A Fractional-Order Modeling Method for a Single-Phase Power Spring

Abstract

The invention discloses a fractional-order modeling method for a single-phase power spring, belonging to the technical field of power electronic system modeling. The modeling method includes establishing fractional-order mathematical models of a fractional-order filter inductor and a fractional-order filter capacitor, a fractional-order mathematical model of the single-phase power spring, and a fractional-order small-signal model of the single-phase power spring. Aiming at the problem that the current models of single-phase power springs are usually integer-order models established by using integer-order inductors and capacitors, and there are large deviations between the modeling and analysis results and the actual system, considering the fractional-order characteristics of actual inductors and capacitors, the above models are established, which can be conveniently used for the analysis of the operating characteristics of single-phase power springs and the design of their fractional-order controllers, improving the operating performance and control effect. Compared with the integer-order model, the fractional-order model can achieve high-precision simulation of the output waveform, with a smaller difference from the actual system, and can better reflect the essence of the actual system.

Description

Technical Field

[0001] The present invention relates to the technical field of power electronic system modeling, and particularly relates to a fractional-order modeling method for a single-phase power spring. Background Art

[0002] With the large-scale and high-proportion integration of renewable energy power generation into the power grid, it may lead to the mismatch between load demand and power generation, causing problems such as harmonic pollution, voltage fluctuation, and frequency flicker. The large power grid has a certain self-regulation ability for voltage fluctuation; while the small isolated microgrid has a weak regulation ability, and voltage fluctuation will have an adverse impact on electrical equipment, and in severe cases, it will damage key equipment. In view of the above situation, the Shu Yuen (Ron) Hui team proposed the concept of "Electric Spring (ES)".

[0003] The electric spring is a distributed voltage control technology. Its core idea is to adjust the non-critical loads in the power grid to respond to the fluctuations of intermittent renewable energy power generation, so as to maintain the stability of critical loads. It changes the traditional operation mode in which the power generation of the power system is restricted by the power consumption, and forms a new operation mode in which the power consumption automatically matches the power generation.

[0004] At present, the models of electric springs are all linear integer-order models established by using integer-order inductors and integer-order capacitors. For example:

[0005] 1) "T.Yang, T.Liu, J.Chen, S.Yan and S.Y.R.Hui, Dynamic Modular Modeling of Smart Loads Associated With Electric Springs and Control", IEEE Transactions on Power Electronics., 2018, 33(12): 10071-10085. ("Dynamic Modular Modeling of Smart Loads Associated With Electric Springs and Control") published in the IEEE Transactions on Power Electronics in 2018. This literature proposed a modular and dynamic electric spring model, which improved the flexibility of the model. However, combining the controller design with the circuit dynamic characteristics makes the model too complex.

[0006] 2) "Yang, Y., Ho, S.S., Tan, S.C., Hui, S.Y.R., Small-signal model and stability of electric springs in power grids", IEEE Trans. Smart Grid., 2018, 9: 857-865. ("Small-signal model and stability analysis of electric springs in power grids") This literature developed a frequency-domain-based linearized state-space model of electric springs for the stability analysis of distribution networks with multiple electric springs, but this model is not compatible with the standard stability models of other components.

[0007] 3) "Areed, E.F., Abido, M.A. and Al-Awami, A.T., Switching model analysis and implementation of electric spring for voltage regulation in smart grids", Iet Generation Transmission & Distribution., 2017, 11: 3703-3712. ("Switching model analysis and implementation of electric spring for voltage regulation in smart grids") This literature proposed a vector control-based linearized state-space model of electric springs that can be integrated with other components into the stability models of distribution networks and microgrids, but ignored the dynamic characteristics of the DC link voltage.

[0008] In summary, the existing technologies have the following problems:

[0009] 1) The modeling methods for single-phase electric springs are all based on linear integer-order mathematical models established with integer-order filter inductors and integer-order filter capacitors, ignoring the fact that the impedances of actual inductors and actual capacitors exhibit fractional-order characteristics, resulting in inaccurate models.

