt-ampere characteristic observed from the node port to the system side is represented as linearity, that is, the expression of the formula (2) is simplified into a linear expression. However, due to the nonlinearity of elements in the power system, the volt-ampere characteristic presented by the port is also nonlinear, so that the external volt-ampere characteristic described by thevenin equivalent parameters solved through network linearization has a certain deviation from the real characteristic of the port.
Aiming at the problem, another idea is adopted for linearizing the volt-ampere characteristic of the node port, namely, a sample point on a real volt-ampere characteristic curve of the node port is quickly obtained and is a reference point of the real volt-ampere characteristic, the linear characteristic with the minimum volt-ampere characteristic deviation with the actual node port under the current section is obtained through a fitting method, and then the optimal Thevenin equivalent parameter is obtained. The optimal Thevenin equivalent parameters are mainly used for on-line static voltage stability margin evaluation and can also be used for the aspects of interconnection power grid boundary equivalence, optimization of transmission and distribution cooperative calculation, simplification of large-system calculation and the like.
The optimal Thevenin equivalent parameter calculation method comprises the following steps:
a port volt-ampere characteristic corresponding to the Thevenin equivalent parameter required to be optimized by the node i is assumed to be shown as a formula (5).
In the formula (I), the compound is shown in the specification,
and
is the voltage and current phasor of the node i;
and
is an optimization of node iThevenin equivalent potential and reactance.
Equation (6) is obtained by equaling the real part and the imaginary part on both sides of equation (5):
in the formula (I), the compound is shown in the specification,
the real and imaginary parts of the optimal thevenin equivalent potential for node i,
for node i the real and imaginary parts, U, of the optimal Thevenin equivalent impedanceiR、UiXFor the real and imaginary parts of the voltage phasors at node I, IiR、IiXThe real and imaginary parts of the current phasors for node i.
The node voltage phasor and the real part and the imaginary part of the current phasor of the operating point under the current state section
Substituting formula (6) to give formula (7):
the following formula is obtained from formula (6) and formula (7):
further finishing to obtain:
if the sample point on the real volt-ampere characteristic curve can be quickly obtained, the current vector of the sample point is brought to the right side of the formula (9), the voltage vector is brought to the left side of the formula (9), and the optimization target is that the deviation between the voltage vector of the sample point and the voltage vector of the sample point calculated by the Thevenin equivalent parameters of the current vector of the sample point is minimum, namely the deviation between the left side and the right side of the formula (9) is minimum. The optimization target is as follows:
in the formula, QiAnd solving an optimization objective function when thevenin equivalent parameters are obtained for the node i, wherein the subscript i represents the ith node, the subscript j represents the jth sample, and n represents the number of sample points.
Order to
VIiX、VUiR、VUiXIs a vector of the same form as equation (11).
Order to
Then equation (9) can be expressed as equation (15):
AZ＝B (15)
the extremum condition of equation (10):
will be in the current stateNode voltage vector and current vector estimated values of node port volt-ampere characteristic sample points obtained under the section are carried into the formula (12) and the formula (13), and the real part of the optimal Thevenin equivalent impedance of the node i can be obtained by solving the equation set formula (16)
And imaginary part
Then the real part and the imaginary part of the optimal Thevenin equivalent impedance are brought into the formula (7) to obtain the real part of the optimal Thevenin equivalent potential
And imaginary part
Calculating a volt-ampere characteristic curve sample point based on a sensitivity method:
calculating the optimal Thevenin equivalent parameters according to the minimum volt-ampere characteristic expression deviation of the node port requires sampling the actual volt-ampere characteristic curve. If the conventional load flow calculation is carried out on each sample point, the calculation time is long, and the real-time requirement of thevenin equivalent parameter online identification cannot be met. Therefore, the node port volt-ampere characteristic curve of the current section is rapidly sampled by adopting a sensitivity-based method, and the calculation flow is optimized and accelerated according to the characteristics of the method.
