CHB converter harmonic compensation method based on Kalman filtering

Abstract

The invention provides a CHB converter harmonic compensation method based on Kalman filtering, and the method comprises the steps: constructing a continuous time state space model according to a topological structure of a CHB converter; discretizing the continuous time state space model to obtain a discrete time dynamic model; acquiring a voltage harmonic component of the CHB converter, and expanding a system state in the discrete time dynamic model according to the voltage harmonic component to obtain an expanded state space model; and carrying out harmonic error compensation design on the extended state space model through a Kalman filter to obtain a state observer dynamic model, and carrying out harmonic compensation on the CHB converter through the state observer dynamic model to carry out further control so as to solve the steady-state error problem of the CHB converter under the optimal control framework.

Description

Technical Field

[0001] This invention relates to the field of directional control technology, and more specifically, to a harmonic compensation method for CHB converters based on Kalman filtering. Background Technology

[0002] For current control strategies in series H-bridge (CHB) converters based on voltage-oriented control (VOC) methods, model predictive control (MPC) has gradually become a powerful alternative in the power converter field due to the rapid development of digital control platforms and the improvement of computing power. Compared with traditional controllers, MPC has advantages such as simple design, fast dynamic response, ease of handling multi-input multi-output systems, and the ability to directly introduce constraints and nonlinear factors. Therefore, it is widely regarded as an important development direction for future power converter control. To maintain the fast dynamic performance of MPC, an indirect MPC strategy has been proposed. In this method, the objective of the optimization problem is shifted from the switching state to the calculation of the modulation signal or duty cycle; in other words, a PWM stage is introduced into the predictive model instead of directly enumerating all switching combinations. This type of method reduces the computational burden and maintains good steady-state performance. In recent years, modulation-type MPC and sequential PS-PWM MPC have become important directions in CHB converter research, and experimental results show that they achieve a good balance between control accuracy and computational complexity.

[0003] However, the steady-state performance of both direct and indirect MPC is highly dependent on the accuracy of the prediction model. In practical applications, parameter uncertainties and unmodeled disturbances are unavoidable, such as parameter deviations in arm inductance and resistance, the impact of submodule capacitor voltage ripple, and dynamic changes in battery characteristics, all of which significantly reduce the performance of the optimal control strategy.

[0004] Against this backdrop, research on MMC (Multi-Mode Capacitor) indicates that when the instantaneous capacitor voltage ripple cannot be accurately measured, high-order harmonic components will appear in the circulating current, thus affecting the output current quality. To address this, a feedforward compensation method based on capacitor voltage measurement has been proposed, combined with online estimation of inductor parameters to reduce steady-state error. However, this type of method has extremely high requirements for sampling frequency, measurement noise, and real-time performance of capacitor voltage measurement, limiting its application in practical systems.

[0005] In the CHB-SL-BESS application of a secondary energy storage system based on a series H-bridge converter, the problem is even more complex. Because the internal resistance and capacity of the secondary batteries dynamically change over time during aging, the uncertainty of the model parameters is not only more significant but also exhibits marked nonlinear changes during charge-discharge cycles. This means that if the control strategy lacks the ability to compensate for dynamic parameter changes, steady-state current deviations and harmonic distortions will inevitably occur. Although some studies have proposed introducing online estimation methods for battery internal resistance or efficiency into the control system to mitigate these effects, such solutions often require additional measurement circuitry or high-bandwidth communication interfaces to acquire the voltage and current signals of each battery pack in real time, thus increasing system complexity and cost. Summary of the Invention

[0006] In view of this, the present invention proposes a harmonic compensation method for CHB converter based on Kalman filtering to solve the problems existing in the prior art.

[0007] To achieve the above objectives, this invention proposes a harmonic compensation method for CHB converters based on Kalman filtering, comprising: Based on the topology of the CHB converter, a continuous-time state-space model is constructed. Discretize the continuous-time state-space model to obtain a discrete-time dynamic model; Obtain the voltage harmonic components of the CHB converter, and extend the system state in the discrete-time dynamic model based on the voltage harmonic components to obtain an extended state-space model; Harmonic error compensation is designed for the extended state-space model using a Kalman filter to obtain a dynamic model of the state observer. The CHB converter is then subjected to harmonic compensation using the dynamic model of the state observer for further control.

[0008] Optionally, the continuous-time state-space model is:

[0009] in, Represents state variables, , , This represents the state coefficient matrix, input coefficient matrix, and disturbance input matrix. The output voltage of the CHB arm The resulting control input, It is a vector containing the PCC voltage.

[0010] Optionally, the discrete-time dynamic model is:

[0011]

[0012]

[0013]

[0014] Where k represents the number of steps, A represents the system matrix, B represents the input matrix, and M represents the disturbance input matrix. Represents a 3x3 identity matrix. Indicates the sampling period.