[0010] 2) None of the existing inventions have proposed a solution to establish a non-linear fractional-order model considering the fractional-order characteristics of actual inductors and actual capacitors to improve the accuracy of the model. Summary of the Invention

[0011] The technical problem to be solved by the present invention is that the traditional modeling method for single-phase electric springs does not consider the fractional-order characteristics of actual inductors and actual capacitors, resulting in inaccurate models.

[0012] The technical solution of the present invention is as follows:

[0013] A fractional-order modeling method for a single-phase power spring. The topological structure involved in this method includes a single-phase power spring, a non-critical load, a critical load, a line impedance, and a renewable energy microgrid. Among them, the topological structure of the single-phase power spring includes a DC power supply, a main inverter, a fractional-order filter inductor, and a fractional-order filter capacitor. The input end of the main inverter is connected in parallel with the DC power supply. The positive pole of the output end of the main inverter is connected in series with the fractional-order filter inductor. The other end of the fractional-order filter inductor is connected to the positive pole of the fractional-order filter capacitor, the positive pole of the critical load, and the line impedance. The other end of the line impedance is connected to the positive pole of the renewable energy microgrid. The negative pole of the output end of the main inverter is connected to the negative pole of the fractional-order filter capacitor and the positive pole of the non-critical load. The negative pole of the non-critical load is connected to the negative pole of the critical load and the negative pole of the renewable energy microgrid;

[0014] The fractional-order modeling method includes establishing the fractional-order mathematical models of the fractional-order filter inductor and the fractional-order filter capacitor, the fractional-order mathematical model of the single-phase power spring, and the fractional-order small-signal model of the single-phase power spring. The specific steps are as follows:

[0015] Step 1, Sampling

[0016] Sample the following parameters: the voltage of the DC power supply and denote it as the DC-side voltage V dc , the voltage v in at the output port of the inverter; the impedance Z NC of the non-critical load, the voltage v NC across the non-critical load, the current i NC flowing through the non-critical load, the impedance Z CL of the critical load, the voltage across the critical load and denote it as the critical load voltage v s , the current i CL flowing through the critical load, the line impedance Z g , the voltage v g at the grid connection point of the renewable energy power generation, the infeed current i g ; the inductance value L of the fractional-order filter inductor, the current flowing through the fractional-order filter inductor and denote it as the fractional-order filter inductor current i Lα , the voltage v L across the fractional-order filter inductor, the capacitance value C β of the fractional-order filter capacitor, the current i Cβ flowing through the fractional-order filter capacitor, the voltage across the fractional-order filter capacitor and denote it as the fractional-order filter capacitor voltage v ES , where α is the order of the fractional-order filter inductor and satisfies 0 < α < 1, and β is the order of the fractional-order filter capacitor and satisfies 0 < β < 1;

[0017] Step 2, establish the fractional-order mathematical models of the fractional-order filter inductor and the fractional-order filter capacitor

[0018] Considering the fractional-order characteristics of the actual inductor and the actual capacitor, based on the fractional-order calculus theory, establish the fractional-order mathematical models of the fractional-order filter inductor and the fractional-order filter capacitor, and their expressions are as follows:

[0019]

[0020] In the formula, d α i Lα / dt α is the fractional-order differential form of the current i of the fractional-order filter inductor, taking the order as α, the order of the fractional-order filter inductor, d Lα v β / dt ES is the fractional-order differential form of the voltage v of the fractional-order filter capacitor, taking the order as β, the order of the fractional-order filter capacitor; β ES s s

[0021] Step 3, establish the fractional-order mathematical model of the single-phase power spring

[0022] According to the mathematical models of the fractional-order filter inductor and the fractional-order filter capacitor, combined with Kirchhoff's voltage and current laws, establish the fractional-order mathematical model of the single-phase power spring, and its expression is as follows:

[0023]

[0024] In the formula, σ is the switching function of the power switch tube in the main inverter. When 0 < t < dT s , take σ = 1. When dT s <t < T s , take σ = 0, where T S is the switching period, t is the running time, and d is the duty cycle;

[0025] Step 4, establish the fractional-order small-signal model of the single-phase power spring

[0026] The single-phase power spring with fractional-order inductor and fractional-order capacitor introduced is a non-linear system. Linearize it through small-signal analysis to simplify the analysis process, and establish the fractional-order small-signal model of the single-phase power spring. Its expression is as follows:

[0027]

[0028] In the formula, is the perturbation of the current i of the fractional-order inductor Lα , is the perturbation of the voltage v across the fractional-order capacitor ES , is the disturbance quantity of the DC-side voltage V dc , is the disturbance quantity of the grid voltage v g , is the disturbance quantity of the critical load voltage v s , is the disturbance quantity of the duty cycle d is the fractional-order inductor current disturbance quantity takes the fractional-order differential form with the order being the fractional-order inductor order α; is the voltage disturbance quantity across the fractional-order capacitor takes the fractional-order differential form with the order being the fractional-order capacitor order β; A is the state matrix, B is the input matrix, C is the output matrix, and E is the output-input matrix.

[0029] Preferably, the non-critical load refers to a dissipative load, and the critical load refers to a load with precise requirements for the terminal voltage.

[0030] Preferably, the expressions of the state matrix A, input matrix B, output matrix C, and output-input matrix E in step 4 are respectively:

[0031]

[0032] where D is the duty cycle at the static operating point.

[0033] Compared with the prior art, the beneficial effects of the present invention are:

[0034] 1. Traditional modeling methods for single-phase power springs are all based on linear integer-order mathematical models established with integer-order filter inductors and integer-order filter capacitors, ignoring the fact that the impedances of actual inductors and actual capacitors exhibit fractional-order characteristics, resulting in inaccurate models. The present invention takes into account the fractional-order characteristics of actual inductors and actual capacitors, proposes a fractional-order modeling method for single-phase power springs, which is different from traditional integer-order modeling methods, is very simple and innovative;

[0035] 2. The present invention provides a fractional-order circuit simulation model of a single-phase power spring. Compared with the integer-order circuit simulation model, the fractional-order model has a smaller deviation between the high-precision simulation waveform of the single-phase power spring and the actual system waveform, and can more accurately describe the essence of the actual system. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is the main circuit structure diagram of the single-phase power spring with fractional-order characteristics adopted by the present invention.

[0037] Figure 2 is the Bode plot of the fractional-order transfer function under different cases of the order α.

[0038] Figure 3 The step response curves of the fractional-order transfer function for different orders α

[0039] Figure 4 The Bode plots of the fractional-order transfer function for different orders β

[0040] Figure 5 The step response curves of the fractional-order transfer function for different orders β

[0041] Figure 6 Is the equivalent circuit model of the fractional-order inductor and the fractional-order capacitor

[0042] Figure 7 Is the key load voltage waveform of the simulation output of the integer-order model of the single-phase power spring

[0043] Figure 8 Is the key load voltage waveform of the simulation output of the fractional-order model of the single-phase power spring

[0044] Figure 9 Is the change curve graph of the difference ΔU1 (ΔU2) between the integer-order model (fractional-order model) and the actual system Specific implementation manners

[0045] Next, the technical solution of the present invention will be clearly and completely described in conjunction with the accompanying drawings.

[0046] Figure 1 Is the main circuit structure diagram involved in the single-phase power spring adopted by the present invention. It can be seen from this figure that the topological structure involved in the modeling method of the present invention includes a single-phase power spring, a non-critical load, a critical load, a line impedance, and a renewable energy microgrid. Among them, the topological structure of the single-phase power spring includes a DC power supply, a main inverter, a fractional-order filter inductor, and a fractional-order filter capacitor. The input end of the main inverter is connected in parallel with the DC power supply. The positive pole of the output end of the main inverter is connected in series with the fractional-order filter inductor. The other end of the fractional-order filter inductor is connected to the positive pole of the fractional-order filter capacitor, the positive pole of the critical load, and the line impedance. The other end of the line impedance is connected to the positive pole of the renewable energy microgrid. The negative pole of the output end of the main inverter is connected to the negative pole of the fractional-order filter capacitor and the positive pole of the non-critical load. The negative pole of the non-critical load is connected to the negative pole of the critical load and the negative pole of the renewable energy microgrid.

[0047] In this embodiment, the non-critical load refers to a dissipative load, and the critical load refers to a load with precise requirements for the terminal voltage.