The node port volt-ampere characteristic curve of the current section is rapidly sampled, and the method specifically comprises the following steps: firstly, calculating the elements of the inverse matrix of the Jacobian matrix participating in calculation, then setting different load power fluctuation values, calculating the node voltage amplitude and phase angle corresponding to the power fluctuation values, further calculating the current under corresponding power, and obtaining the sampling point of the volt-ampere characteristic
The correction formula of Newton method load flow calculation is as follows:
in the formula, J is a Jacobian matrix corresponding to the section of the current state;
the active and reactive variable quantity of the node is obtained;
is the variation of the phase angle and amplitude of the node voltage.
The fluctuation of the active power and the reactive power of the ith load node is made to be delta PiAnd Δ QiThe variation of the voltage amplitude and phase angle is DeltaUiAnd Δ θi. The inverse matrix of the jacobian matrix of the formula (17) is multiplied by the node power fluctuation value to obtain the node voltage amplitude and the phase angle variation, which are as follows:
an estimated value of the voltage and current vector of the ith load node after the jth fluctuation can be obtained:
in the formula of Ui0And thetai0The voltage amplitude and the phase angle value of the current section node i are obtained; pi0And Qi0Is a current sectionThe active power and the reactive power value of the node i;
and
the estimated values of the voltage amplitude and the phase angle variation of the ith load node after the jth fluctuation;
and
the estimated values of the voltage amplitude and the phase angle of the ith node load after the jth fluctuation; delta PijAnd Δ QijFluctuating the variation of the active power and the reactive power for the ith load node at the jth time;
and
and the current and voltage phasor estimated value of the ith load node after the jth fluctuation.
Multiple fluctuations of the same node only require matrix multiplication of formula (18), and multiple sets of estimated values of voltage and current phasors are obtained through formulas (19) to (22). The real part of the equivalent impedance of the node i Thevenin can be obtained by taking the estimated values into the formula (12) and the formula (13) and solving the formula (16) of the equation set
And imaginary part
Then the real part and the imaginary part of thevenin equivalent impedance are brought into the formula (7) to obtain the real part of thevenin equivalent potential
And imaginary part
Although the speed of the sensitivity-based volt-ampere characteristic sample point calculation is higher than that of the load flow calculation, for a large system, the Jacobian matrix is large in scale, and the calculation of the sample point for multiple times is too heavy for online application. Therefore, the calculation method is optimized according to the characteristics of the calculation process, and the calculation speed is obviously improved so as to adapt to the real-time requirement of online application.
Optimization of a volt-ampere characteristic sample point calculation method: equation (18) for calculating the sample point on the node port voltage-current characteristic curve has two characteristics: firstly, the variation of other load nodes in the delta P is zero; secondly, only the voltage amplitude and the phase angle variation corresponding to the load node need to be solved. The calculation method based on the two characteristic sample points can be optimized, so that the calculation speed is accelerated.
The optimization steps are as follows: firstly, carrying out Gaussian elimination solving on column elements of a Jacobian matrix inverse matrix corresponding to a node to be solved, wherein because the variable quantity of other load nodes in the delta P is constant to zero, only two elements which really participate in calculation are arranged in the column, and then taking out the corresponding elements and non-zero elements in the delta S to calculate so as to obtain corresponding voltage amplitude and phase.
