FTESO-based parallel inverter system recursive nonsingular terminal sliding mode control method

Abstract

The invention provides an FTESO-based parallel inverter system recursive nonsingular terminal sliding mode control method, and the method comprises the following steps: firstly, designing FTESO to estimate system load disturbance; and secondly, designing an RNTSM control law for the inverter system based on FTESO compensation, so that the inverter system can completely overcome the influence of load interference, and the robustness of filtering parameter perturbation is enhanced. According to the method, the recursive terminal sliding mode surface formed by combining the non-singular fast terminal sliding mode and the integral sliding mode is adopted, it can be ensured that the output voltage of an inverter system rapidly tracks the reference voltage of the inverter system within finite time, and system buffeting is effectively weakened.

Description

Technical Field

[0001] The present invention belongs to the technical field of control of parallel inverter systems in new energy microgrids, and relates to a recursive nonsingular terminal sliding mode control method for parallel inverter systems based on FTESO. Background Art

[0002] A microgrid is defined as a low-voltage network composed of multiple distributed energy sources (DGs), a parallel inverter system of energy storage, and loads, etc., which supports the integration of distributed generation into a traditional large-capacity power parallel inverter system and operates in an islanded or grid-connected mode. Since the capacity of a single inverter is limited, DGs usually use multiple inverters in parallel to connect new energy to an AC microgrid. Therefore, in order to improve the power quality of the AC microgrid, it is of great engineering significance to study the control method of its parallel inverter system.

[0003] The traditional voltage control scheme for parallel inverter systems in an island usually adopts proportional-integral-derivative control. Although the parallel inverters can equally divide the load power, when suffering from various internal and external disturbances, the dynamic performance of the output voltage of the parallel inverter system is poor. To improve the dynamic and steady-state performance of the parallel inverter system, advanced control methods such as model predictive control, adaptive control, control, and intelligent control have been successfully applied one after another. In addition, the sliding mode control (SMC) method has been widely used in microgrids due to its excellent performance such as fast dynamic response and strong robustness. However, there are obvious technical defects of chattering and steady-state error in the control of parallel inverter systems based on traditional SMC.

[0004] To solve the above problems, some existing technicians have adopted an adaptive SMC strategy to reduce the steady-state error of the output voltage of the islanded microgrid, but the dynamic response speed of the SMC parallel inverter system based on a linear sliding mode surface is not fast enough. For this reason, for a grid-connected photovoltaic inverter, some existing technicians have constructed an adaptive terminal sliding mode control based on two terminal sliding mode surfaces to achieve finite-time control of the parallel inverter system, but it ignores the singularity problem; some existing technicians have also proposed a nonsingular terminal sliding mode control (NTSMC) method for a two-stage photovoltaic parallel inverter system affected by interference, thus avoiding the singularity problem. However, the steady-state tracking error of this parallel inverter system is not satisfactory.

[0005] At present, the Extended State Observer (ESO) can estimate the states and disturbances of a parallel inverter system in real time and has been widely used in the engineering field. In response to filter parameter variations and grid disturbances, some existing technicians have adopted an active disturbance rejection control method based on a Linear Extended State Observer (LESO) to achieve robust control of the output current of a grid-connected parallel inverter system. However, the observation accuracy of its LESO is relatively low. To improve the observation accuracy, some existing technicians have used a non-linear ESO based on the hyperbolic tangent function to estimate the unknown disturbances of the parallel inverter system. However, the convergence time of its observation error is uncontrollable. To improve the steady-state performance of the parallel inverter system and accelerate the dynamic response speed of the parallel inverter system, on the basis of inheriting the advantages of NTSMC, the present invention proposes a recursive non-singular terminal sliding mode control method for a parallel inverter system based on FTESO to solve the above technical defects. Summary of the Invention

[0006] The object of the present invention is to provide a recursive non-singular terminal sliding mode control method for a parallel inverter system based on FTESO in view of the problems existing in the prior art, and to solve the problems that traditional observers cannot converge within a finite time, traditional SMC has a slow response and a large steady-state error.

[0007] The present invention adopts a recursive terminal sliding mode surface composed of a non-singular terminal sliding mode and an integral sliding mode, and proposes a Recursive Nonsingular Terminal Sliding Mode control (RNTSMC) strategy.

[0008] Since the SMC parallel inverter system only truly has robustness after entering the sliding mode, but it is still very sensitive to load disturbances in the reaching phase. To ensure that the entire SMC process is robust to load disturbances, the present invention conceives a strategy that can quickly and accurately estimate load disturbances and compensate for them.

[0009] To improve the observation accuracy and ensure the finite-time convergence of the observation error, based on the finite-time convergence concept of a sliding mode differentiator, the present invention designs a Finite Time Extended State Observer (FTESO) to enable it to quickly and accurately estimate load disturbances in real time, and through feedforward compensation for it, to ensure the robustness of the entire SMC process to load disturbances.