[0048] ​​​The dissipative loads include electric kettles, electric furnaces, lighting systems, etc. The critical loads include medical devices for monitoring vital signs, data centers, monitoring and ventilation equipment in coal mines, etc.

[0049] Considering the fractional-order characteristics of the filtering inductor and the filtering capacitor, a fractional-order modeling method for a single-phase power spring is provided. The fractional-order modeling method includes establishing the fractional-order mathematical models of the fractional-order filtering inductor and the fractional-order filtering capacitor, the fractional-order mathematical model of the single-phase power spring, and the fractional-order small-signal model of the single-phase power spring. The specific steps are as follows:

[0050] Step 1, Sampling

[0051] Sample the following parameters: the voltage of the DC power supply and denote it as the DC-side voltage V dc , the voltage v in at the output port of the inverter; the impedance Z NC of the non-critical load, the voltage v NC across the non-critical load, the current i NC flowing through the non-critical load, the impedance Z CL of the critical load, the voltage across the critical load and denote it as the critical load voltage v s , the current i CL flowing through the critical load, the line impedance Z g , the voltage v g at the grid connection point of renewable energy generation, the infeed current i g ; the inductance value L α of the fractional-order filtering inductor, the current flowing through the fractional-order filtering inductor and denote it as the fractional-order filtering inductor current i Lα , the voltage v L across the fractional-order filtering inductor, the capacitance value C β of the fractional-order filtering capacitor, the current i Cβ flowing through the fractional-order filtering capacitor, the voltage across the fractional-order filtering capacitor and denote it as the fractional-order filtering capacitor voltage v ES , where α is the order of the fractional-order filtering inductor and satisfies 0 < α < 1, and β is the order of the fractional-order filtering capacitor and satisfies 0 < β < 1.

[0052] In the example of the present invention, take V dc = 480V, L α = 3mH, C β = 50Mf.

[0053] Step 2, Establish the fractional-order mathematical models of the fractional-order filtering inductor and the fractional-order filtering capacitor

[0054] Considering the fractional-order characteristics of the actual inductor and the actual capacitor, based on the fractional-order calculus theory, establish the fractional-order mathematical models of the fractional-order filtering inductor and the fractional-order filtering capacitor, and their expressions are:

[0055]

[0056] where d α i Lα / dt α is the fractional - order derivative of the fractional - order filter inductor current i Lα taking the fractional - order differential form with the order being the fractional - order of the fractional - order filter inductor α, d β v ES / dt β is the fractional - order derivative of the fractional - order filter capacitor voltage v ES taking the fractional - order differential form with the order being the fractional - order of the fractional - order filter capacitor β.

[0057] Step 3, establish the fractional - order mathematical model of the single - phase power spring

[0058] According to the mathematical models of the fractional - order filter inductor and the fractional - order filter capacitor, combined with Kirchhoff's voltage and current laws, establish the fractional - order mathematical model of the single - phase power spring, and its expression is:

[0059]

[0060] where σ is the switching function of the power switch in the main inverter. When 0 < t < dT s , take σ = 1. When dT s <t < T s , take σ = 0. Among them, T S is the switching period, t is the running time, and d is the duty cycle.

[0061] Step 4, establish the fractional - order small - signal model of the single - phase power spring

[0062] The single - phase power spring with fractional - order inductor and fractional - order capacitor is a non - linear system. Through small - signal analysis, it is linearized to simplify the analysis process, and the fractional - order small - signal model of the single - phase power spring is established, and its expression is:

[0063]

[0064] where is the perturbation of the fractional - order inductor current i Lα , is the perturbation of the voltage across the fractional - order capacitor v ES , is the perturbation of the DC - side voltage V dc , is the perturbation of the grid voltage v g , is the perturbation of the critical load voltage v s , is the disturbance quantity of the duty cycle d, is the disturbance quantity of the fractional-order inductor current takes the fractional-order differential form with the order being the fractional-order inductor order α; is the disturbance voltage across the fractional-order capacitor takes the fractional-order differential form with the order being the fractional-order capacitor order β; A is the state matrix, B is the input matrix, C is the output matrix, and E is the output-input matrix.