Fig. 1 shows an optimized calculation method of the node state quantity change amount, in which PV represents a PV node, PQ represents a PQ node, M PV nodes and R PQ nodes are total, and the position of the ith load node in the PQ node is assumed to be C. Since the variation of other load nodes in Δ P is always zero when the active power of the ith load node fluctuates, as shown by Δ S in the figure, the triangular portion of Δ S is a portion whose variation is not zero. From FIG. 1, it is evident that the inverse J of the Jacobian matrix is used-1When multiplied by the node power fluctuation Δ S, J-1Only the PQC-th column element in (a) corresponds to a non-zero element in Δ S, i.e., the column to which the triangle portion corresponds. Other columns are multiplied by zero element of Δ S when they participate in the operation, and therefore may not be considered. Only the voltage amplitude and the phase angle variation, J, corresponding to the load node need to be solved-1In which only two rows corresponding to the amplitude phase angle of PQC need to be consultedThe row corresponding to the operation, i.e. the circle part. Finally, only J is required to be added-1The two elements corresponding to the medium pentagon are respectively multiplied by the triangular elements in the delta S to obtain the obtained voltage amplitude and the phase angle variation. Due to J-1For dense matrices, the acceleration method of FIG. 1 can greatly accelerate the computation time of the matrix multiplication.
The jacobian matrix inversion process is shown in fig. 2. J is a jacobian matrix and the black parts represent non-zero elements. J. the design is a square-1The inverse of the Jacobian matrix, the triangle portion is the column corresponding to the C-th PQ node. D is a black portion of 1 and a triangular portion is a column corresponding to the C-th PQ node in the M + 2R-dimensional unit matrix. The inversion process of the jacobian matrix is converted into the solution of a linear equation, as shown in formula (23).
In the formula (I), the compound is shown in the specification,
h column of the inverse of the Jacobian matrix J, dhIs a unit column vector with the h-th element being 1.
Since only one column corresponding to the C-th PQ node in the inverse Jacobian matrix is needed to be solved, the column corresponding to the triangle in the matrix D is selected to use the formula (23) to obtain the selected column in the inverse Jacobian matrix, which is J in the graph IV-1Labeled as the column to which the triangle corresponds.
The first optimization requires only one column of the inverse of the solution Jacobian matrix rather than all, which greatly reduces the amount of computation. The second optimization is that the obtained list of elements does not need to be completely involved in calculation, and only two related elements are taken for calculation, so that the calculation is further accelerated. By adopting the two optimization methods, a lot of calculation amount can be reduced, and the on-line calculation requirement of Thevenin equivalent parameters is met.
Measuring the precision of thevenin equivalent parameters: the traditional Thevenin equivalent parameter calculation method starts from the definition of Thevenin equivalence to measure the precision of the Thevenin equivalent parameters obtained by calculation, but only measures the amplitude error of a voltage estimation value and an actual value after node load fluctuation, and the error measurement limitation is caused because the difference between voltage phase angles is not considered.
The present disclosure provides a new method for measuring accuracy of thevenin equivalent parameters. Keeping the loads of other nodes unchanged, adding random disturbance to a load node i, and setting the load after the disturbance of the load node i as Sik(k ═ 1,2 …, m), the total number of perturbations is m. With SikThe power of the original node i is replaced to carry out load flow calculation to obtain the standard value of the node voltage phasor
Then the S is mixedikEstimation value of node voltage phasor calculated by two-node system with Thevenin equivalent parameter correspondence
Measuring the accuracy of thevenin equivalent parameters by the ratio of the phasor difference modulus of the estimated value of the node voltage and the standard value to the modulus of the node voltage phasor standard value, and calculating the ratio of the phasor difference modulus of the estimated value of the node voltage and the standard value to the modulus of the node voltage phasor standard valueikThe method is defined as the relative error between the estimated value of the node voltage phasor obtained by adopting Thevenin equivalent parameter calculation and the standard value of the node voltage phasor obtained by actual load flow calculation under the k-th load random disturbance of the ith node. The following formula:
simulation calculation example: and (3) verifying the precision of thevenin equivalent parameters of different sample point numbers and sample intervals:
an IEEE3 machine 9 node system and a New England10 machine 39 node system are selected as test systems. This section tests the accuracy of thevenin equivalent parameters of the method provided by the disclosure in sample sampling intervals of different numbers of voltammetric characteristic samples and different power fluctuation ranges. Directly calculating to obtain thevenin equivalent parameter of the node i according to the data of the current section by using the method
Zi1. The frequency of load disturbance is 30 times, the power fluctuation range adopts +/-30%, and the relative error e between the estimated value of the node voltage phasor corresponding to the 30-time disturbance of each node and the standard value is obtained by applying the methodik(k ═ 1,2 …, 30). Because the number of the sample points can influence the calculated amount, different numbers of the sample points and different power fluctuation ranges are taken as sample sampling intervals to observe the influence on the error of thevenin equivalent parameters. The number of sample points is set to range from 1 to 19, and the sampling interval of the samples is from 20% to 140% of the set power fluctuation range (± 30%). Firstly, averaging the relative errors obtained by 30 times of disturbance of each node to obtain an average value of the relative errors of each node, and averaging the average value of the relative errors of each node to be used as the error of thevenin equivalent parameters corresponding to a specific sample point number and a sampling interval. The thevenin equivalent parameter errors of different sample point numbers and sampling intervals of the 9-node system and the 39-node system are shown in fig. 3 and 4.
