A method for judging harmonic instability of transformer under saturation condition

Abstract

The present invention provides a method for distinguishing harmonic instability under transformer saturation conditions, and relates to the technical field of transformers. The method includes, under transformer saturation conditions, based on the basic magnetization curve, combining the flux formula and the full current law to deduce the flux-current curve; then combining the AC and DC fluxes on the core to deduce the excitation current expression, and Fourier decomposition to obtain the second harmonic component of the excitation current. Combined with the AC and DC harmonic transfer characteristics of high-voltage direct current transmission, the magnitude of the aforementioned second harmonic component is compared with the sum of the second harmonic currents generated twice subsequently, and whether the harmonic is unstable is determined based on the result. In order to make the judgment result more accurate, hysteresis can be considered. This method can quickly and effectively calculate the harmonic value generated by transformer saturation when the transformer saturation working state estimation is complex and the amount of calculation is large, and further quickly determine whether harmonic instability occurs in the DC transmission system.

Description

Technical Field

[0001] The present invention relates to the technical field of transformers, and in particular to a method for determining harmonic instability under transformer saturation conditions. Background Art

[0002] Since the 21st century, the HVDC transmission system has made great progress, but it has also brought many DC bias problems. The rapid development of HVDC transmission technology has made DC transmission widely used in large-capacity long-distance power transmission. At the same time, the DC bias and harmonic instability problems existing in high-voltage transmission are also increasingly affecting the stable and safe operation of the power system. The use of a single pole loop or bipolar unbalanced operation in the HVDC transmission system, as well as geomagnetically induced currents (GIC) and asymmetric loads caused by solar plasma winds may cause surface potential differences. This potential difference will generate DC current on the surface that enters the transformer winding through the grounding wire, which may cause DC bias of the transformer. When a disturbance occurs in the DC transmission converter station, the transformer bias may be aggravated, which in turn causes harmonic instability in the DC transmission system.

[0003] When the transformer is DC biased, the DC current injection generates DC flux, causing the transformer to work in the nonlinear region. The excitation current becomes an asymmetric peak wave, with the main feature of a sharp increase in the peak value on one side of the positive and negative half-cycles and a decrease in the peak value on the other side. At this time, there are a large number of second harmonics in the excitation current. Transformer saturation causes distortion of the excitation current and increases the harmonic content. Harmonic transmission in the DC transmission system may increase the saturation of the transformer, increase the harmonics generated by the transformer, and form positive feedback. The transformer core is saturated in half a cycle, and a large number of harmonics are generated, causing the transformer to vibrate more, and a series of other problems, which seriously affect the safe and stable operation of the system.

[0004] At present, the AC and DC harmonic instability judgment method considering transformer saturation is complex and difficult to judge quickly and effectively. Summary of the invention

[0005] In view of the above problems, the purpose of the present invention is to provide a convenient and quick method for determining harmonic instability under saturation conditions, so as to solve the relatively complex technical problem of existing determination methods.

[0006] The technical solution adopted by the present invention is as follows:

[0007] S1: The basic magnetization curve is combined with the magnetic flux φ and the excitation current i to calculate the piecewise linear φ-i curve.

[0008]

[0009] in,

[0010]

[0011]

[0012] In the formula, φ r Remanence B r The corresponding magnetic flux, φ r =B r S,i m ,φ m are the coordinates of the knee point of the basic magnetization curve on the piecewise linear φ-i curve, B is the magnetic induction intensity, S is the magnetic field area, H is the magnetic field intensity, l is the effective magnetic circuit length, and N is the number of turns of the magnetization circuit;

[0013] S2: Calculate the excitation current:

[0014]

[0015] In the formula, φ ac ,φ dc are the AC magnetic flux and DC magnetic flux of the transformer core respectively, ω is the angular frequency, is the initial phase angle,

[0016]

[0017] in,

[0018]

[0019] Where U m is the AC voltage peak value, f is the transformer operating frequency, N 1 is the number of turns of the transformer primary winding, R m is the magnetic circuit reluctance;