[0010] According to the above technical concept, the present invention adopts the following technical solutions:

[0011] Recursive Nonsingular Terminal Sliding Mode Control Method for Parallel Inverter System Based on FTESO, including the following steps:

[0012] For a parallel inverter system composed of n inverters, construct the dynamic model of the inverter main circuit, and its expression is as follows:

[0013]

[0014] In the formula, is the first state variable; is the second state variable; is the derivative of; is the derivative of; is the filter inductor; is the filter capacitor; is the DC input voltage; is the average duty cycle of the inverter, that is, the control signal of the parallel inverter system; is the load disturbance;

[0015] Among them, , , The expressions of are as follows:

[0016]

[0017]

[0018]

[0019] In the formula, is the capacitor voltage of the inverter; is the first derivative of; is the derivative of; is the load current;

[0020] Specifically, set the load disturbance to be bounded, and its derivative satisfies , is a bounded positive real number.

[0021] is to quickly estimate the disturbance within a finite time and ensure the estimation accuracy. Design FTESO, and its expression is as follows:

[0022]

[0023] Let ; ;

[0024] In the formula, is the estimated value of; is the third state variable; is the first gain of FTESO; is the second gain of FTESO; is the third gain of FTESO; is a positive real number between 0 and 1; is a monotonically increasing smooth function, and its expression is , which can eliminate the chattering caused by the sign function ; is the load disturbance 's first derivative.

[0025] The convergence analysis of FTESO is as follows:

[0026] Let the observation error be , then the FTESO error equation is:

[0027]

[0028] In the formula, is the first observation error; is the first observation error 's first derivative; is the second observation error; is 's second observation error 's first derivative; is the third observation error; is the third observation error 's first derivative;

[0029] Define the auxiliary column vector , and its expression is as follows:

[0030]

[0031] Therefore, we have:

[0032]

[0033] In the formula, is the absolute value of the first observation error ;

[0034] Then the convergence analysis of FTESO is equivalent to 's convergence analysis, and the latter's convergence analysis is given by Theorem 1.

[0035] Theorem 1: There exist and , such that It can converge within a finite time and satisfy:

[0036]

[0037] Wherein, is the minimum singular value of matrix ; is the minimum singular value of the positive definite symmetric matrix ; , , , matrix and the positive definite symmetric matrix are respectively:

[0038] ,

[0039] Proof: Let , , taking the derivative of gives:

[0040]

[0041] Wherein:

[0042]

[0043] The characteristic equation of matrix is:

[0044]

[0045] Wherein, is a complex variable; is a matrix; is the identity matrix of the same dimension as matrix ;

[0046] According to the Routh criterion, select the parameter that satisfies the inequality to ensure that matrix A is a Hurwitz matrix. Therefore, there exists a symmetric positive definite matrix Q that satisfies the following Lyapunov equation:

[0047]

[0048] Wherein, represents the matrix transpose; is a symmetric positive definite matrix;

[0049] Define the Lyapunov function:

[0050]

[0051] Substitute , the matrix and the positive definite symmetric matrix into the above formula, we can get:

[0052]

[0053] Since satisfies the following inequality:

[0054]

[0055] In the formula, is the minimum eigenvalue of the positive definite symmetric matrix ; is the maximum eigenvalue of the positive definite symmetric matrix ;

[0056] According to the above formula, the following inequality can be obtained:

[0057]

[0058] Derive to get:

[0059]

[0060] In the formula, , ;

[0061] Let , ;

[0062] According to the Cauchy-Schwarz inequality, the following inequality can be obtained:

[0063]

[0064] According to the above formula, we have:

[0065]

[0066] For the symmetric positive definite matrix , the following inequality holds:

[0067]

[0068] Except , the matrix A is a non-singular matrix, so we can get:

[0069]

[0070] Decompose the matrix ( ) into:

[0071]

[0072] In the formula:

[0073]

[0074] Except outside, are all non-singular matrices. Therefore, the following inequality holds:

[0075]

[0076] Furthermore, it can be obtained that:

[0077]

[0078] In the following, according to the two distribution cases in the above formula, the convergence domain and its convergence time will be discussed respectively.

[0079] Case 1: When , there is , and it can be obtained that:

[0080]

[0081] Define:

[0082]

[0083]

[0084] Thus, it can be obtained that:

[0085]

[0086] Since and are both bounded, therefore, there must exist such that holds;

[0087] The inequality about can also be written as:

[0088]

[0089] According to the finite-time convergence lemma, within the finite time , and will both converge and respectively satisfy: , ;

[0090] The convergence time satisfies:

[0091]

[0092] Case 2: When , there is ;

[0093] Thus, it can be obtained that:

[0094]

[0095] Definition:

[0096]

[0097]

[0098] Thus:

[0099]

[0100] Assume the following inequality holds:

[0101]

[0102] It can be obtained that:

[0103]

[0104] Furthermore, the following inequality is obtained:

[0105]