[0015] Optionally, the extended state-space model is:

[0016]

[0017] in, , , This represents the expanded state coefficient matrix, input coefficient matrix, and disturbance input matrix. This represents the expanded system state at time k. Indicates the corresponding control input, Represents the perturbation matrix. Indicates the output matrix. This is the system output.

[0018] Optionally, in the extended state-space model,

[0019]

[0020] in It includes the frequency that affects each arm. The vector of voltage perturbation in radians per second. This indicates the expanded system state. This represents a permutation of system states. This represents the voltage disturbance vector sequence, where k represents time and h represents the harmonic order. It is the fundamental frequency. It is its phase.

[0021] Optionally, the dynamic model of the state observer is:

[0022]

[0023] in, Indicates the predicted estimated state. This represents the predicted output, where k represents time. It is the observer gain matrix.

[0024] Optionally, for the observer gain matrix Calculated using a steady-state Kalman filter, where:

[0025]

[0026] in, To estimate the covariance matrix in steady state, These are the covariance matrices of process noise and sensor noise, respectively.

[0027]

[0028] in, and This represents the estimated voltage disturbance value of the modeling motor and the disturbance coefficient of the current sensor.

[0029] Optionally, the control input of the optimal control strategy of the state observer dynamic model for:

[0030] Among them, control input compensation Where D represents the perturbation input matrix, This represents the estimated perturbation value in the αβ coordinate system. This indicates the required CHB output voltage to maintain the robotic arm current under the desired steady-state operating conditions.

[0031] On the other hand, the present invention provides a harmonic compensation system for a CHB converter based on Kalman filtering, for performing the above-described method.

[0032] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention proposes a harmonic compensation method based on Kalman filtering (KF) to address the steady-state error problem of CHB converters under optimal control framework. This method utilizes an extended state-space model to jointly model arm current and harmonic voltage disturbances as system states. By designing a steady-state KF observer, instantaneous voltage disturbances can be estimated in real time from current measurements, and the estimated values ​​are fed back to the current prediction model, thereby improving steady-state current tracking performance and arm voltage reference. Compared with existing compensation methods, this KF strategy has two significant advantages: First, it can simultaneously suppress fundamental frequency errors caused by parameter mismatch and high-order harmonic interference caused by module capacitor voltage ripple, improving the compensation capability for dynamic parameter changes and enhancing control accuracy; second, this method has low hardware dependence, requiring only the measurement of average submodule capacitor voltage, and has low sampling frequency requirements, reducing system complexity and cost, and can be seamlessly integrated with existing BMS hardware architectures, thus making it more feasible in engineering. Attached Figure Description

[0033] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. In the drawings: Figure 1 This is a schematic diagram of the Δ-CHB converter topology in an embodiment of the present invention; Figure 2 This is a schematic diagram of the equivalent circuit of the enhanced state-space model of the CHB converter topology with perturbation in an embodiment of the present invention. Figure 3 This is a block diagram of the KF observer used for steady-state error compensation in an embodiment of the present invention.

[0034] Figure 4 The figure shows the experimental results under constant arm power reference values ​​P1=P3 = 500W, P2 = 200W, Q = 0VAr, and PR-based compensation strategy. Figure 5 The experimental results of the KF harmonic compensator are shown in the figure, with constant arm power reference values ​​P1=P3 = 500W, P2 = 200W, Q = 0VAr. Figure 6 The figure shows the experimental results of LQR for the Kalman filtering strategy under transient conditions. Detailed Implementation

[0035] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the disclosure to those skilled in the art. It should be noted that, unless otherwise specified, the embodiments and features described herein can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0036] This embodiment proposes a harmonic compensation method for CHB converters based on Kalman filtering, such as... Figure 1 As shown, it includes: This invention proposes a harmonic compensation method based on Kalman filtering (KF) to address the steady-state error problem of CHB converters under optimal control framework. This method utilizes an extended state-space model to jointly model arm current and harmonic voltage disturbances as system states. By designing a steady-state KF observer, instantaneous voltage disturbances can be estimated in real time from current measurements, and the estimated values ​​are fed back to the current prediction model, thereby improving steady-state current tracking performance and arm voltage reference. Compared with existing compensation methods, this KF strategy has two significant advantages: First, it can simultaneously suppress fundamental frequency errors caused by parameter mismatch and high-order harmonic interference caused by module capacitor voltage ripple, improving the compensation capability for dynamic parameter changes and enhancing control accuracy; second, this method has low hardware dependence, requiring only the measurement of average submodule capacitor voltage, and has low sampling frequency requirements, reducing system complexity and cost, and can be seamlessly integrated with existing BMS hardware architectures, thus making it more feasible in engineering.