As can be seen from fig. 3 and 4, when the sampling interval and the number of the sample points change, the error of the davinan equivalent parameter remains in the same order of magnitude, and it can be seen that the method provided by the present disclosure is not greatly affected by the sampling interval and the number of the samples. With the increase of the number of samples, the error of thevenin equivalent parameters has a descending trend. However, when the number of sample points is about 10, the Thevenin equivalent error is basically kept unchanged, which is influenced by the fact that the port volt-ampere characteristic measured by the Thevenin equivalent parameters is linear, and the fitting error cannot be continuously reduced by increasing the number of sample points, which indicates that the optimal Thevenin equivalent parameters are found at the moment. The number of subsequent simulated sample points is 10.
The size of the sampling interval needs to balance the error influence of the range of the sample points and the sensitivity method, the larger the range is, the larger the nonlinear range covered by the sample points is, the better the fitting effect is, but the error of the sensitivity method and the real trend result is increased due to the increased range. To balance the two factors, the following simulations all use samples with a sampling interval of 0.5 times the power fluctuation range (i.e., + -15% of the current load).
Comparing and verifying thevenin equivalent parameter precision with other wide area measurement methods:
"Voltage Stability Monitoring Based on the Concept of Coupled Single-Port Circuit," in IEEE Transactions on Power Systems, vol.26, No.4, pp.2154-2163, and Nov.2011.doi: 10.1109/TPWRS.2011.2154366.
Document [24] Yuanzhi, Yuexi, salty, Liangjun, Liudawei, and a wide-area Thevenin equivalent parameter online calculation method considering the reactive out-of-limit of a generator [ J ] power system automation, 2016,40(11):53-60+67.
Since the documents [22] and [24] have verified that the accuracy of the wide-area measurement thevenin equivalent parameter is better than that of the local-area measurement thevenin equivalent parameter, the present disclosure is no longer contrasted with the accuracy of the local-area measurement thevenin equivalent parameter. This section mainly verifies the accuracy of documents [22], [24] and the method provided by the present disclosure in thevenin equivalent parameter identification. In order to compare the equivalent parameter identification method of wide-area measurement Thevenin based on the concept of coupling single port, which is disclosed in the disclosure, with the equivalent parameter identification method of wide-area measurement Thevenin based on the concept of coupling single port, which is disclosed in the document [22], and the equivalent effect of the parameter calculation method of modifying the node admittance matrix of the system by using the equivalent load as impedance, which is disclosed in the document [24], on the volt-ampere characteristic of the node port, simulation analysis is performed on the New England10 machine 39 node system.
Method according to the disclosure and document [22] based on data of a current section]、[24]Calculating to obtain thevenin equivalent parameter of load node i
Zi1And
Zi2and
Zi3. The number of the sample points adopted by the method is 10, the size of the sample interval is 0.5 times of the disturbance interval, and the load disturbance frequency is 30 times. Using the above method, each node is disturbed 30 timesAnd the estimated value and the standard value of the corresponding node voltage vector. And obtaining the estimated value and the standard value of the node current vector through the disturbance power of the node and the standard value and the estimated value of the node voltage vector. Through the calculation, the relation between the node voltage vector and the current vector, namely the volt-ampere characteristic relation of the node port can be obtained. The real current-voltage characteristic and the imaginary current-voltage characteristic of the node 29 are shown in fig. 5 and 6.