[0106] According to the finite-time lemma, will converge to the region from the region in finite time , that is:

[0107]

[0108] The convergence time satisfies:

[0109]

[0110] Combining the above two analysis cases, is decreasing with respect to time and converges to the region within finite time , then there is:

[0111]

[0112] Therefore, select appropriate parameters such that is sufficiently large, then will be sufficiently small, and satisfy:

[0113]

[0114] Furthermore, in order to enhance the robustness of the parallel inverter system to the changes in the LC filter parameters and achieve the finite-time convergence and high-precision control of the output voltage of the parallel inverter system, a first-layer non-singular fast terminal sliding mode surface is designed; in order to further reduce the steady-state tracking error of the parallel inverter system, a second-layer recursive terminal sliding mode surface is designed; based on the first-layer non-singular fast terminal sliding mode surface and the second-layer recursive terminal sliding mode surface , an RNTSMC control law is designed based on FTESO; a feed-forward control law is designed based on FTESO; combining the RNTSMC control law and the feed-forward control law , the control signal of the parallel inverter system is finally obtained. Specifically:

[0115] The expression of the first-layer non-singular fast terminal sliding mode surface is as follows:

[0116]

[0117] In the formula, , are both positive real numbers; is a real number greater than 1; is the capacitor voltage tracking error, is the derivative of the capacitor voltage tracking error;

[0118] Among them:

[0119]

[0120]

[0121] The first derivative of

[0122]

[0123] In the formula, is the second derivative of; is the capacitor voltage reference value obtained by droop control; is the second derivative of; is the RNTSMC control law;

[0124] Among them, The expression of

[0125]

[0126] The second - layer recursive terminal sliding surface The expression of

[0127]

[0128] In the formula, is a positive real number; and are both positive odd numbers, and ; is the integral with respect to time;

[0129] Substituting the expression of into the above formula, we can get:

[0130]

[0131] The first - order differential of the above formula is denoted as:

[0132]

[0133] To reduce the chattering of the parallel inverter system and ensure the convergence time of the parallel inverter system, the following sliding - mode reaching law of the parallel inverter system is selected :

[0134]

[0135] In the formula, is the fourth gain of FTESO; is the fifth gain of FTESO; is a positive real number between 0 and 1; , , are positive real numbers.

[0136] Based on the parallel inverter system with FTESO compensation, the RNTSMC control law The expression of

[0137]

[0138] Based on FTESO, the feed - forward control law , the expression of which is as follows:

[0139]

[0140] To compensate for the disturbances caused by load variations in a parallel inverter system, a control signal is designed , and its expression is as follows:

[0141]

[0142] Since there is a recursive relationship between the first-layer nonsingular fast terminal sliding mode surface and the second-layer recursive terminal sliding mode surface , then, first, the parallel inverter system will reach the second-layer recursive terminal sliding mode surface in a finite time. Next, it will slide from the second-layer recursive terminal sliding mode surface to the first-layer nonsingular fast terminal sliding mode surface in a finite time. Finally, it will slide on the first-layer nonsingular fast terminal sliding mode surface such that the capacitor voltage tracking error converges to zero in a finite time; therefore, RNTSMC can ensure that the capacitor voltage of the parallel inverter system tracks its reference voltage in a finite time, and the stability analysis process is as follows.

[0143] First, define the following Lyapunov function:

[0144]

[0145] The time derivative of

[0146]

[0147] According to Lyapunov theory, based on the finite-time convergence lemma, the parallel inverter system will approach the second-layer recursive terminal sliding mode surface in a finite time, and its convergence time satisfies the following inequality:

[0148]

[0149] When the parallel inverter system reaches the second-layer recursive terminal sliding mode surface , from the expression of the second-layer recursive terminal sliding mode surface , we can obtain . The above results show that the first-layer nonsingular fast terminal sliding mode surface and change synchronously.

[0150] Next, define the following Lyapunov function:

[0151]

[0152] The time derivative of

[0153]

[0154] Therefore, will converge to zero, and will also converge to zero synchronously. According to the finite-time convergence lemma, the parallel inverter system will converge from the second-layer recursive terminal sliding mode surface to the first-layer non-singular fast terminal sliding mode surface in finite time, and its convergence time satisfies the following inequality:

[0155]

[0156] When the parallel inverter system slides to the first-layer non-singular fast terminal sliding mode surface , from the expression of the first-layer non-singular fast terminal sliding mode surface , we can obtain .

[0157] Finally, define the following Lyapunov function:

[0158]

[0159] The time derivative of

[0160]

[0161] Therefore, the capacitor voltage tracking error will converge to zero from the first-layer non-singular fast terminal sliding mode surface in finite time, and its convergence time satisfies the following inequality:

[0162]

[0163] In summary, the RNTSMC control law proposed in the present invention can not only ensure the stability of the parallel inverter system, but also ensure that the capacitor voltage tracking error converges to zero within a finite time .