[0037] To further verify the effectiveness of the proposed KF harmonic compensation strategy, this invention constructed an SL-BESS experimental platform based on a Δ-CHB converter. Under conditions of significant parameter uncertainty and measurement error, three different optimal control schemes were experimentally compared. Experimental results show that the proposed method can significantly reduce steady-state current error and output harmonic distortion while maintaining fast dynamic performance, providing a reliable and practical control solution for the high-performance operation of CHB-SL-BESS.

[0038] The above technical solution is described in detail below: CHB Converter Model and Optimal Control Design like Figure 1 As shown, the arm is made of n H-bridge switches (HB-SMs) are connected in series, with the inductance of the arm filter being L and the equivalent resistance of the arm being... rThe DC side of HB-SMs consists of a floating capacitor, which can be used independently or connected to a battery or DC-DC conversion stage, depending on application requirements, and is suitable for Δ-CHB converters. The AC side of HB-SMs has two pairs of power switches, allowing for three different output voltage levels. Each AC terminal is connected to one phase at the point of common coupling (PCC).

[0039] Continuous-time dynamic model To describe the current dynamics of the CHB converter, the arm current can be considered as the system state, i.e.:

[0040] in, Represents state variables, This represents the current at time t in the 1st, 2nd, and 3rd arms.

[0041] based on Figure 1 The CHB converter topology shown can be used to obtain the following continuous-time state-space model:

[0042] in The output voltage of the CHB arm The resulting control input, The control input represents the voltage formation under different arms. It is a vector containing the PCC voltage, where the subscripts a, b, and c represent different phases at the common coupling point. Furthermore,

[0043] in, , , This represents the state coefficient matrix, input coefficient matrix, and disturbance input matrix. This represents a 3x3 identity matrix.

[0044] For general n - Unit CHB converter arm, total output voltage This can be expressed as the individual output voltage of each HB-SM. The sum, such as

[0045] The number of CHB converters on each arm is n, and j represents the CHB converter number.

[0046] In a Δ-CHB converter, circulating current can be reduced without affecting the grid current. This circulating current is injected into the arm current. This circulating current allows for control of power imbalances between arms, enabling different power references for each arm while maintaining balanced AC current at the converter output. The relationship between arm current, grid current, and circulating current can be expressed by the following formula:

[0047] in Represents the coefficient matrix and , , , This represents the three currents in the power grid.

[0048] Discrete-time dynamic model To implement the optimal control strategy on a digital control platform, it is typically necessary to establish a discrete-time dynamic model of the CHB converter arm current. Therefore, zero-order hold (ZOH) discretization can be applied to (3.2) to obtain a discrete-time state-space model with a sampling period of [missing information]. Its expression is:

[0049] and

[0050]

[0051]

[0052] Furthermore, at each sampling time, the arm output voltage can be expressed as .

[0053] Where k represents the number of steps, A represents the system matrix, B represents the input matrix, and M represents the disturbance input matrix.

[0054] Therefore, the optimal control strategy for controlling the CHB converter defines the allowable range of values ​​for the arm voltage, which serves as the control input. For example, in the FCS-MPC strategy, the control input... It is typically considered as a discrete voltage level that can be obtained at the arm output voltage, i.e. , where discrete sequence Conversely, in the optimal control strategy considering the modulation phase, the possible arm output voltages consist of a set of continuous values, defined as... , This represents the stable voltage. In either case, (3.6) can serve as the predictive model for the CHB converter in the standard optimal control strategy.

[0055] Optimal control problem and steady-state input design Several standard FCS-MPC and indirect MPC strategies are based on the computational ability to minimize the following cost function. Optimal control input

[0056]

[0057] in This is the required CHB output voltage to maintain the robotic arm current under the desired steady-state operating conditions. Therefore, the weighting factor... This allows designers to adjust the closed-loop performance of the controller; the superscript * indicates the parameters required for steady-state operation. This represents the weighting coefficient.

[0058] It should be noted that when the system state is close to its reference value, that is... At this point, the first term of the cost function (3.10) is almost zero. Therefore, the second term becomes the dominant term. track In steady state. In this sense, the appropriate steady-state reference voltage for the CHB converter arm can be obtained by simply replacing the system state reference in the dynamic model (3.6) and solving the system of equations. As shown below:

[0059] Substituting the model matrices (3.7), (3.8), and (3.9) into (3.11) yields:

[0060] Auxiliary variables Furthermore, the arm current reference value can be obtained by taking into account the grid current reference. and circulating current reference The calculation method is as follows:

[0061] Among them, the grid current reference can be based on traditional p - q The theoretical reference can be obtained by following the time-active power balance strategy proposed in the paper.

[0062] Ideally, (3.12) should perfectly track the required steady-state arm current reference value. However, in practical applications, the steady-state current may deviate from its reference value due to prediction model errors. These errors may stem from model parameter uncertainties, measurement errors, and discretization approximations. Therefore, the proposed Kalman filter strategy aims to compensate for these errors, improve the current prediction and steady-state control input reference in the dynamic model of the CHB converter, thereby enhancing the performance of the standard optimal control scheme.