[0065] In this embodiment, the expressions of the state matrix A, input matrix B, output matrix C, and output-input matrix E are respectively:

[0066]

[0067] where D is the duty cycle of the static operating point.

[0068] In this embodiment, the fractional-order transfer function from the duty cycle to the output critical load voltage is also plotted to perform stability analysis on the fractional-order model of the single-phase power spring to verify the effectiveness of the model. The fractional-order transfer function from the duty cycle to the output critical load voltage is denoted as the fractional-order transfer function This fractional-order transfer function is established through the following steps:

[0069] S1. To effectively improve the efficiency of system characteristic analysis and control design, using the Laplace transform, the fractional-order small-signal model of the single-phase power spring in the time domain is converted to the s domain, and the disturbance quantity of the duty cycle d in the s domain is denoted as the disturbance quantity of the critical load voltage v in the s domain s is denoted as

[0070] S2. The main function of the power spring system is to maintain the stability of the critical load voltage v s A fractional-order transfer function is established to reflect the performance of the system, and its expression is:

[0071]

[0072] S3. Plot the Bode plot and step response curve of the fractional-order transfer function and analyze the stability and dynamic performance of the system.

[0073] Figure 2 is the Bode plot of the fractional-order transfer function under different cases of the order α, Figure 3 is the Bode plot of the fractional-order transfer function The step response curve, Figure 4 For the fractional-order transfer function under different orders β The Bode plot, Figure 5 For the fractional-order transfer function under different orders β The step response curve, according to Figures 2 to 5 Analyze the influence of the fractional-order α and order β on the system stability and dynamic performance. It can be seen that: as the values of the fractional-order α and order β increase, the peak gain increases and the phase margin decreases; as the values of the fractional-order α and order β increase, the peak time and settling time of the step response increase, and the dynamic performance deteriorates; therefore, the fractional-order α and order β will affect the system stability and dynamic performance, and the greater the deviation of the order α and order β from 1, the deeper the influence.

[0074] In the embodiment of the present invention, in order to prove the beneficial effects of the present invention, dynamic response simulations are also carried out.

[0075] In the simulation, a simulation circuit of the single-phase power spring based on the integer-order model and a simulation circuit of the single-phase power spring based on the fractional-order model are built. In the simulation circuit of the integer-order single-phase power spring, its filter inductor and capacitor adopt the integer-order model; in the simulation circuit of the single-phase power spring based on the fractional-order model, α = 0.8, β = 0.8, that is, the filter inductor adopts the fractional-order filter inductor L with order α of 0.8 0.8 and the filter capacitor adopts the fractional-order filter capacitor C with order β of 0.8 0.8 . The fifth-order equivalent circuit model of the fractional-order filter inductor L 0.8 and the fractional-order filter capacitor C 0.8 is as shown in Figure 6 , which includes equivalent resistors R0, R1, R2, R3, R4, R5, equivalent inductors L1, L2, L3, L4, L5, equivalent capacitors C1, C2, C3, C4 and equivalent capacitor C P .

[0076] The key load voltage waveform of the simulation output of the single-phase power spring based on the integer-order model is as shown in Figure 7 ; the key load voltage waveform of the simulation output of the single-phase power spring based on the fractional-order model is as shown in Figure 8 .

[0077] In order to further compare the deviations between the integer-order model and the fractional-order model and the actual system, the change curves of the relative differences ΔU1 (ΔU2) between the output voltages of the integer-order model (fractional-order model) and the actual system output voltage at the same moment are as shown in Figure 9 . FromFigure 9 It can be seen that, compared with the integer-order model, the difference between the output voltage of the fractional-order model and the output voltage of the actual system is smaller. The fractional-order model can capture and describe the dynamic characteristics of the system more meticulously and reflect the essence of the actual system. That is, the correctness and necessity of the fractional-order model established by the present invention are verified.

