Analysis Method for the Stability Region Boundary of a Pulse Load Power System

Abstract

The present application is an analysis method for the stability domain of a pulsed load power system, which relates to the field of power electronics technology, including establishing an equivalent simplified circuit for a pulsed load power system, wherein the simplified circuit is a DC / DC boost circuit with a pulsed load, and the load is controlled by a pulse switch to generate pulse characteristics to simulate the operating state of a PLPS system. The operating domain of the pulsed load power system is analyzed from two perspectives. One is to decompose the input pulse signal into upper and lower steps, and perform time domain analysis based on the dynamic change response GPL(s) of the pulsed load. The other is to regard the input pulse signal as a whole for harmonic expansion, and derive the response analytical expression of each harmonic based on the resonance principle; finally, the two are combined to obtain the analytical domain of the pulsed load power system. The present application adopts a direct analytical method to analyze the stability of the PLPS system, deriving the corresponding time domain analytical expression and stability criterion from two different ideas, and then combining the two to make up for the simplified part of each other, and finally obtaining a reasonable analytical stability domain of the PLPS system.

Description

Technical Field

[0001] The present invention relates to the technical field of PLPS, and in particular to an analysis method for a stability domain boundary, and in particular to an analysis method for a stability domain boundary of a pulse load power system. Background Art

[0002] With advances in power conversion and storage technologies, high-power pulsed weapon systems such as radar, electromagnetic catapults, and electromagnetic guns have become increasingly widely used in defense and military applications. The strong nonlinearity and constant power characteristics of pulsed loads make pulsed load power systems (PLPS) less stable and more complex than conventional load power systems. Due to the lack of a clear stability boundary to guide their design, these devices often experience instability and collapse during operation. Research on the stability boundary of PLPS systems is a prerequisite for ensuring safe and stable system operation.

[0003] A large number of studies have shown that it is not enough to characterize the overall stability of the PLPS system only by single-mode stability. The modal switching characteristics and the mutual influence of sub-modes must also be comprehensively considered, but this will greatly increase the complexity of stability research.

[0004] In power system stability research, some methods use direct analytical methods to address specific problems that conventional methods cannot analyze. Based on direct analytical methods, some methods have revealed the impact of factors such as grid strength, active power, and PLL on the stability of outer-loop active power control in weak grids. The advantage of direct analytical methods lies in their ability to directly reveal the inherent correlations between system characteristics and key parameters through analytical expressions. For some special systems or operating conditions where traditional stability analysis methods are no longer applicable, analytical methods can provide an alternative approach, directly characterizing the key correlations of the problem. However, the disadvantage of analytical methods is that the mathematical derivation process is difficult and difficult to advance, requiring appropriate and reasonable simplification. Summary of the Invention

[0005] This invention proposes a method for analyzing the stability region of a pulse load power system.

[0006] The technical solutions of the present invention are as follows:

[0007] The method for analyzing the stability domain of a pulse load power system includes establishing an equivalent simplified circuit of the pulse load power system, wherein the simplified circuit is a boost circuit controlled by a controller, and a periodic square wave g is applied to the simplified circuit by the controller. PL(t) , periodic square wave g PL The duty cycle of (t) is D PL , with a period of T PL , pulse frequency f PL =1 / T PLFrom the simplified circuit, we can get the expression of capacitor voltage uc and the response G to the dynamic change of pulse load in two modes. PL (s),

[0008] Analyze the domain boundary of the pulse load power system and respond to the dynamic changes of the pulse load PL (s) Perform Fourier decomposition and derive the analytical response expression of each harmonic from the perspective of resonance principle; finally, combine the two to obtain the analytical domain boundary of the pulse load power system.

[0009] As a further optimization of this solution, it is obtained from the simplified circuit that a periodic square wave g is applied to the simplified circuit by the controller. PL(t) , periodic square wave g PL The duty cycle of (t) is D PL , with a period of T PL , pulse frequency f PL =1 / T PL ,get

[0010]

[0011] Where d1 = 1-d g , d g is the switching signal S g The duty cycle, R C is the equivalent resistance of the normal load, P PL It's S PL The pulse energy constant in the on state is expressed as the capacitor voltage uC as the output variable. According to formula (1), it is written as a Latent transformation to obtain U C (s)=G s (s)U s (s)+G PL (s)P PL (s)

[0012]

[0013] Where Gs(s) represents the response to the power supply, and GPL(s) represents the response to the dynamic changes of the pulse load. When the pulse load is in the state and only ΔuC is generated, ΔUC(s) = GPL(s) PPL(s).