[0063] The impact of capacitor voltage ripple on control performance As can be seen from (3.12), r and L The parameter errors and the measurement errors of the PCC voltage directly affect the reference value of the steady-state control input. However, the ripple of the SM capacitor voltage is also an additional cause of steady-state error, depending on... Application in the SM converter. In some traditional CHB converter optimization control strategies, the impact of SM capacitor voltage fluctuations on the possible arm output voltage level is ignored, or this is also true during the modulation stage, for FCS-MPC and indirect MPC strategies. Therefore, steady-state errors will appear in the arm output voltage, and these errors are proportional to the sum of the amplitudes of the SM capacitor voltage fluctuations, as analyzed in detail below.

[0064] Assuming the arm voltage is evenly distributed among the SMs, and the SM capacitor voltage is equal to the average BP voltage of the balancing SoC, then the actual steady-state output voltage of each HB-SM is... for:

[0065] in This is the instantaneous voltage of the SM capacitor. This represents the average BP voltage. The arm designation indicates the arm number; in this analysis, it is assumed that the voltage of each SM is equal. Therefore, substituting (3.14) into (3.4) yields the arm output voltage. It can be represented as:

[0066] Considering the balanced SM capacitor voltage From DC voltage value and SM capacitor ripple Composition, that is Substituting this expression into (3.15), we get:

[0067] in This indicates the output voltage error of the arm caused by the sum of voltage ripple from the SM capacitor.

[0068] Due to the single-phase nature of the CHB converter arm, the voltage ripple of the SM capacitor inherently contains oscillations with frequencies at twice the fundamental frequency and integers thereof. However, harmonics above the fourth order are generally negligible in the voltage ripple of the SM capacitor. Therefore, the sum of the voltage ripple of the SM capacitor can be approximated as:

[0069] in, This represents the amplitude of the sum of the second-order ripple components of all capacitor voltages on the x-th bridge arm. Let x represent the amplitude of the sum of the fourth ripple components of all capacitor voltages on the x-th bridge arm. This indicates the phase angle.

[0070] Furthermore, substituting (3.17) into (3.16) yields an approximate expression for the instantaneous arm output voltage error:

[0071] in . Let the modulation index be denoted. Finally, using the trigonometric identity expansion of the product of sines into a sum (3.18), we obtain:

[0072] Therefore, neglecting the SM capacitor voltage ripple when applying the optimal control input of CHB will lead to voltage errors, which can be decomposed into the main first, third, and fifth harmonic components. These voltage errors propagate through the CHB arm current, increasing the total harmonic distortion (THD) and affecting the steady-state performance of the converter.

[0073] Although the BMS measures the BP battery voltage for safety reasons, the sampling time of this embedded system is typically around 1Hz, and its measurements usually involve a low-pass filtering stage. Therefore, these measurements cannot be effectively utilized by the Δ-CHB converter current controller, and an additional SM capacitor voltage sensor is required to compensate for the SM capacitor voltage ripple through a similar feedforward term, which increases the hardware complexity and cost of the converter.

[0074] Steady-state error compensation method based on Kalman filtering: Kalman Filtering Principles and State Estimation This section introduces a KF harmonic compensator for unbiased optimal control of CHB converter current. The proposed KF strategy achieves this by estimating the voltage drops in the converter arms, which the standard converter dynamic model (3.6) cannot describe. Therefore, the estimated voltages can be used to enhance the steady-state control input reference in the optimal control scheme and improve current prediction, eliminating steady-state current errors even when the ripple of the SM capacitor is not measured.

[0075] Suppose that the symbol used in (3.19) is The core idea of ​​the proposed Kalman filter steady-state error compensation strategy is to introduce a sinusoidal voltage disturbance model composed of multiple harmonic components, which affects the state dynamics of each arm current. These voltages represent the equivalent voltage drops on the CHB converter arms that are not considered in the standard converter model (3.6), and may be caused by modeling errors or external disturbances. A steady-state Kalman filter is then designed to estimate these voltage drops and predict the arm current values.

[0076] As a general case, sinusoidal voltage harmonic components with constant amplitude and frequency can be represented in a single-phase αβ coordinate system as:

[0077]

[0078] in This represents the h-th harmonic voltage disturbance of the α-axis component in the stationary αβ coordinate system. Let d represent the h-th harmonic voltage disturbance of the β-axis component in the stationary αβ coordinate system, where d represents the amplitude of the harmonic voltage disturbance and h represents the harmonic order. It is the fundamental frequency. Its phase indicates that different voltage harmonic components are not necessarily the same. Furthermore, by analyzing (3.20) and (3.21)... Differentiating the voltage yields the dynamic equation describing this harmonic component:

[0079] in, express differential, This represents the h-th harmonic voltage disturbance in the stationary αβ coordinate system.