In fig. 5 and 6, voltage 1 is a standard value obtained by the calculation of the power flow of the original system, voltage 2 is a calculated value of the node voltage of the two-node system corresponding to the davinan equivalent parameter solved by the method proposed in the document [24], voltage 3 is a calculated value of the node voltage calculated by the method proposed in the present disclosure, and voltage 4 is a calculated value of the node voltage calculated by the method proposed in the document [22 ]. It can be seen from fig. 5 and 6 that the real current-voltage characteristic and the imaginary current-voltage characteristic of the node port can be well measured by the method provided by the present disclosure. Although the voltage 3 shows a nonlinear characteristic in the current-voltage characteristics of the real part and the imaginary part, it still shows a linear characteristic in the current-voltage characteristic relationship of the phasor of the formula (1). From the analysis, the Thevenin equivalent parameters obtained by the method disclosed by the invention can be well equivalent to the power network in which the node port looks.
In addition to the volt-ampere characteristic analysis of the slave node port, the relative error e of the ith load node defined by the 2.3 node in the kth load random disturbance is adoptedikSo as to quantitatively measure the precision of thevenin equivalent parameters. And averaging the relative errors obtained by the 30 times of disturbance of each node to obtain the average value of the relative errors of each node, wherein the average value of the relative errors of each node is shown in fig. 7.
It is obvious from fig. 7 that the precision of thevenin equivalent parameters of the method provided by the present disclosure is obviously higher than that of the methods of documents [22] and [24] on each node.
Verifying the accuracy of Thevenin equivalent parameters under different tidal current sections: in order to verify that the method provided by the disclosure can obtain accurate Thevenin equivalent parameters under various power flow sections, the section adopts a plurality of load increasing modes on an IEEE3 machine 9 node system and a New England10 machine 39 node system to calculate errors of Thevenin equivalent parameters under various power flow sections.
As shown in tables 1 and 2, four load increase modes are adopted for obtaining different tidal current sections in this section, which are respectively: all the loads are increased according to the original proportion; part of heavy load nodes are increased according to the original proportion; part of light load nodes are increased according to the original proportion; and parts of heavy-load and light-load nodes are increased according to the original proportion.
Different load increasing modes of table 19 node system
Different load increasing modes of table 239 node system
Method according to the disclosure and document [22] based on data of a current section]、[24]Calculating to obtain thevenin equivalent parameter of load node i
Zi1And
Zi2and
Zi3. The number of the sample points adopted by the method is 10, the sampling interval of the samples is 0.5 times of the power fluctuation range (namely +/-15% of the current load), and the load disturbance frequency is 30 times. Obtaining the relative error e between the estimated value of the node voltage phasor and the standard value corresponding to the 30-time disturbance of each node by using the methodik(k ═ 1,2 …, 30). Firstly, the methodAnd averaging and maximizing the relative errors obtained by the 30-time disturbance of each node to obtain the average value and the maximum value of the relative errors of each node, averaging and maximizing the relative errors of each node to obtain the average value and the maximum value of the relative errors corresponding to a single power flow section, and finally averaging and maximizing the relative errors of each power flow section to obtain the average value and the maximum value of the relative errors corresponding to all the power flow sections. The average value and the maximum value of the relative errors corresponding to all the power flow sections of the 9-node system and the 39-node system in different growth modes are shown in tables 3 and 4.