[0014] As a further optimization of this scheme, GPL(s) has two modes, i=1 for pulse loading model and i=2 for pulse load reduction model.

[0015]

[0016] As a further optimization of this solution, in the analysis of the domain boundary of the pulse load power system, the load increase and decrease are regarded as step responses, and its zero-state response ΔuC(t)=yPLi(t)

[0017] Where yPLBi(t) is the outer envelope of yPLi(t),

[0018] Response amplitude K PLi =k i r Li P PL / (d1 2 R Li +r),

[0019] Angular displacement

[0020] Undamped angular frequency

[0021] Damping coefficient

[0022] Damped oscillation angular frequency is the system oscillation angular frequency, f d (=ω d / 2π) is the "natural frequency" of the system, if ξ i →0, then f d ≈f n .

[0023] As a further optimization of this scheme, according to the convergence state of pulse load operation,

[0024]

[0025] Δu C_max2 =y PLB2 (0),ξ2>0

[0026] Δu C_max =max{Δu C_max1 ,Δu C_max2} (4).

[0027] As a further optimization of this scheme, the starting and ending values of the loading mode and the unloading mode are obtained.

[0028]

[0029] Among them, y PLB2 '(t) is the initial value of Y 2s The outer envelope of mode 2, ① when ξ1>0, both modes 1 and 2 converge and the system is stable; ② when ξ1<0, mode 1 diverges. At this time, if Y2e >Y 1s , the waveform amplitude in the next cycle will continue to increase, the waveform will continue to diverge, and the system will be unstable; Y 2e ≤Y 1s , then the amplitude of the waveform in the next cycle will decrease or remain stable. Therefore, the stability criterion of the "metastable state" of the PLPS system is: 2e ≤Y 1s .

[0030] As a further optimization of this solution, the capacitor voltage uC is expressed as a differential equation:

[0031]

[0032] in,

[0033] Expand the pulse signal gPL(t) into a Fourier series

[0034]

[0035] Among them, ω PL =2π / f PL , A0=D PL , A i =2sin(iD PL π) / iπ,

[0036] Substituting the expansion into equation (6) we get

[0037]

[0038] in,

[0039] As a further optimization of this scheme, the i-th resonance is written in standard form:

[0040]

[0041] Among them, the undamped angular frequency The value is determined by the system structure; h is the resonance damping coefficient, and the weighted average value ξ of the two modal values is taken h =D PL ξ1+(1-D PL )ξ2; let γ i is 0, the i-th simple harmonic response y hi (t) and the overall system response y h (t),

[0042] y hi (t) = b1β i Hi sin(iω PL t+θ i )

[0043]

[0044] Among them, b1H i is the input amplitude, β i is the amplification factor, b1β i h i That is, the i-th harmonic in the DC bus u C The amplitude of the fluctuation generated at PL / ω n , angular displacement θ i

[0045]

[0046] make (Y0 is Δu in formula (4) C_max )

[0047] get,

[0048]

[0049] Δu C_max There are two cases for analytical expressions: ①f PL =f n / N(T PL =NT n ), that is, when N resonance occurs or near it, there is β N H N >1,Y0β N H N >>Y0; when no resonance occurs, β i H i <1,Y0>>Y 0s β N H N ,thereby

[0050] Δu C_max =max{Y0β N H N ,Y0} (13)

[0051] Let Δu C The stability threshold is δ max , we can get the stability criterion of the comprehensive modal step and resonance analytical expression

[0052] max{Y0β N H N , Y0}≤δ max (IV)

[0053] Based on the above analytical expressions and criteria, the stability region of the PLPS system is drawn.