[0164] The beneficial effects of the present invention are as follows:

[0165] For the parallel inverter system affected by load disturbances, filter parameter perturbations, etc., the present invention proposes an RNTSMC control method based on FTESO. To resist the adverse effects of load disturbances, an FTESO capable of estimating load disturbances is designed and feed-forward compensated; at the same time, to enhance the immunity to the adverse effects of filter parameter perturbations on the parallel inverter system, an RNTSM control law is designed.

[0166] The present invention can ensure that the output voltage of the parallel inverter system accurately tracks its reference voltage within a limited time, reduce its total harmonic distortion rate, and effectively resist internal and external disturbances of the parallel inverter system, thereby improving the robustness of the parallel inverter system, having a fast response ability, and realizing the fast equal sharing of load power by the parallel inverters; specifically:

[0167] (1) Compared with the LESO, the proposed FTESO of the present invention has higher observation accuracy, and the strategy based on ESO feedforward compensation enables the parallel inverter system to completely resist load disturbances;

[0168] (2) The proposed RNTSMC of the present invention can not only significantly reduce the chattering of the parallel inverter system and improve the control accuracy of the parallel inverter system, but also has strong robustness to filter parameter perturbations;

[0169] (3) Since the FTESO and RNTSMC can respectively make the observation error and the capacitor voltage tracking error converge within a limited time, therefore, the present invention can enable the parallel inverter system to have a faster response ability. Description of the Drawings

[0170] Figure 1 is the control block diagram of the parallel inverter system in this embodiment;

[0171] Figure 2 is the RNTSMC control block diagram based on the FTESO in this embodiment;

[0172] Figure 3 is the comparison schematic diagram of the capacitor voltages of the parallel inverter system 1 and the parallel inverter system 2 under the perturbation of the filter inductor parameters;

[0173] Figure 4 is the comparison schematic diagram of the inverter filter capacitor voltages of the parallel inverter system 1 and the parallel inverter system 2 when the filter capacitor parameters change;

[0174] Figure 5 is the comparison schematic diagram of the capacitor voltages of the parallel inverter system 1 and the parallel inverter system 2 when the reference voltage changes;

[0175] Figure 6 is the comparison schematic diagram of the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 2 when the reference voltage changes;

[0176] Figure 7 is the schematic diagram of the capacitor voltage and load current of the parallel inverter system 1 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.1s and 0.2s respectively;

[0177] Figure 8Schematic diagram of the capacitor voltage and load current of the parallel inverter system 3 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.1s and 0.2s respectively;

[0178] Figure 9 Schematic diagram of the comparison of the filtering capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 3 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.1s and 0.2s respectively;

[0179] Figure 10 Schematic diagram of the power distribution of the two inverters in the parallel inverter system 1 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.1s and 0.2s respectively;

[0180] Figure 11 Schematic diagram of the power distribution of the two inverters in the parallel inverter system 3 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.1s and 0.2s respectively;

[0181] Figure 12 Schematic diagram of the comparison of the capacitor voltage observation errors of the parallel inverter system 1 and the parallel inverter system 4 under the condition that the load suddenly increases by 2KW + 1KVar at 0.15s;

[0182] Figure 13 Schematic diagram of the comparison of the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 4 under the condition that the load suddenly increases by 2KW + 1KVar at 0.15s;

[0183] Figure 14 Schematic diagram of the capacitor voltage and load current of the parallel inverter system 1 under the perturbation of the filter inductor parameters;

[0184] Figure 15 Schematic diagram of the capacitor voltage and load current of the parallel inverter system 2 under the perturbation of the filter inductor parameters;

[0185] Figure 16 Schematic diagram of the capacitor voltage and load current of the parallel inverter system 1 under the perturbation of the filter capacitor parameters;

[0186] Figure 17 Schematic diagram of the capacitor voltage and load current of the parallel inverter system 2 under the perturbation of the filter capacitor parameters;

[0187] Figure 18 Schematic diagram of the comparison of the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 2 when the droop control reference voltage changes;

[0188] Figure 19Schematic diagrams of the capacitor voltage and load current of the parallel inverter system 1 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.06s and 0.12s respectively;

[0189] Figure 20 Schematic diagrams of the capacitor voltage and load current of the parallel inverter system 3 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.06s and 0.12s respectively;

[0190] Figure 21 Schematic diagram of the comparison of the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 3 under the condition that the load suddenly increases and decreases by 2KW + 1KVar at 0.06s and 0.12s respectively;

[0191] Figure 22 Schematic diagram of the comparison of the capacitor voltage observation errors of the parallel inverter system 1 and the parallel inverter system 4 under the condition that the load suddenly increases by 2KW + 1KVar at 0.1s;

[0192] Figure 23 Schematic diagram of the comparison of the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 4 under the condition that the load suddenly increases by 2KW + 1KVar at 0.1s. Detailed implementation manners

[0193] The technical solutions of the present invention will be described in detail below in conjunction with the accompanying drawings and the implementation methods.