Table 39 relative error corresponding to all power flow sections of node system
Relative error corresponding to all power flow sections of table 439 node system
The numbers in the first column in tables 3 and 4 represent the load increase pattern. From tables 3 and 4, it can be seen that the maximum value and the average value of the relative error corresponding to all the flow sections of the method provided by the disclosure on the IEEE3 machine 9 node system and the New England10 machine 39 node system are obviously smaller than those of the methods in the documents [22] and [24 ]. Therefore, Thevenin equivalent parameters calculated by the method under various power flow sections have high precision, and the high precision can be maintained even under the power flow section with high system nonlinearity when the load is heavy. The high-precision explanation of the dimensionality equivalent parameters under the condition of traversing each power flow section and various random disturbances shows that the method provided by the disclosure can provide relatively accurate static equivalent parameters for online static stability analysis.
And (3) fast verification of computation of thevenin equivalent parameters: the simulation shows that the method provided by the invention can greatly improve the accuracy of Thevenin equivalent parameter identification based on a single-state section. On-line identification of thevenin equivalent parameters needs to meet the requirement of rapidity, and the section tests the time required by the method disclosed by the invention and the document [22] method for solving thevenin equivalent parameters of all load nodes in a New England10 machine 39 node system, a 118 node system, a 300 node system and a 1354 node system provided by MATPOWER software. The parameter calculation time before and after acceleration and in document [22] of the disclosed method is shown in table 3. The simulation of the section is completed on MATLAB, the CPU is Xeon E5-2667v4, and the memory is 64 GB.
TABLE 5 Thevenin equivalent parameter calculation time before and after acceleration
It can be seen from the table that the Thevenin equivalent parameter calculation method provided by the disclosure has greatly reduced calculation time after optimization and acceleration, the calculation time is less than that of the document [22], and the requirement of online operation can be met. If the method disclosed by the invention is only applied to the key load node for static voltage stability monitoring, the calculation time of the parameters can be further reduced. The calculation time of the method can be further prolonged by writing the program into compiled languages such as C language and the like and performing parallel calculation on the matrix inversion algorithm.
Based on single-state section data, in order to obtain more accurate Thevenin equivalent parameters, the method for calculating the optimal Thevenin equivalent parameters according to the minimum volt-ampere characteristic deviation of the node port is provided. Different from a method for carrying out linear processing on a current power flow section, other sample points on the volt-ampere characteristic of a node port are calculated through a sensitivity method, the minimum deviation of the volt-ampere characteristic corresponding to the Thevenin parameter of the node and the real volt-ampere characteristic is taken as a target function, and the optimal Thevenin equivalent parameter of the node is solved. In order to meet the real-time requirement of calculation, the parameter calculation process is optimized according to the characteristics of the sample point generation process, and the calculation speed is accelerated. And a new method for measuring the accuracy of thevenin equivalent parameters is adopted, so that the limitation of equivalent effect evaluation is avoided.
The simulation calculation example verifies that the method provided by the disclosure can greatly improve the identification precision of thevenin equivalent parameters, and meanwhile, the identification time of thevenin equivalent parameters can meet the requirement of online identification through optimization. The method can provide faster and more accurate static equivalent parameters for online static stability analysis under the condition that the access of large-scale clean energy brings stronger randomness and fluctuation to the system operation mode.
Example two
An optimal Thevenin equivalent parameter calculation system based on node port volt-ampere characteristics comprises:
a sample point acquisition module for rapidly acquiring node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method;
an optimization target module, wherein the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
EXAMPLE III
The present embodiment aims to provide a computing device, including a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein the processor executes the program to implement the following steps, including:
rapidly obtaining node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method;
the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
Example four
An object of the present embodiment is to provide a computer-readable storage medium.
A computer-readable storage medium, on which a computer program is stored which, when executed by a processor, performs the steps of:
rapidly obtaining node voltage vectors and current vector estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method;
the optimization target is as follows: the deviation between the voltage vector calculated by the sample point current vector through the Thevenin equivalent parameters and the voltage vector of the sample point is minimum;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
The steps involved in the apparatuses of the above second, third and fourth embodiments correspond to the first embodiment of the method, and the detailed description thereof can be found in the relevant description of the first embodiment. The term "computer-readable storage medium" should be taken to include a single medium or multiple media containing one or more sets of instructions; it should also be understood to include any medium that is capable of storing, encoding or carrying a set of instructions for execution by a processor and that cause the processor to perform any of the methods of the present invention.