[0054] The working principle and beneficial effects of the present invention are:

[0055] This application uses the time domain analytical method to analyze the stability of the PLPS system. Taking the DC bus voltage uC as the output, the pulse input signal is processed using two approaches respectively, and the corresponding time domain analytical expression and stability criterion are derived. The two are then combined to complement each other's simplified parts, and finally a reasonable analytical stability domain boundary of the PLPS system is obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0057] Figure 1 is the equivalent simplified circuit of PLPS;

[0058] Figure 2 This is the waveform diagram for pulse load operation;

[0059] Figure 3 β i With λ and ξ i Graph of changes in

[0060] Figure 4 Comparison of three-dimensional views of two domains;

[0061] Figure 5 for Figure 4 Comparison of the top views of the two domains;

[0062] Figure 6 for Figure 4 Comparison of the left rear views of the two domain boundaries;

[0063] Figure 7 for Figure 4 Comparison of the right views of the two domain boundaries. DETAILED DESCRIPTION

[0064] The following will be combined with the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the embodiments described 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 any creative efforts are within the scope of protection of the present invention.

[0065] In order to ensure the effectiveness of the pulse peak power of the pulse load equipment, a controller is generally used to ensure the constancy of the pulse power peak. Therefore, the pulse load often exhibits the characteristic of "pulse constant energy".

[0066] As the instruction manual Figure 1As shown, considering Figure 1 The DC boost circuit model shown is an average model because S g The switching frequency is much higher than the pulse load switch S PL The pulse frequency of the control signal g PL (t) is a periodic square wave, T PL is the pulse period, pulse frequency f PL =1 / T PL , D PL is the pulse duty cycle. Figure 1 In the model, we can get

[0067]

[0068] Where d1 = 1-d g , d g is duty cycle ofS g .R C is the equivalent resistance of conventional load; P PL is the pulse constant energy, when S PL When turned on.

[0069] Capacitor voltage u C (The DC bus voltage u dc ) is the output variable, which can be written as a pull-type transformation according to formula (1):

[0070] U C (s)=G s (s)U s (s)+G PL (s)P PL (s)

[0071]

[0072] Among them, G s (s) represents the response to the power supply; G PL (s) represents the response to the dynamic change of pulse load. When the pulse load is in the state, only Δu C .ΔU C (s)=G PL (s)P PL (s). G PL (s) has two modes: i = 1 corresponds to the pulse loading model; i = 2 corresponds to the pulse unloading model,

[0073]

[0074] The whole system is decomposed by Fourier transform, and the analytical expression of each harmonic response is derived from the perspective of resonance principle; finally, the two are combined to obtain the analytical domain of PLPS.

[0075] First, we discuss the time domain analysis of the modal step. Due to the constant power characteristics of the pulse, both load addition and load reduction can be regarded as step responses, and its zero-state response Δu C (t) = y PLi (t) (i=1 for loading / i=2 for unloading):

[0076]

[0077] Among them, y PLBi (t) is y PLi The outer envelope of (t),

[0078] Response amplitude K PLi =k i r Li P PL / (d1 2 R Li +r ) ,

[0079] Angular displacement

[0080] Undamped angular frequency

[0081] Damping coefficient (load mode, when P PL = 0, ξ1 = ξ2)

[0082]

[0083]

[0084] Damped oscillation angular frequency is the system oscillation angular frequency, f d (=ω d / 2π) is called the "natural frequency" of the system. If ξ i →0, then f d ≈f n .

[0085] from Figure 2 It can be seen that: ① Under the load reduction mode, R L2 >0,ξ2>0, the waveform can always converge,Δu C_max =|y PL2 (T d / 4)|≈|y PLB2 (0)|. The loading mode needs to be discussed: ②ξ1>0, y PL1 (t) Oscillation decay, Δu C_max =|y PL1 (T d / 4)|≈|y PLB1 (0)|;③ξ1=0,yPL1 (t) Periodic oscillation does not decay, Δu C_max =|y PL1 (nT PL +T d / 4)|≈|y PLB1 (0)|;④ξ1<0,y PL1 (t) Oscillation divergence, Δu C_max ≈|y PLB1 (D PL T PL )|.