[0194] Embodiment

[0195] As Figure 1 shown, the recursive nonsingular terminal sliding mode control method for the parallel inverter system based on FTESO includes the following steps:

[0196] For the parallel inverter system composed of n inverters with uncertainties such as filter parameters and load changes, a dynamic model of the inverter main circuit is constructed, and its expression is as follows:

[0197]

[0198] In the formula, is the first state variable; is the second state variable; is the derivative of; is the derivative of; is the filter inductor; is the filter capacitor; is the DC input voltage; is the average duty cycle of the inverter, that is, the control signal of the parallel inverter system; is the load disturbance;

[0199] Among them, 、 、 The expressions of are respectively:

[0200]

[0201]

[0202]

[0203] In the formula, is the capacitor voltage of the inverter; is the first derivative of; is the first derivative of; is the load current;

[0204] Specifically, assuming that the load disturbance is bounded, and its derivative satisfies , is a bounded positive real number.

[0205] is to quickly estimate the disturbance within a finite time and ensure the estimation accuracy. Design the FTESO, and its expression is as follows:

[0206]

[0207] Let ; ;

[0208] In the formula, is the estimated value of ; is the third state variable; is the first gain of the FTESO; is the second gain of the FTESO; is the third gain of the FTESO; is a positive real number between 0 and 1; is a monotonically increasing smooth function, and its expression is , which can eliminate the chattering caused by the sign function; is the first derivative of the load disturbance .

[0209] As Figure 2 shown, to enhance the robustness of the parallel inverter system to the changes of LC filter parameters and achieve the finite-time convergence and high-precision control of the output voltage of the parallel inverter system, design the first layer non-singular fast terminal sliding mode surface ; To further reduce the steady-state tracking error of the parallel inverter system, a second-layer recursive terminal sliding surface is designed. ; Based on the first-layer nonsingular fast terminal sliding surface and the second-layer recursive terminal sliding surface , an RNTSMC control law is designed based on FTESO ; A feedforward control law is designed based on FTESO ; Combining the RNTSMC control law and the feedforward control law to obtain the control signal of the parallel inverter system .

[0210] The expression of the first-layer nonsingular fast terminal sliding surface is as follows:

[0211]

[0212] Wherein, , are all positive real numbers; is a real number greater than 1; is the capacitor voltage tracking error; is the derivative of the capacitor voltage tracking error;

[0213] Among them:

[0214]

[0215]

[0216] The first derivative of

[0217]

[0218] Wherein, The expression of

[0219]

[0220] Wherein, is The second derivative of; is the reference value of the inverter capacitor voltage obtained through droop control; is The second derivative of; is the RNTSMC control law.

[0221] The expression of the second-layer recursive terminal sliding surface is as follows:

[0222]

[0223] In the formula, is a positive real number; and are both positive odd numbers, and ; is the integral with respect to time;

[0224] Substituting the expression of into the above formula, we can get:

[0225]

[0226] The first-order differential of the above formula is:

[0227]

[0228] To reduce the chattering of the parallel inverter system and ensure the convergence time of the parallel inverter system, the following sliding mode reaching law of the parallel inverter system is selected :

[0229]

[0230] In the formula, is the fourth gain of FTESO; is the fifth gain of FTESO; is a positive real number between 0 and 1; , , are positive real numbers.

[0231] The RNTSMC control law designed for the parallel inverter system based on FTESO compensation has the following expression:

[0232]

[0233] To compensate for the disturbance caused by the load change in the parallel inverter system, the control signal is designed, and its expression is as follows:

[0234]

[0235] In the formula, is the feedforward control law.

[0236] To verify the effectiveness and superiority of the algorithm proposed in the present invention, taking the parallel connection of two inverters as an example, four sets of parallel inverter systems as shown in Figure 3 were built in MATLAB / Simulink, which are respectively called parallel inverter system 1, parallel inverter system 2, parallel inverter system 3, and parallel inverter system 4. The control methods applied to the above parallel inverter systems are as follows: this embodiment is RNTSMC based on FTESO (abbreviated as FTESO-based RNTSMC), SMC based on FTESO (abbreviated as FTESO-based SMC), NFTSMC based on FTESO (abbreviated as FTESO-based NFTSMC), and RNTSMC based on LESO (abbreviated as LESO-based RNTSMC). The four sets of parallel inverter systems adopt exactly the same inverters, line impedances, and droop controllers, and their electrical parameters and control parameters are shown in Table 1 respectively.

[0237] Table 1 Electrical parameters and control parameters

[0238]

[0239] To verify that parallel inverter system 1 has strong robustness, fast response speed, and low chattering, it is compared with parallel inverter system 2.