Those skilled in the art will appreciate that the modules or steps of the present invention described above can be implemented using general purpose computer means, or alternatively, they can be implemented using program code that is executable by computing means, such that they are stored in memory means for execution by the computing means, or they are separately fabricated into individual integrated circuit modules, or multiple modules or steps of them are fabricated into a single integrated circuit module. The present invention is not limited to any specific combination of hardware and software.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Although the embodiments of the present invention have been described with reference to the accompanying drawings, it is not intended to limit the scope of the present invention, and it should be understood by those skilled in the art that various modifications and variations can be made without inventive efforts by those skilled in the art based on the technical solution of the present invention.
Claims (9)
1. An optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics is characterized by comprising the following steps:
rapidly obtaining node voltage phasor and current phasor estimated values of node port volt-ampere characteristic sample points under a current state section by a sensitivity-based method; the method specifically comprises the following steps: firstly, solving elements of a Jacobian matrix which participate in calculation, then setting different load power fluctuation values, solving a node voltage amplitude value and a phase angle which correspond to the power fluctuation values, further calculating current under corresponding power, and obtaining a sampling point of volt-ampere characteristics;
the optimization target is as follows: the voltage phasor deviation between the voltage phasor calculated by the Thevenin equivalent parameter of the sample point current phasor and the sample point voltage phasor is minimum; specifically, the optimization target is as follows:
namely, QiSolving an optimization objective function when thevenin equivalent parameters are obtained for a node i, wherein a subscript i represents an ith node, a subscript j represents a jth sample, and n represents the number of sample points;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
2. The optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics as claimed in claim 1, wherein the method for measuring the precision of Thevenin equivalent parameters comprises the following steps: and measuring the accuracy of the Thevenin equivalent parameters by keeping the loads of other nodes unchanged, adding fluctuation quantity to the loads at the calculation nodes, and calculating the modulus value of the difference between the node voltage phasor obtained by calculating the Thevenin equivalent parameters and the node voltage phasor obtained by calculating the actual load flow.
3. The optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics as claimed in claim 2, wherein the measurement of the Thevenin equivalent parameter precision specifically comprises: adding random disturbance to a load node i, and setting the load after the load node i is disturbed as SikK is 1,2 …, m, and the total number of perturbations is m;
with SikThe power of the original node i is replaced to carry out load flow calculation to obtain the standard value of the node voltage phasor
Then the S is mixedikEstimation value of node voltage phasor calculated by two-node system with Thevenin equivalent parameter correspondence
And measuring the accuracy of the Thevenin equivalent parameters by the proportion of the phasor difference modulus of the estimated value of the node voltage and the standard value to the modulus of the node voltage phasor standard value.
4. The optimal Thevenin equivalent parameter calculation method based on node port volt-ampere characteristics as claimed in claim 1, wherein the sample point used for calculating the node port volt-ampere characteristic curve has two characteristics: firstly, the variation of other load nodes in the active power delta P of the load node is zero; secondly, only the voltage amplitude and the phase angle variation corresponding to the load node need to be solved.