[0086] In summary, ① to ④, except for ξ1<0, Δu C_max With pulse characteristics t=D PL T PL The other three cases are all determined by the initial state t=0. And Δu C_max All can be enclosed by external PLBi (t) approximate expression, we get

[0087]

[0088] The derivation of the above formula is based on the independent calculation of the loading and unloading modes, and the mutual influence of the two modes needs to be further considered. Figure 2 (b) It can be seen that the two modes have a periodic pattern of initial values:

[0089] Set the initial value of mode 1 to Y 1s , the final value is Y 1e ; It can be seen that Y 1e is the initial value of mode 2, that is, Y 2s =Y 1e , based on Y 2s Get the final value Y of mode 2 2e ; Y 2e Then it is the initial value of mode 1. The calculation process is:

[0090]

[0091] Among them, y PLB2 '(t) is the initial value of Y 2s The outer envelope of mode 2. ① When ξ1>0, both modes 1 and 2 converge and the system is stable; ② When ξ1<0, mode 1 diverges. At this time, if Y 2e >Y 1s , the waveform amplitude in the next cycle will continue to increase, the waveform will continue to diverge, and the system will be unstable; Y 2e ≤Y 1s , then the amplitude of the waveform in the next cycle will decrease or remain stable. Therefore, the stability criterion of the "metastable state" of the PLPS system can be obtained as follows:

[0092] Y 2e ≤Y1s (Criterion III)

[0093] Both criteria II and III give the pulse characteristic P PL 、D PL 、T PL The associated inequality, the former is based on the voltage threshold δ max The latter is bounded by stability.

[0094] The following discusses the time domain analysis of the resonance mechanism, based on ΔU C (s)=G PL (s)P PL (s), which can be written as a differential equation as follows

[0095]

[0096] in,

[0097] The pulse signal g PL (t) is expanded into a Fourier series

[0098]

[0099] Among them, ω PL =2π / f PL , A0=D PL , A i =2sin(iD PL π) / iπ, Due to the constant power characteristics of the pulse, g PL (t) contains a large number of high-order harmonics. Substituting the expanded formula into formula (6) we get

[0100]

[0101] in,

[0102] Where the constant term A0r=D PL r will make u C Produces a very small steady-state deviation; the summation term in the formula produces a resonant response, so formula (6) can be regarded as a resonant frequency of f PL Resonant system. Take the i-th resonance and write it in standard form:

[0103]

[0104] Among them, the undamped angular frequency The value is determined by the system structure; h is the resonance damping coefficient, Section 2 ξ iCorresponding to two modal values respectively, this section regards the pulse signal as a whole and takes its weighted average value ξ h =D PL ξ1+(1-D PL )ξ2; let γ i is 0, the above formula is a simple harmonic (simplest excitation) system, and the i-th simple harmonic response y hi (t) and the overall system response y h (t)

[0105] y hi (t) = b1β i H i sin(iω PL t+θ i )

[0106]

[0107] Among them, b1H i is the input amplitude, β i is the amplification factor, b1β i h i That is, the i-th harmonic in the DC bus u C The amplitude of the fluctuation generated at . Let λ=iω PL / ω n , angular displacement θ i

[0108]

[0109] Plotting β i With λ and ξ i The changing trend of Figure 3 As shown, it can be seen that β i It is largest near λ=1, and ξ i The smaller it is, the closer the maximum value is to λ=1, at this time θ i =π / 2. That is, when λ=1(iω PL =ω n ), the amplitude of the i-th harmonic is amplified by β i times.

[0110] According to formula (3), the maximum loading mode Δu without considering resonance is C_max , let it be Y0, that is Formula (7) shows that f[g PL (t)]≈ΣH i sin(iD PL t), then

[0111]

[0112] It can be seen that when N resonances occur, βN >>1, other harmonic β i <1, then Δu C The Nth harmonic amplitude Y0β N H N is dominant, and other harmonics are attenuated, i.e.