[0080] In this way, the zero-order hold method can be used to discretize (3.22) to obtain a dynamic model of the voltage disturbance harmonic components affecting the armature current, i.e.:

[0081]

[0082] As analyzed in the previous section, to completely eliminate the steady-state error of the armature current, voltage disturbances consisting of the first, third, and fifth harmonics must be compensated in each armature. Therefore, the following extended system states are defined:

[0083]

[0084] in h=1, 3, 5, is the frequency that affects each arm. The vector of voltage per second in radians affects a delta-bridge multilevel converter. The extended state-space model represents the extended system state. Furthermore, considering (3.25) and the dynamic models (3.6) and (3.23), the proposed extended state-space model describes the dynamic relationship between the converter and the disturbance coupling, and can be expressed as:

[0085]

[0086] and

[0087]

[0088] and and output matrix D represents the perturbation input matrix. Represents a partial matrix. Represents the state transition matrix. Represents a submatrix. This represents the perturbation matrix. Finally, as... Figure 2 As shown.

[0089] exist Figure 2 The equivalent continuous-time circuit for enhancing the state-space model of the CHB converter is provided, in which... .

[0090] Error Compensation Strategy Design Based on Kalman Filter The observability matrix of the proposed linear extended state-space model (3.30) is full rank. Therefore, a state observer can be designed to estimate the robot arm voltage disturbance based on the robot arm current measurement. Thus, consider the following dynamic model of the state observer:

[0091]

[0092] in It is the observer gain matrix. Indicates the predicted estimated state. This represents the predicted output. This matrix must be designed to ensure the closed-loop observer matrix... It is Schur stable, meaning that the norm of all its eigenvalues ​​is strictly less than one. Therefore, this paper proposes to use a steady-state Kalman filter to calculate the observer gain matrix. This can be obtained offline by solving the following discrete-time algebraic Riccati equation. :

[0093]

[0094] in To estimate the covariance matrix in steady state, These are the covariance matrices of process noise and sensor noise, respectively.

[0095] The process noise covariance matrix represents the uncertainty in the dynamic model (3.25), reflecting the modeling error that cannot be represented by the proposed voltage harmonic interference model. Conversely, the sensor noise covariance matrix reflects the variability of the current sensor measurements. Since the extended system state consists of motor current and sinusoidal voltage interference, and considering that the motor current is the only measurement output used for state estimation, the covariance matrix of the proposed Kalman filter is defined as follows:

[0096] in and These are positive constants, used to model the relative uncertainties of the motor voltage interference estimate and the current sensor, respectively.

[0097] First, to design a KF observer, one can perform numerous measurements on a current sensor in an experimental setup, fix the input values, and then calculate the covariance of the dataset to obtain... The value can then be adjusted. The value is used to achieve the ideal closed-loop performance of the state observer.

[0098] smaller A high value implies that the predictive model is very accurate. Therefore, the observer relies more on the model's predictions rather than rapidly adjusting the state estimate of the voltage disturbance based on the current prediction error. Conversely, a larger value indicates a more accurate prediction. A value indicating significant uncertainty in the dynamic model leads to larger corrections by the observer based on the estimation error. However, a larger value... The value also increases the extent to which sensor noise is transmitted into the state estimation.

[0099] Finally, note that modifications are possible. To avoid numerical rounding errors in the digital control platform when adjusting the observer bandwidth or obtaining near-zero values ​​in experimental measurements. However, it must be considered that increasing... A higher value of ...

[0100] like Figure 3As shown, the predicted voltage disturbance estimate can be extracted from the extended system state and used to calculate the required control input compensation to offset its impact on the robot arm's output voltage, as detailed below:

[0101] Adding the control input compensation (3.34) to (3.12) yields the compensated steady-state control input reference value, which includes the overall modeling error of the CHB converter. Therefore, the control input required for the optimal control strategy needs to be... for:

[0102] at last, Figure 3 The implementation block diagram of the proposed KF strategy is shown. Note that implementing this KF requires not only the use of an enhanced steady-state control input reference (3.35), but also the input of the estimated predicted current into the optimal control, since the state prediction takes into account the effects of unmeasured disturbances.

[0103] Please note that in the FCS-MPC scheme, predictions must be made using model (3.30) to achieve standard delay compensation techniques.

[0104] Main parameters of the experimental platform Experimental verification Experimental platform and testing scheme The main parameters of the experimental platform provided are shown in Table 1. Table 1

[0105] Experiments were conducted to analyze the performance of the proposed KF harmonic compensator. A Δ-CHB converter was set up. The device consisted of a CHB-SL-BESS prototype, in which the HB-SM was directly connected to the battery pack. These battery packs were assembled from 18650 lithium-ion batteries from recycled electric bicycle batteries using 24S2P and 24S3P battery configurations. The main parameters of the experimental setup and controller are summarized in Table 1.