5. An optimal Thevenin equivalent parameter calculation system based on node port volt-ampere characteristics is characterized by comprising the following steps:
a sample point acquisition module for rapidly acquiring node voltage phasor and current phasor estimated values of node port volt-ampere characteristic sample points under the current state section by a sensitivity-based method; the method specifically comprises the following steps: firstly, solving elements of a Jacobian matrix which participate in calculation, then setting different load power fluctuation values, solving a node voltage amplitude value and a phase angle which correspond to the power fluctuation values, further calculating current under corresponding power, and obtaining a sampling point of volt-ampere characteristics;
an optimization target module, wherein the optimization target is as follows: the voltage phasor deviation between the voltage phasor calculated by the Thevenin equivalent parameter of the sample point current phasor and the sample point voltage phasor is minimum; specifically, the optimization target is as follows:
namely, QiSolving an optimization objective function when thevenin equivalent parameters are obtained for a node i, wherein a subscript i represents an ith node, a subscript j represents a jth sample, and n represents the number of sample points;
and the solving module is used for solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, so that the real part and the imaginary part of the optimal Thevenin equivalent potential are obtained.
6. The optimal Thevenin equivalent parameter calculation system based on node port volt-ampere characteristics as claimed in claim 5, wherein the method for measuring the precision of Thevenin equivalent parameters comprises the following steps: and measuring the accuracy of the Thevenin equivalent parameters by keeping the loads of other nodes unchanged, adding fluctuation quantity to the loads at the calculation nodes, and calculating the modulus value of the difference between the node voltage phasor obtained by calculating the Thevenin equivalent parameters and the node voltage phasor obtained by calculating the actual load flow.
7. The optimal Thevenin equivalent parameter calculation system based on node port volt-ampere characteristics as claimed in claim 6, wherein the measurement of the Thevenin equivalent parameter precision is as follows: adding random disturbance to a load node i, and setting the load nodei the disturbed load is SikK is 1,2 …, m, and the total number of perturbations is m;
with SikThe power of the original node i is replaced to carry out load flow calculation to obtain the standard value of the node voltage phasor
Then the S is mixedikEstimation value of node voltage phasor calculated by two-node system with Thevenin equivalent parameter correspondence
And measuring the accuracy of the Thevenin equivalent parameters by the proportion of the phasor difference modulus of the estimated value of the node voltage and the standard value to the modulus of the node voltage phasor standard value.
8. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the processor when executing the program performs the steps of:
rapidly obtaining node voltage phasor and current phasor estimated values of node port volt-ampere characteristic sample points under a current state section by a sensitivity-based method; the method specifically comprises the following steps: firstly, solving elements of a Jacobian matrix which participate in calculation, then setting different load power fluctuation values, solving a node voltage amplitude value and a phase angle which correspond to the power fluctuation values, further calculating current under corresponding power, and obtaining a sampling point of volt-ampere characteristics;
the optimization target is as follows: the voltage phasor deviation between the voltage phasor calculated by the Thevenin equivalent parameter of the sample point current phasor and the sample point voltage phasor is minimum; specifically, the optimization target is as follows:
namely, QiOptimization objective for solving Thevenin equivalent parameter for node iA function, wherein index i represents the ith node, index j represents the jth sample, and n represents the number of sample points;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
9. A computer-readable storage medium, on which a computer program is stored, which program, when executed by a processor, carries out the steps of:
rapidly obtaining node voltage phasor and current phasor estimated values of node port volt-ampere characteristic sample points under a current state section by a sensitivity-based method; the method specifically comprises the following steps: firstly, solving elements of a Jacobian matrix which participate in calculation, then setting different load power fluctuation values, solving a node voltage amplitude value and a phase angle which correspond to the power fluctuation values, further calculating current under corresponding power, and obtaining a sampling point of volt-ampere characteristics;
the optimization target is as follows: the voltage phasor deviation between the voltage phasor calculated by the Thevenin equivalent parameter of the sample point current phasor and the sample point voltage phasor is minimum; specifically, the optimization target is as follows:
namely, QiSolving an optimization objective function when thevenin equivalent parameters are obtained for a node i, wherein a subscript i represents an ith node, a subscript j represents a jth sample, and n represents the number of sample points;
and solving and calculating the real part and the imaginary part of the optimal Thevenin equivalent impedance based on the objective function optimization, thereby obtaining the real part and the imaginary part of the optimal Thevenin equivalent potential.
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