[0113] Δu C_maxh =max{Y0β N H N sin(iω PL t+θ i )}=Y0β N H N (10)

[0114] In summary, Δu C_max There are two cases for analytical expressions: ①f PL =f n / N(T PL =NT n ), that is, when N resonance occurs or near it, there is β N H N >1,

[0115] Δu C_max The resonance analytical formula (10) shows that: Y0β N H N >>Y0; when no resonance occurs, β i H i <1,Δu C_max Expressed by the step analytical formula (4): Y0>>Y 0s β N H N . Comprehensively obtain

[0116] Δu C_max =max{Y0β N H N ,Y0} (13)

[0117] Let Δu C The stability threshold is δ max , we can get the stability criterion of the comprehensive modal step and resonance analytical expression

[0118] max{Y0β N H N , Y0}≤δ max (IV)

[0119] Therefore, according to the resonance analytical formula (9) and the step analytical formula (4), as well as criterion IV, the stability region boundary of the PLPS system can be drawn.

[0120] Characterize the PLPS stability domain,

[0121] in accordance with Figure 1 A MATLAB / Simulink simulation model of the PLPS system was established, and the same parameter settings as those of the experimental platform were selected, as shown in Table 1 below.

[0122] Table 1 Test platform parameters / Simulink simulation model parameters

[0123] <![CDATA[u s ]]> r L C <![CDATA[d1]]> <![CDATA[R C ]]> <![CDATA[T PL / s]]> <![CDATA[D PL ]]> <![CDATA[P PL / KW]]> 400V 10mΩ 2.6mH 940uF 1 / 3 30Ω 0:0.02:1 0:0.01:1 35:0.5:0

[0124] As the instructions attached Figure 4-7 As shown, according to criterion IV, P PL 、T PL and D PL As the axis, the analytical stability domain B is obtained as follows Figure 6 To verify the accuracy of the drawn domain, a Simulink model was used to simulate each set of parameters with the same step size to obtain the stable domain A, which can be used as a reference domain for the comparison of domain B.

[0125] There is a significant resonance phenomenon in the stable domain, and the law can be based on

[0126]

[0127] exist Figure 6 There are N “resonance traps” on the resonance band of the Nth resonance, and the lowest value is located at max{β N H N}, that is, D PL =(2N-1) / 2i (N is a positive integer). It can be seen that Δu C_max With D PL It is not a monotonic relationship.

[0128] exist Figure 7 In the example, the lowest value of the “trap” increases with the number of times T PL decreases as the value of

[0129]

[0130] This paper mathematically deduces the PLPS system from the perspectives of dual-mode switching and resonance mechanism, and comprehensively obtains the maximum fluctuation value of the DC bus voltage Δu C_max A time-domain analytical expression for the PLPS stability domain was obtained, and a relatively accurate stability bound was plotted. The correctness of the analytical bound was verified by comparing it with the ergodic simulation bound of a Simulink model. This method can quickly plot the stability bound of the PLPS and directly characterize the relationship between stability and various parameter values. This provides a theoretical basis for the parameter design and power matching of pulsed load devices.

[0131] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for analyzing the stability region of a pulse load power system, characterized in that: The invention comprises establishing an equivalent simplified circuit of a pulse load power system, wherein the simplified circuit is a boost circuit controlled by a controller, and applying a periodic square wave g to the simplified circuit by the controller. PL(t) , periodic square wave g PL The duty cycle of (t) is D PL , with a period of T PL , pulse frequency f PL =1 / T PL From the simplified circuit, we can get the expression of capacitor voltage uc and the response G to the dynamic change of pulse load in two modes. PL (s), the two modes include a pulse loading mode and a pulse unloading mode; The domain boundary of the pulse load power system is analyzed to obtain the modal step analytical expression; Response to dynamic changes in pulse load G PL (s) Perform Fourier decomposition and derive the response expression of each harmonic from the perspective of resonance principle to obtain the resonance expression. Finally, the resonance expression and the modal step expression are combined to obtain the analytical domain of the pulse load power system.