[0106] In addition, the experimental setup included the grid simulator REGATRON TC30.528.43-ACS and the OPAL-RT OP4510 control platform, in which optimal current control and the proposed KF strategy were implemented, both running on the same CPU core.

[0107] Steady-state performance test and benchmark comparison To evaluate the steady-state performance of the proposed KF harmonic compensator, observer experiments were conducted with three different optimal control schemes: 1) Finite Control Set Model Predictive Control (FCS-MPC), 2) Phase Shift Modulation-Based Model Predictive Control (PS-MPC), and 3) Standard Linear Quadratic Regulator (LQR). Furthermore, these controllers employed a PR-based compensation strategy to compare their performance with that of the proposed KF harmonic compensator. Table 2 lists the main parameters of the optimal controllers and their corresponding observers.

[0108] Table 2

[0109] To increase the modeling error of the experiments sought in this benchmark analysis, the converter parameters were modified. The values ​​r = 1Ω and L = 5mH were assigned to the optimal current controller and observer, introducing 100% and 50% parameter errors for the resistor and inductor, respectively. Furthermore, the FCS-MPC and LQR strategies did not measure the SM capacitor voltage, instead assuming each SM capacitor voltage was equal to 80V. This assumption is not feasible for the PS-MPC strategy, as it requires the SM capacitor voltage to achieve the SM power imbalance required for SL-BESS applications. However, only the average SM capacitor voltage was passed to the PS-MPC strategy, achieved by applying a low-pass filter with a cutoff frequency of 1Hz to the SM capacitor voltage measurements.

[0110] like Figure 4 As shown, in the experimental results under constant arm power reference values ​​P1=P3=500W, P2=200W, Q=0VAr, and the PR-based compensation strategy, (a)-(e) show the results of FCS-MPC, (f)-(j) show the results of PS-MPC, and (k)-(o) show the results of LQR. The last row shows the i values ​​with and without the compensation strategy. a Harmonic spectrum Experimental results show Figure 4 In the middle, compensation based on PR; in Figure 5 The proposed KF strategy is shown in the figure. Steady-state compensation is activated after the 30ms test in each experiment. Therefore, the first row of both figures, showing the converter current tracking, clearly demonstrates that the standard optimal controller without steady-state error compensation cannot effectively track the current reference value due to interference caused by model parameter mismatch and SM capacitor voltage fluctuations. Details of each experiment will be provided in subsequent sections.

[0111] FCS-MPC This section compares two different FCS-MPC strategies. First, an FCS-MPC scheme was implemented. This FCS-MPC strategy uses an additional PR controller in each branch to enhance the current prediction model and compensate for steady-state errors.

[0112] like Figure 5 The constant arm power reference values ​​shown are P1=P3=500W, P2=200W, Q=0VAr, and the experimental results of the KF harmonic compensator are shown in (a)-(e) for FCS-MPC, (f)-(j) for PS-MPC, and (k)-(o) for LQR. The last line shows the i values ​​with and without compensation strategy. a Harmonic spectrum The experimental results of this control scheme are as follows: Figure 4 As shown in (a)-(e) in the figure. Secondly, the FCS-MPC using the proposed KF harmonic compensator was also tested, and the experimental results are shown in […]. Figure 5 (a)-(e).

[0113] CHB converter current for each strategy, such as Figure 4 Figures 5(a) and 5(a) show that these steady-state error compensation techniques effectively reduce the current tracking error at the fundamental frequency.

[0114] also, Figure 4 (e) and (e) of 5 demonstrate the i of their respective FCS-MPC strategies a Harmonic spectrum. These figures show that, before enabling the steady-state error compensation strategy, the FCS-MPC scheme exhibits the lowest third and fifth harmonic interference components in the output current, compared to PS-MPC and LQR. Therefore, the FCS-MPC strategy is less sensitive to high-frequency interference caused by SM capacitor voltage fluctuations.

[0115] The proposed KF harmonic compensator outperforms the PR-based strategy in suppressing the third and fifth voltage harmonic components, reducing the total current harmonic distortion (WTHD) from 0.7% to 0.55%. However, the FCS-MPC strategy exhibits a low WTHD, likely due to its different cost function that only penalizes the current tracking error, resulting in a more aggressive current tracking approach.

[0116] Based on these results, it can be concluded that the PR and KF-based compensation strategies provide similar steady-state performance in the FCS-MPC scheme. In this case, even assuming the SM capacitor voltage remains constant, compensating for voltage disturbances caused by higher-order harmonics may not be critical for steady-state error compensation at high sampling frequencies.