Claims

1. A fractional order modeling method for a single-phase power spring, the topology involved in the method includes a single-phase power spring, a non-critical load, a critical load, a line impedance and a renewable energy microgrid, wherein: The topological structure of the single-phase power spring includes a DC power supply, a main inverter, a fractional-order filter inductor and a fractional-order filter capacitor. The input end of the main inverter is connected in parallel with the DC power supply, the positive pole of the output end of the main inverter is connected in series with the fractional-order filter inductor, the other end of the fractional-order filter inductor is connected to the positive pole of the fractional-order filter capacitor, the positive pole of the critical load, and the line impedance, and the other end of the line impedance is connected to the positive pole of the renewable energy microgrid; the negative pole of the output end of the main inverter is connected to the negative pole of the fractional-order filter capacitor and the positive pole of the non-critical load, and the negative pole of the non-critical load is connected to the negative pole of the critical load and the negative pole of the renewable energy microgrid; It is characterized in that the fractional-order modeling method includes establishing a fractional-order mathematical model of a fractional-order filter inductor and a fractional-order filter capacitor, a fractional-order mathematical model of a single-phase power spring, and a fractional-order small signal model of a single-phase power spring, and the specific steps are as follows: Step 1: Sampling The following parameters are sampled: The voltage of the DC power supply is recorded as the DC side voltage V dc , inverter output port voltage v in ; Impedance Z of non-critical load NC , the voltage across the non-critical load v NC , the current i flowing through the non-critical load NC , the impedance of the critical load is Z CL , the voltage across the critical load is recorded as the critical load voltage v S , the current i flowing through the critical load CL , line impedance Z g , the voltage of the renewable energy power generation grid connection point v g , grid current i g ; The inductance value L of the fractional-order filter inductor α , the current flowing through the fractional-order filter inductor is recorded as the fractional-order filter inductor current i Lα , the voltage across the fractional-order filter inductor is v L , the capacitance value C of the fractional filter capacitor β , the current i flowing through the fractional filter capacitor Cβ , the voltage across the fractional filter capacitor is recorded as the fractional filter capacitor voltage v ES , where α is the order of the fractional filter inductor and satisfies 0<α<1, β is the order of the fractional filter capacitor and satisfies 0<β<1; Step 2: Establish the fractional-order mathematical model of the fractional-order filter inductor and fractional-order filter capacitor Considering the fractional-order characteristics of actual inductance and actual capacitance, based on the fractional-order calculus theory, the fractional-order mathematical model of fractional-order filter inductance and fractional-order filter capacitance is established, and its expression is: Where, d α i Lα / dt α is the fractional-order filter inductor current i Lα Take the fractional order differential form of the fractional order filter inductor order α, d β v ES / dt β is the fractional filter capacitor voltage v ES Take the fractional differential form of the order of the fractional filter capacitor β; Step 3: Establish a fractional-order mathematical model of a single-phase electric spring According to the mathematical model of fractional-order filter inductor and fractional-order filter capacitor, combined with Kirchhoff's voltage and current law, the fractional-order mathematical model of single-phase power spring is established, and its expression is: Where σ is the switching function of the power switch in the main inverter. <t<dT s When σ=1, when dT s <t <T s When σ=0, T S is the switching cycle, t is the operating time, and d is the duty cycle; Step 4: Establish a fractional-order small signal model of a single-phase power spring The single-phase electric spring with fractional-order inductance and fractional-order capacitance is a nonlinear system. It is linearized by small signal analysis to simplify the analysis process. The fractional-order small signal model of the single-phase electric spring is established, and its expression is: In the formula, is the fractional inductor current i Lα The disturbance amount, is the voltage across the fractional capacitor v ES The disturbance amount, is the DC side voltage V dc The disturbance amount, is the grid voltage v g The disturbance amount, is the critical load voltage v s The disturbance amount, is the disturbance of duty cycle d, is the fractional-order inductor current disturbance Take the fractional differential form of the fractional inductor order α; is the voltage disturbance across the fractional capacitor Take the fractional differential form of the fractional capacitor order β; A is the state matrix, B is the input matrix, C is the output matrix, and E is the output input matrix.

2. The fractional order modeling method of a single-phase power spring according to claim 1, characterized in that: The non-critical load refers to a dissipative load, and the critical load refers to a load that requires precise terminal voltage.

3. The fractional order modeling method of a single-phase power spring according to claim 1 is characterized in that: The expressions of the state matrix A, input matrix B, output matrix C, and output input matrix E in step 4 are respectively: Where D is the duty cycle of the static operating point.

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