[0020] S3: The excitation current is decomposed by Fourier to obtain the second harmonic component I of the excitation current under transformer saturation conditions. ac2 :

[0021]

[0022] S4: Calculate the second harmonic voltage U ac2 :

[0023] U ac2 =Z ac2 I ac2 ;

[0024] In the formula, Z ac2 is the second harmonic impedance on the AC side;

[0025] Second harmonic voltage U ac2 Transmitted to the DC side to generate fundamental frequency voltage Udc1 ,

[0026]

[0027] The DC side generates a fundamental frequency current I dc1 ,

[0028]

[0029] In the formula, Z dc1 is the fundamental frequency harmonic impedance on the DC side;

[0030] DC side fundamental frequency current I dc The second harmonic current I' is generated when it is transmitted to the AC side. ac2 And DC current I ac0 ,

[0031]

[0032] S5: Calculate the DC current I based on S1-S3 ac0 Second harmonic current that saturates the transformer

[0033]

[0034] like If the harmonic instability occurs, it is determined that harmonic instability occurs; otherwise, it is determined that harmonic instability does not occur.

[0035] In addition, when considering hysteresis, we have,

[0036] S1: The basic magnetization curve is combined with the magnetic flux φ and the excitation current i to obtain a piecewise linear φ-i curve considering hysteresis.

[0037]

[0038] in,

[0039]

[0040]

[0041] In the formula, φ r Remanence B r The corresponding magnetic flux, φ r =B r S, δ is the directional coefficient, i m ,φ m are the coordinates of the knee point of the basic magnetization curve on the piecewise linear φ-i curve, B is the magnetic induction intensity, S is the magnetic field area, H is the magnetic field intensity, l is the effective magnetic circuit length, and N is the number of turns of the magnetization circuit;

[0042] S2: Calculate the hysteresis excitation current,

[0043]

[0044] Among them, where φ ac ,φ dc are the AC magnetic flux and DC magnetic flux of the transformer core respectively, ω is the angular frequency, is the initial phase angle,

[0045]

[0046] in,

[0047]

[0048] Where U m is the AC voltage peak value, f is the transformer operating frequency, N 1 is the number of turns of the transformer primary winding, R m is the magnetic circuit reluctance;

[0049] S3: The second harmonic component I of the hysteresis excitation current under transformer saturation condition is obtained by Fourier decomposition of the hysteresis excitation current ac2 ;

[0050] S4: Calculate the second harmonic voltage U ac2 :

[0051] U ac2 =Z ac2 I ac2 ;

[0052] In the formula, Z ac2 is the second harmonic impedance on the AC side;

[0053] Second harmonic voltage U ac2 Transmitted to the DC side to generate fundamental frequency voltage U dc1 ,

[0054]

[0055] The DC side generates a fundamental frequency current I dc1 ,

[0056]

[0057] In the formula, Z dc1 is the fundamental frequency harmonic impedance on the DC side;

[0058] DC side fundamental frequency current I dc The second harmonic current I' is generated when it is transmitted to the AC side. ac2 And DC current I ac0 ,

[0059]

[0060] S5: Calculate the DC current I based on S1-S3 ac0 Second harmonic current that saturates the transformer

[0061] like If the harmonic instability occurs, it is determined that harmonic instability occurs; otherwise, it is determined that harmonic instability does not occur.

[0062] The method provided by the present invention can quickly and effectively calculate the harmonic value when the transformer saturation working state estimation is complex and the amount of calculation is large, and further quickly determine whether the DC power transmission system has harmonic instability. In addition, considering hysteresis can make the judgment more accurate and reduce the judgment error rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 FIG. 4 is a schematic diagram of a piecewise linear simplified magnetization curve (ie, a φ-i curve) in one embodiment of the present invention.

[0064] Figure 2 It is a schematic diagram of the corresponding relationship between the φ-i curve and the magnetic flux in one embodiment of the present invention.

[0065] Figure 3 It is a schematic diagram of the corresponding relationship of the excitation current waveform without considering the hysteresis in one embodiment of the present invention.

[0066] Figure 4 The figure is a schematic diagram of the corresponding relationship of the hysteresis excitation current waveform in one embodiment of the present invention.