[0117] PS-MPC The M 2 The experimental results of PC technology are in Figure 4 (f)-(j) and Figure 5Compensation strategies based on PR and KF are shown in (f)-(j), respectively. Although a high sampling frequency of 12kHz is used in PS-MPC, its steady-state performance is severely affected because the average SM capacitor voltage is considered instead of the actual measured SM capacitor voltage when calculating the optimal modulation signal. Figure 4 (j) and Figure 5 The harmonic current spectrum in (j) shows that, without a compensation strategy, there are significant harmonic components at 150Hz and 250Hz. This disadvantage of PS-MPC compared to FCS-MPC stems from the higher weighting factor. λ u This is to address higher frequency measurement noise. Therefore, under this control scheme, Calculation errors are more likely to lead to a decrease in steady-state current tracking performance.

[0118] PR-based compensation strategies cannot effectively mitigate third and fifth harmonic voltage disturbances. Therefore, these disturbances will behave like... Figure 4 As shown in (j), it propagates into the armature current. On the other hand, the proposed KF harmonic compensator enables the PS-MPC strategy to completely suppress the fundamental and higher-order voltage disturbances. Therefore, the proposed KF harmonic compensator outperforms the PR-based compensation strategy in this optimal control scheme, reducing the total harmonic distortion (THD) of the output current and the weighted THD by 1.29% and 0.6%, respectively [see...]. Figure 4 [i in 5 and (i in 5)].

[0119] A linear quadratic regulator (LQR) was implemented to obtain the modulation voltages of the CHB converter arms. These voltages were applied to the HB-SM by implementing the proposed OVA-PS-PWM strategy and sampling technique introduced in the previous chapter. The proposed modulation strategy allows for current control at a lower sampling frequency of 4kHz while maintaining the same PWM carrier frequency as the PS-MPC strategy. However, the slower sampling frequency and the assumption that the SM capacitor voltage is constant affect the steady-state performance of the LQR without compensation. In fact, the LQR without steady-state error compensation performs the worst in suppressing interference from the third and fifth harmonic voltages, such as... Figure 4 As shown in (o) in 5 and (o) in 6.

[0120] After enabling the steady-state compensation strategy, the results obtained are similar to those of PS-MPC. On the one hand, the LQR based on the PR compensation strategy proposed in the prior art cannot compensate for high-order harmonic interference in the input converter current. On the other hand, the LQR using the proposed KF harmonic compensator significantly reduces these voltage interferences.

[0121] Table 3 summarizes the amplitudes of the third and fifth harmonic components for each control strategy. These results demonstrate that the proposed KF harmonic compensator outperforms the uncompensated and PR-based schemes in suppressing interference at these specific frequencies and is applicable to each optimal control scheme.

[0122] Table 3

[0123] Furthermore, Table 4 summarizes the output current harmonic distortion (THD), root mean square error (RMSE) of armature current reference tracking, and computational burden observed on the control platform for each optimal control strategy in terms of steady-state performance. Table 4 and... Figure 5 As shown in (a) of the table, among all tested control strategies, the LQR strategy using the KF harmonic compensator combined with the OVA-PS-PWM stage achieved the lowest RMSE and output current THD. However, as can be seen from Table 3, when combined with the KF strategy, the amplitudes of the third and fifth harmonic components of the PS-MPC and LQR schemes are the same. Therefore, LQR performs slightly better than PS-MPC.

[0124] Table 4

[0125] Through the modulation stage, OVA-PS-PWM provides a variable PWM carrier phase offset angle, improving the total harmonic distortion (THD) of the output voltage by minimizing harmonic components at twice the carrier frequency and its harmonics. This modulation strategy is superior to traditional PS-PWM, especially when there are differences in the DC voltage or AC modulation signal of HB-SMs, as shown in the previous chapter, which is also the case for the SL-BESS prototype used in this study.

[0126] Regarding the computational burden of each control scheme (see Table 4), it is important to emphasize that the difference in execution time between the PR-based steady-state error compensation strategy and the proposed KF harmonic compensator is negligible, less than 1 microsecond. This result is attributed to the lower computational complexity of the proposed steady-state KF, where the observation gain matrix is ​​constant and computed offline. Furthermore, the dimension of the extended model matrices (24) is independent of the number of SMs, and these matrices are sparse. Therefore, the implementation of state prediction and correction can be optimized to reduce the number of floating-point operations.

[0127] like Figure 6 As shown, in the LQR experimental results of the Kalman filtering strategy under transient conditions, The transient response of the ad circulating current step change; the eh active power reference change; the jl transient process of the power flow reversing from -1.2kW to 1.2kW; the experiments show that the proposed KF harmonic compensator can eliminate the steady-state current error of the CHB converter in both direct and indirect MPC schemes, even with significant parameter uncertainties and SM capacitor voltage measurement errors. Furthermore, the proposed KF harmonic compensator outperforms existing PR-based schemes in modulation-optimal control techniques because it achieves major suppression of the desired high-order voltage harmonic components.