[0240] It is designed that the dynamic model of the parallel inverter in parallel inverter system 2 is the same as that of parallel inverter system 1, and its SMC control law based on FTESO is designed as follows:

[0241] Select the linear sliding surface :

[0242]

[0243] The first-order differential of the above formula is formula (1):

[0244]

[0245] Select the constant velocity reaching law formula (2):

[0246]

[0247] In the formula, is the reaching law gain.

[0248] Let formula (1) be equal to formula (2), and the SMC control law based on FTESO in parallel inverter system 2 can be obtained as:

[0249]

[0250] In the formula, the parameters and Selected separately as and respectively.

[0251] To ensure a fair comparison, the FTESO with the same parameters is adopted in the parallel inverter system 1 and the parallel inverter system 2. Due to space limitations, only the dynamic responses of the first inverter in the parallel inverter system 1 and the parallel inverter system 2 are compared below.

[0252] Comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 2 when the filter inductor parameter changes:

[0253] The filter inductor parameter changes from in a stepwise manner to at 0.1 s and from in a stepwise manner to at 0.2 s. Under these working conditions, Figure 3 and Table 2 respectively give the capacitor voltage responses and their THD values (Total Harmonic Distortion) of the parallel inverter system 1 and the parallel inverter system 2.

[0254] From Figure 3 it can be seen that when the filter inductor changes before and after 0.1 s and 0.2 s, the capacitor voltages of the parallel inverter system 1 (red line) and the parallel inverter system 2 (yellow line) can track their reference voltages (blue line). The above results show that both have strong robustness to the perturbation of the filter inductor parameter; however, from Figure 3 the local enlarged view, it can be seen that the chattering amplitude of the parallel inverter system 1 is significantly smaller than that of the parallel inverter system 2. At the same time, from Table 2, it can be seen that the THD value of the capacitor voltage of the parallel inverter system 1 is smaller than that of the parallel inverter system 2 under different filter inductor parameters.

[0255] Table 2 THD values of capacitor voltages under different inductor parameters

[0256]

[0257] Comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 2 when the filter capacitor parameter is perturbed:

[0258] The filter capacitor parameter changes from in a stepwise manner to at 0.1 s and from in a stepwise manner to at 0.2 s. Under these working conditions, Figure 4 and Table 3 respectively give the capacitor voltage responses and their THD values of the parallel inverter system 1 and the parallel inverter system 2.

[0259] FromFigure 4 It can be seen that when the filtering capacitor changes before and after two moments of 0.1 s and 0.2 s, the capacitor voltages of the parallel inverter system 1 (red line) and the parallel inverter system 2 (yellow line) can both track their reference voltages (blue line). The above results indicate that both of them have strong robustness to the perturbation of the filtering capacitor parameters; however, from Figure 4 the local enlarged view and Table 3, it can be clearly seen that compared with the parallel inverter system 2, the parallel inverter system 1 can effectively weaken the chattering.

[0260] Table 3 THD values of capacitor voltages under different capacitor parameters

[0261]

[0262] Regarding the comparison of the response speeds of the parallel inverter system 1 and the parallel inverter system 2 when the droop control reference voltage changes suddenly:

[0263] Droop control reference voltage Under the condition that it suddenly decreases from 220 V to 110 V at 0.15 s, and Figure 6 respectively give the capacitor voltage response and its tracking error.

[0264] From it can be seen that when the droop control reference voltage changes suddenly before and after 0.15 s, the capacitor voltages of the parallel inverter system 1 (red line) and the parallel inverter system 2 (yellow line) can both quickly track their reference voltages, which indicates that both of them have strong robustness to the change of the reference voltage; but from Figure 6 the local enlarged view, it can be seen that the convergence times of the parallel inverter system 1 and the parallel inverter system 2 are approximately 0.00025 s and 0.00051 s respectively, indicating that the parallel inverter system 1 has a faster response speed.

[0265] To verify that the parallel inverter system 1 has higher tracking accuracy, it is compared with the parallel inverter system 3 under load disturbance. Among them, the expression of the NFTSMC control law based on FTESO selected for the parallel inverter system 3 is as follows:

[0266]

[0267] To ensure a fair comparison, the parallel inverter system 1 and the parallel inverter system 3 adopt FTESO with the same parameters.

[0268] Under the condition that the load suddenly increases by 2 kW + 1 kVar at 0.1 s and suddenly decreases by the same amount at 0.2 s, as can be seen from Figures 7 and 8, when the load suddenly increases at 0.1 s, the capacitor voltages of both can track their reference voltages, and the amplitude of the load current increases accordingly; when the load suddenly decreases at 0.2 s, the capacitor voltages of both can also track their reference voltages, and the amplitude of the load current decreases accordingly. This is due to the fact that both are designed with FTESO that can perform feedforward compensation for load disturbances, thus ensuring that the parallel inverter system can resist load interference. However, from Figure 9 it can be seen that the capacitor voltage tracking error of parallel inverter system 1 is 0.12 V, which is significantly smaller than 0.23 V of parallel inverter system 3. It is precisely because of the use of a recursive terminal sliding mode surface with an integral term that the control accuracy of parallel inverter system 1 is improved. Since the load of the parallel inverter system is resistive-inductive, so Figure 7 and 8 the load current (blue line) lags behind the capacitor voltage (black line).