[0067] Figure 5 FIG. 4 is a schematic diagram of a switch function in an embodiment of the present invention.

[0068] Figure 6 Schematic diagram of AC and DC side harmonic transfer characteristics in one embodiment of the present invention.

[0069] Figure 7 The figure is a schematic diagram of a harmonic instability criterion process considering transformer saturation in one embodiment of the present invention. DETAILED DESCRIPTION

[0070] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0071] Example 1

[0072] Step 1: Propose a piecewise linear simplified hysteresis curve and function model for fitting the magnetization curve.

[0073] The simplified single-loop piecewise linear φ-i curve is obtained based on the magnetization curve, where the double-loop curve is the simplified curve after considering hysteresis, and the single-loop curve is the simplified curve without considering hysteresis. The linear parameters of the piecewise curve are respectively determined by the magnetic field intensity at the knee point of the magnetization curve (H m , B m ) and residual magnetism B r The parameters are obtained. The magnetic field intensity H and magnetic induction intensity B are linearly related to the magnetic flux φ and the excitation current i respectively. As shown in the following formula:

[0074]

[0075] In the formula, φ is the magnetic flux, B is the magnetic induction intensity, S is the magnetic field area, i is the excitation current, H is the magnetic field intensity, l is the effective magnetic path length, and N is the number of turns of the magnetization circuit. Therefore, the simplified magnetic flux-current curve corresponding to the magnetization curve can be obtained as follows Figure 1 shown.

[0076] The expression of the piecewise linearization curve of flux-current without considering hysteresis is:

[0077]

[0078] Point A(B) is the knee point of the basic magnetization curve, and its coordinates are (i m ,φ m ). The slopes of the two-stage segmented curves are φ r Remanence B r The corresponding magnetic flux is φ r =B r S.

[0079] The expression of the piecewise linearization curve of flux-current considering hysteresis is:

[0080]

[0081] in, δ is the directional coefficient,

[0082] When the transformer is saturated, the core contains both DC flux and AC flux. The specific analysis is as follows:

[0083] When DC current is injected into the transformer winding, the magnetic flux can be expressed as:

[0084]

[0085] Instantaneous value of AC excitation voltage:

[0086]

[0087] Among them, U mis the peak value of AC voltage, and the induced electromotive force of the transformer primary winding is:

[0088]

[0089] In the formula, E 1m is the maximum value of the induced electromotive force of the primary winding;

[0090] Effective value of the induced electromotive force of the primary winding:

[0091]

[0092] in: The maximum value of AC magnetic flux is obtained by combining:

[0093]

[0094] Where f is the transformer operating frequency, N 1 is the number of turns of the transformer primary winding, R m is the magnetic circuit reluctance;

[0095] And,

[0096] Hl=NI dc =φ dc R m ;

[0097] Find:

[0098]

[0099]

[0100] In the formula, μ is the magnetic permeability. In summary, the transformer core flux expression is:

[0101]

[0102] That is, the transformer core flux is related to the AC excitation voltage and the DC excitation current; where ω is the angular frequency, is the initial phase angle.

[0103] Step 2: Figure 2 As shown in the figure, when the transformer is saturated, the excitation current changes along the excitation loop trajectory. When hysteresis is not considered, the waveform of the excitation current generated is as follows: Figure 2 (a) is shown; when hysteresis is considered, the waveform of the generated excitation current is as follows Figure 2 (b) as shown.

[0104] The excitation current expression can be obtained from the transformer core flux expression and flux-current expression obtained in step 1, as shown below:

[0105] The expression of excitation current without considering hysteresis is:

[0106]

[0107] in θ 2 =π-θ 1 , the corresponding relationship between magnetic flux and current waveform is as follows Figure 3 shown.

[0108] Consider the hysteresis excitation current expression:

[0109]

[0110] where k 1 , k 2 , k 3 The simplified magnetization curve parameters are obtained from step 1. When the external voltage and DC current of the transformer are known, the simplified result of the excitation current can be obtained. The schematic diagram of the corresponding waveform relationship between the magnetic flux and the current is as follows: Figure 4 shown.