2. The method for analyzing the stability region of a pulse load power system according to claim 1, characterized in that: From the simplified circuit, it is obtained that a periodic square wave g is applied to the simplified circuit by the controller. PL(t)， Periodic square wave g PL The duty cycle of (t) is D PL , with a period of T PL , pulse frequency f PL =1 / T PL ,get Where d1 = 1-d g , d g is the switching signal S g The duty cycle, R C is the equivalent resistance of the normal load, P PL It's S PL The pulse energy constant in the on state is expressed as the capacitor voltage U C is the output variable, which can be written as a pull-type transformation according to formula (1) to obtain Among them, G PL (s) represents the response to the dynamic change of pulse load. When the pulse load is in the state, only ΔU C , ΔU C (s)=G PL (s)P PL (s).

3. The method for analyzing the stability region of a pulse load power system according to claim 2, characterized in that: G PL (s) has two modes, i=1 is the pulse loading model, i=2 is the pulse unloading model, 4. The method for analyzing the stability region of a pulse load power system according to claim 3, characterized in that: The domain analysis of the pulse load power system includes considering the load increase and decrease as a step response, and its zero-state response ΔU C (t) = y PLi (t), Among them, y PLBi (t) is y PLi The outer envelope of (t), Response amplitude K PLi =k i r Li P PL / (d1 2 R Li +r), Angular displacement Undamped angular frequency Damping coefficient Damped oscillation angular frequency is the system oscillation angular frequency, f d =ω d / 2π is the natural frequency of the system, if ξ i →0, then f d ≈f n , f n is the undamped natural frequency.

5. The method for analyzing the stability region of a pulse load power system according to claim 4, characterized in that: According to the convergence state of pulse load operation, Thu C_max2 =|y PLB2 (0)|,ξ2>0 Δu C_max =max{Δu C_max1 ,Δu C_max2 } (4)。 6. The method for analyzing the stability region of a pulse load power system according to claim 5, characterized in that: According to the initial and final values of loading mode and unloading mode, Y 1s =y PLB1 (0)→Y 1e =y PLB1 (τ1)→Y 2s =Y 1e → Y 2e =max{|y PLB2 '(τ2)|} Among them, y PLB2 '(t) is the initial value of Y 2s The outer envelope of mode 2, ① when ξ1>0, both modes 1 and 2 converge and the system is stable; ② when ξ1<0, mode 1 diverges. At this time, if Y 2e >Y 1s , the waveform amplitude in the next cycle will continue to increase, the waveform will continue to diverge, and the system will be unstable; Y 2e ≤Y 1s , then the amplitude of the waveform in the next cycle will decrease or remain stable; thus, the metastable stability criterion of the PLPS system is: Y 2e ≤Y 1s .

7. The method for analyzing the stability region of a pulse load power system according to claim 6, characterized in that: The capacitor voltage U C Expressed as a differential equation, in, The pulse signal g PL (t) is expanded into a Fourier series Among them, oh PL =2π / f PL ,A0=D PL ,A i =2sin(iD PL π) / iπ, Substituting the expansion into equation (6) we get in, 8. The method for analyzing the stability region of a pulse load power system according to claim 7, characterized in that: Take the i-th resonance and write it in standard form: Among them, the undamped angular frequency The value is determined by the system structure; h is the resonance damping coefficient, and the weighted average value ξ of the two modal values is taken h =D PL ξ1+(1-D PL )ξ2; let γ i is 0, the i-th simple harmonic response y hi (t) and the overall system response y h (t), y hi (t)=b1β i H i sin(iω PL t+θ i ) Among them, b1H i is the input amplitude, β i is the amplification factor, b1β i h i That is, the i-th harmonic in U C The amplitude of the fluctuation generated at PL / ω n , angular displacement θ i make Y0 is Δu in formula (4) C_max get, Δu C_max There are two cases for analytical expressions: ①f PL =f n / N,T PL =NT n , that is, when N resonance occurs or near it, there is β N H N >1,Y0β N H N >>Y0; when no resonance occurs, β i H i <1,Y0>>Y0β N H N ,thereby Thu C_max =max{Y0β N H N ,Y0} (13) Let Δu C The stability threshold is δ max , we can get the stability criterion of the comprehensive modal step and resonance analytical expression max{Y0β N H N ,Y0}≤δ max (IV) By analyzing equations (4), (9) and criterion IV, the stability region of the PLPS system is drawn.

Images