[0269] From Figure 10 and 11 it can be seen that when the load suddenly increases and decreases at 0.1 s and 0.2 s, whether it is parallel inverter system 1 or parallel inverter system 3, the active power and reactive power response curves of the two parallel inverters almost coincide. The above results show that the two parallel inverters evenly divide the load power, thus verifying the effectiveness of the droop controller proposed in this embodiment.

[0270] To verify that the proposed FTESO in parallel inverter system 1 has higher observation accuracy, it is compared with parallel inverter system 4.

[0271] The LESO of parallel inverter system 4 is designed as:

[0272]

[0273] where are respectively selected as .

[0274] To ensure a fair comparison, parallel inverter system 1 and parallel inverter system 4 adopt exactly the same RNTSMC.

[0275] Under the condition that the load suddenly increases by 2 kW + 1 kVar at 0.15 s, as can be seen from Figure 12 before and after the load mutation at 0.15 s, the capacitor voltage observation errors of parallel inverter system 1 and parallel inverter system 4 are approximately 0.08 V and 0.2 V respectively. From Figure 13It can be seen that the capacitor voltage tracking errors of the parallel inverter system 1 and the parallel inverter system 4 are approximately 0.12V and 0.15V respectively; the above results show that compared with the LESO of the parallel inverter system 4, the FTESO designed for the parallel inverter system 1 has higher observation accuracy, and at the same time, makes the parallel inverter system 1 have a smaller capacitor voltage tracking error.

[0276] To further verify the effectiveness and superiority of the strategy proposed in this embodiment, 4 sets of parallel inverter systems identical to the previous simulation experiments were built on the StarSim HIL experimental platform, and the control effects of the parallel inverter system 1 were compared with those of the parallel inverter system 2, the parallel inverter system 3, and the parallel inverter system 4 respectively.

[0277] Comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 2 when the filter inductor parameters change:

[0278] The filter inductor changes stepwise from to at 0.06s and from stepwise to at 0.12s. Under the working conditions from Figure 14 and Figure 15 It can be seen that when the filter inductor undergoes two mutations, the waveforms of the capacitor voltages and load currents of the two are almost the same, and the amplitudes of the capacitor voltage (yellow line) of the parallel inverter system 1 and the capacitor voltage (blue line) of the parallel inverter system 2 are both or so; the amplitudes of the load current (purple line) of the parallel inverter system 1 and the load current (green line) of the parallel inverter system 2 are both or so. The above results show that both of them have strong robustness to the perturbation of the filter inductor parameters; however, it can be seen from Figure 14 and 15 that the ripple of the parallel inverter system 1 is significantly smaller than that of the parallel inverter system 2. Therefore, the method proposed in this embodiment can effectively weaken the chattering.

[0279] Comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 2 when the filter capacitor parameters change:

[0280] The filter capacitor changes stepwise from to at 0.06s and from stepwise to at 0.12s. Under the working conditions from Figure 16 and Figure 17It can be seen that when the filter capacitor parameters change before and after two mutations, the capacitor voltages and load current waveforms of the two are almost the same, and the amplitudes of the capacitor voltage (yellow line) of the parallel inverter system 1 and the capacitor voltage (blue line) of the parallel inverter system 2 are both about; the amplitudes of the load current (purple line) of the parallel inverter system 1 and the load current (green line) of the parallel inverter system 2 are both about. The above results show that both of them have strong robustness to the perturbation of the filter capacitor parameters; however, from Figure 16 and 17 it can be clearly seen that the ripple of the parallel inverter system 1 is significantly smaller than that of the parallel inverter system 2. Therefore, the anti-shaking of this embodiment is more effective.

[0281] Regarding the comparison of the response speeds of the parallel inverter system 1 and the parallel inverter system 2 when the droop control reference voltage changes suddenly:

[0282] Under the condition that the droop control reference voltage suddenly decreases from 220V to 110V at 0.1s, it can be seen from the partial enlarged view in Fig. 18 that after the droop control reference voltage changes suddenly, the convergence speed of the capacitor voltage tracking error (yellow line) of the parallel inverter system 1 is significantly faster than that of the capacitor voltage tracking error (blue line) of the parallel inverter system 2.

[0283] Regarding the comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 3:

[0284] Under the condition that the load suddenly increases and decreases by 2kW + 1kVar at 0.06s and 0.12s respectively, it can be seen from Figures 19 - 21 that before and after the two mutations of the load, the capacitor voltages of the parallel inverter system 1 and the parallel inverter system 3 hardly change, while their load currents can mutate correspondingly, indicating that the control method based on FTESO enables the parallel inverter system to have strong anti-load interference ability. At the same time, it can be clearly seen from Figure 21 that the capacitor voltage tracking error (blue) of the parallel inverter system 1 is significantly smaller than that of the capacitor voltage tracking error (yellow) of the parallel inverter system 3 . The above results show that this embodiment has higher control accuracy.