[0111] Step 3: Based on the excitation current expression obtained in step 2, the Fourier analysis can be performed to obtain the expressions of each harmonic in the excitation current. The specific calculation process is as follows:

[0112] For periodic functions that are not sinusoidal functions, they can be expressed using periodic functions, namely:

[0113] f(t)=f(t+nT);

[0114] Where T is the period of the periodic function f(t), and n is a natural number 1, 2, 3…

[0115] Expanding it into a Fourier series is:

[0116]

[0117] in,

[0118]

[0119] Substituting the excitation current curve expression without considering hysteresis into the Fourier series, the DC component is obtained:

[0120]

[0121] Fundamental frequency component:

[0122]

[0123] in;

[0124]

[0125] Find:

[0126]

[0127] The second harmonic component (i.e., the second harmonic component I ac2 ):

[0128]

[0129] in:

[0130]

[0131]

[0132] Find:

[0133] Right now:

[0134] Similarly, the expression of even harmonic components can be obtained as follows:

[0135]

[0136] Where n1=2k+1, k=1,2,3....

[0137] The expression of odd harmonic components is:

[0138]

[0139] Where n2=2k+1, k=1,2,3....

[0140] Similarly, the second harmonic component of the transformer saturated with hysteresis can be obtained.

[0141] Step 4: Based on the second harmonic component of the excitation current harmonic under transformer saturation obtained in step 3 and the AC / DC harmonic transfer characteristics of high-voltage direct current transmission, a harmonic instability criterion considering transformer saturation is obtained.

[0142] According to modulation theory, the relationship between the voltage and current components on the AC and DC sides can be expressed as:

[0143] U dc =U a S ua +U b S ub +U c S uc ;

[0144]

[0145] Among them, Udc ,i dc is the DC side voltage, U a , U b , U c ,i a ,i b ,i c is the three-phase voltage and current on the AC side, S ua , S ub , S uc , S ia , S ib , S ic is the switching function coefficient.

[0146] The switch function is as follows: Figure 5 As shown, its expression is as follows:

[0147]

[0148] In the formula, A n is the Fourier coefficient of the switching function, n is the item number of the Fourier coefficient;

[0149] When commutation occurs, the current rises and falls between the steady-state value and 0, which is usually represented by a curve. To simplify the calculation, the process is linearized.

[0150] That is, the expressions of straight lines AB and CD are:

[0151]

[0152] Among them, λ is the commutation overlap angle, so the Fourier coefficient A of the switching function can be obtained n for:

[0153]

[0154] The commutation voltage to maintain three-phase symmetry can be expressed as:

[0155]

[0156] Among them, ω m is the commutation voltage angular frequency, U am , U bm , U cm are the peak values ​​of the three-phase voltages, α am , α bm , α cm are the initial phase angles of the three-phase voltages respectively. If the three phases are symmetrical, the amplitudes of each phase are equal and the phase difference is

[0157] Substituting the above formula into the modulation theory formula, the DC voltage can be obtained. Considering only the first component of the switching function, the process of converting the AC harmonic voltage component into the DC harmonic voltage component after modulation can be expressed as:

[0158]

[0159] Among them, U dc1 is the DC side harmonic voltage, A n1 The first term of the Fourier coefficients of the switching function, α m is the initial phase angle of the current phase voltage. m When it is twice the rated angular frequency of the power grid, the AC harmonic voltage is the second harmonic voltage, and its frequency transmitted to the DC side is reduced by 1, that is, the fundamental frequency harmonic voltage is generated on the DC side, and the AC second harmonic voltage U ac2 Transmitted to the DC side to generate fundamental frequency voltage U dc1 , then:

[0160]

[0161] Similarly, the expression of the harmonic current transfer from the DC side to the AC side is:

[0162]

[0163] Among them, I dm is the harmonic current on the DC side. Analysis shows that when the fundamental frequency current on the DC side is modulated to the AC side, secondary harmonic current and DC current are generated. That is, the fundamental frequency harmonic current on the DC side I dc1 Transmitted to the AC side to generate DC current I ac0 and second harmonic current I' ac2 ,have:

[0164]

[0165] like Figure 6 As shown, the voltage and current transfer relationship of the AC and DC sides is derived. The second harmonic voltage U ac2 The second harmonic current I ac2 The second harmonic impedance Z ac2 The calculation results are:

[0166] U ac2 =Z ac2 I ac2 ;

[0167] DC side fundamental frequency harmonic current I dc It can be calculated from the DC side fundamental frequency harmonic voltage and DC side fundamental frequency harmonic impedance:

[0168]

[0169] like Figure 7 As shown in the figure, the harmonic instability criterion can be derived by combining the voltage and current transfer relationship of the AC and DC sides. When harmonic instability occurs, the transformer saturates, causing the excitation current to be distorted, generating a large amount of second harmonic current I ac2 ,

[0170]

[0171] The second harmonic current generates a second harmonic voltage on the AC side. The second harmonic current on the AC side is transferred to the DC side to generate a DC side fundamental frequency voltage. The DC side fundamental frequency voltage generates a DC side fundamental frequency current on the DC side. The DC side fundamental frequency current is transferred to the AC side to generate a DC current I ac0 and second harmonic current I' ac2 ,

[0172]

[0173] The DC current saturates the transformer and generates secondary harmonic current

[0174]

[0175] By superimposing the second harmonic current transmitted from the DC side to the AC side and comparing the initial second harmonic current generated by transformer saturation with the superimposed second harmonic current, it can be determined whether harmonic instability occurs. That is, if

[0176]

[0177] If the harmonic instability occurs, it is determined that harmonic instability occurs; otherwise, it is determined that harmonic instability does not occur.

[0178] In the case of considering hysteresis, the piecewise linear φi curve is replaced by the corresponding expression of the basic magnetization curve considering hysteresis, and the method steps applied are the same as the above method.

[0179] The method provided by the present invention can quickly and effectively calculate the harmonic value when the transformer saturation working state estimation is complex and the amount of calculation is large, and further quickly determine whether the DC power transmission system has harmonic instability. In addition, considering hysteresis can make the judgment more accurate and reduce the judgment error rate.

[0180] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the invention.

Claims

1. A method for determining harmonic instability under transformer saturation conditions, characterized in that: include: S1: The basic magnetization curve is combined with the magnetic flux φ and the excitation current i to obtain a piecewise linear φ-i curve considering hysteresis. in, In the formula, φ r Remanence B r The corresponding magnetic flux, φ r =B r S, δ is the directional coefficient, i m ,φ m are the coordinates of the knee point of the basic magnetization curve on the piecewise linear φ-i curve, B is the magnetic induction intensity, S is the magnetic field area, H is the magnetic field intensity, l is the effective magnetic circuit length, and N is the number of turns of the magnetization circuit; S2: Calculate the hysteresis excitation current, Among them, where φ ac ,φ dc are the AC magnetic flux and DC magnetic flux of the transformer core respectively, ω is the angular frequency, is the initial phase angle, in, Where U m is the peak value of the AC voltage, f is the operating frequency of the transformer, N1 is the number of turns of the transformer primary winding, R m is the magnetic circuit reluctance; S3: The second harmonic component I of the hysteresis excitation current under transformer saturation condition is obtained by Fourier decomposition of the hysteresis excitation current ac2 ; S4: Calculate the second harmonic voltage U ac2 : U ac2 =Z ac2 I ac2 ; In the formula, Z ac2 is the second harmonic impedance on the AC side; Second harmonic voltage U ac2 Transmitted to the DC side to generate fundamental frequency voltage U dc1 , The DC side generates a fundamental frequency current I dc1 , In the formula, Z dc1 is the fundamental frequency harmonic impedance on the DC side; DC side fundamental frequency current I dc1 Transmitted to the AC side to generate second harmonic current I' ac2 And DC current I ac0 , S5: Calculate the DC current I based on S1-S3 ac0 Second harmonic current that saturates the transformer like If the harmonic instability occurs, it is determined that harmonic instability occurs; otherwise, it is determined that harmonic instability does not occur.

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