[0285] Regarding the comparison of the control effects of the parallel inverter system 1 and the parallel inverter system 4:

[0286] Under the condition that the load suddenly increases by 2KW + 1KVar at 0.1s, it can be seen from Figure 22 that before and after the load mutation at 0.1s, the observation error of the FTESO proposed in the present invention is about , which has higher observation accuracy compared with LESO ( ); at the same time, fromFigure 23 It can be seen that the capacitor voltage tracking error of the parallel inverter system 1 ( ) is smaller than that of the parallel inverter system 4 ( ). The above results verify that the FTESO in the parallel inverter system 1 has higher observation accuracy than the LESO in the parallel inverter system 4.

[0287] The Matlab / Simulink simulation and StarSim HIL experimental results show that the control method proposed in this embodiment can resist the internal and external disturbances of the parallel inverter system, enabling the parallel inverter system to have satisfactory response performance and achieve load power sharing; specifically:

[0288] (1) Compared with the LESO, the FTESO proposed in this embodiment has higher observation accuracy, and the strategy based on ESO feedforward compensation enables the parallel inverter system to completely resist load disturbances;

[0289] (2) The RNTSMC proposed in this embodiment can not only significantly reduce the chattering of the parallel inverter system and improve the control accuracy of the parallel inverter system, but also has strong robustness to filter parameter perturbations;

[0290] (3) Since the FTESO and RNTSMC can respectively make the observation error and the capacitor voltage tracking error converge within a finite time, this embodiment can enable the parallel inverter system to have a faster response ability.

Claims

1. A recursive non-singular terminal sliding mode control method for parallel inverter systems based on FTESO, characterized in that: The steps include: For the parallel inverter system composed of n inverters, a dynamic model of the inverter main circuit is constructed; In order to quickly estimate the disturbance d in a limited time and ensure the estimation accuracy, FTESO is designed; In order to enhance the robustness of the parallel inverter system to the variation of LC filter parameters and realize the finite time convergence and high precision control of the output voltage of the parallel inverter system, a first layer non-singular fast terminal sliding surface is designed. ; In order to further reduce the steady-state tracking error of the parallel inverter system, the second-layer recursive terminal sliding surface is designed. ; Based on the first layer non-singular fast terminal sliding surface and the second-layer recursive terminal sliding surface , design RNTSMC control law based on FTESO ; Design of feedforward control law based on FTESO ; Combined with RNTSMC control law and feedforward control law Get the control signal of the parallel inverter system .

2. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 1 is characterized in that: The expression of the inverter main circuit dynamic model is as follows: ; In the formula, is the first state variable; is the second state variable; for The derivative of for The derivative of is the filter inductor; is the filter capacity; is the DC input voltage; is the average duty cycle of the inverter, i.e., the control signal of the parallel inverter system; is the load disturbance; in, , , The expressions are: ; ; ; In the formula, is the capacitor voltage; for The derivative of for The derivative of is the load current; Specifically, set the load disturbance is bounded, and its derivative satisfies , is a bounded positive real number.

3. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 2 is characterized in that: The expression of FTESO is as follows: ; make ; ; In the formula, for An estimated value of is the third state variable; is the first gain of FTESO; is the second gain of FTESO; The third gain for FTESO; is a positive real number between 0 and 1; is a monotonically increasing smooth function, and its expression is , which can eliminate the sign function Caused by chattering; Load disturbance The first derivative of .

4. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 2 is characterized in that: The first non-singular fast terminal sliding surface The expression is as follows: ; In the formula, , are all positive real numbers; is a real number greater than 1; is the capacitor voltage tracking error; is the derivative of the capacitor voltage tracking error; in: ; ; The first-order derivative of is: ; The expression is as follows: ; In the formula, for The second derivative of Capacitor voltage reference value provided for droop control; for The second derivative of is the RNTSMC control law.

5. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 4 is characterized in that: The second recursive terminal sliding surface The expression is as follows: ; In the formula, is a positive real number; and are all positive odd numbers, and ; for Integration about time; Will Substituting the expression into the above formula, we can get: ; The first-order differential of the above equation is: ; In order to reduce the jitter of the parallel inverter system and ensure the convergence time of the parallel inverter system, the following parallel inverter system sliding mode reaching law is selected: : ; In the formula, The fourth gain for FTESO; The fifth gain for FTESO; is a positive real number between 0 and 1; , , are all positive real numbers.

6. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 5, characterized in that: The RNTSMC control law The expression is as follows: 。 7. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 3, characterized in that: The feedforward control law The expression is as follows: 。 8. The recursive non-singular terminal sliding mode control method for parallel inverter system based on FTESO according to claim 7, characterized in that: The control signal of the parallel inverter system The expression is as follows: 。

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