[0128] Transient response performance analysis To evaluate the dynamic response of the proposed KF strategy, three transient tests were conducted. Due to its improved steady-state performance, the LQR current control strategy was adopted in this section. It was assumed that the SM capacitor voltage equaled the average battery voltage. The first transient test introduced inter-arm power imbalance through a cyclic current transient. The inter-arm power was modified from P1=P2=P3=400W to P1=500W, P2=200W, and P3=500W, while maintaining a constant output power of 1.2kW. The second experiment involved a linear change in the three-phase active power reference from 0kW to 1.2kW, and the third test involved a power flow reversal from -1.2kW to 1.2kW. In each experiment, the change in each reference current was introduced at t=30ms. The LQR current control combined with the proposed KF harmonic compensator exhibited satisfactory dynamic performance in each test. These experiments demonstrate that the proposed KF strategy does not increase the overshoot of the current control and enables the controller to reach the current reference value without steady-state error or harmonic distortion caused by SM capacitor voltage ripple.

[0129] This invention addresses the operational requirements of CHB converters in cascaded battery energy storage systems (SL-BESS) by proposing and validating a harmonic compensation strategy based on Kalman filtering (KF) to achieve unbiased optimal current control. This method is based on an extended linear state-space model, explicitly introducing different frequency harmonic components affecting the dynamics of the electrical current into the modeling, treating them as equivalent sinusoidal voltage disturbances. By designing a steady-state KF observer, these equivalent disturbances can be estimated in real time without directly measuring the capacitor voltage ripple of the submodules, and the steady-state input reference of the optimal controller can be corrected accordingly, thereby effectively improving the accuracy of electrical current prediction and current tracking performance.

[0130] In the experimental verification phase, this chapter presents a systematic test based on a three-phase CHB-SL-BESS prototype, employing three optimal control strategies as comparison platforms: Finite Control Set Model Predictive Control (FCS-MPC), Modulation-Based Predictive Control (PS-MPC), and Linear Quadratic Regulator (LQR). The experimental results clearly demonstrate that the proposed KF harmonic compensator can significantly reduce steady-state errors caused by parameter uncertainties, modeling errors, and SM capacitor voltage measurement deviations, while maintaining fast response without introducing overshoot in terms of dynamic performance. In other words, this strategy balances steady-state accuracy and transient performance, exhibiting good robustness.

[0131] Further analysis shows that, under the modulation-optimal control framework, the KF compensator outperforms the traditional PR controller-assisted compensation scheme. Particularly when combined with the OVA-PS-PWM modulation method proposed in Chapter 2 of this paper, the LQR controller exhibits the best overall performance, with its output current showing superior total harmonic distortion (THD) and weighted harmonic distortion (WTHD) compared to other control schemes. Notably, even under the assumption of a constant SM capacitor voltage, this combined strategy maintains excellent harmonic suppression capability, further demonstrating the effectiveness and stability of the KF method under non-ideal conditions.

[0132] In summary, the KF harmonic compensation strategy provides a practical solution for improving the steady-state performance of CHB converters under parameter uncertainty and disturbance environments. This method not only integrates naturally with existing optimal control frameworks such as FCS-MPC, PS-MPC, and LQR, but is also applicable to different control structures with and without modulation. Furthermore, because it relies solely on low-frequency sampling of the average submodule capacitor voltage, eliminating the need for high-speed sensors and complex communication interfaces, the proposed scheme significantly reduces the complexity in terms of hardware implementation and measurement requirements. This characteristic is particularly important for the practical engineering application of SL-BESS, laying a solid foundation for the subsequent realization of efficient, low-cost, and safe energy storage system control.

[0133] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A harmonic compensation method for a CHB converter based on Kalman filtering, characterized in that, include: Based on the topology of the CHB converter, a continuous-time state-space model is constructed. Discretize the continuous-time state-space model to obtain a discrete-time dynamic model; Obtain the voltage harmonic components of the CHB converter, and extend the system state in the discrete-time dynamic model based on the voltage harmonic components to obtain an extended state-space model; Harmonic error compensation is designed for the extended state-space model using a Kalman filter to obtain a dynamic model of the state observer. The CHB converter is then subjected to harmonic compensation using the dynamic model of the state observer for further control.

2. The method according to claim 1, characterized in that, The continuous-time state-space model is as follows: ， in, Represents state variables, , , This represents the state coefficient matrix, input coefficient matrix, and disturbance input matrix. The output voltage of the CHB arm The resulting control input, It is a vector containing the PCC voltage.

3. The method according to claim 2, characterized in that, The discrete-time dynamic model is as follows: ， ， ， ， Where k represents the number of steps, A represents the system matrix, B represents the input matrix, and M represents the disturbance input matrix. Represents a 3x3 identity matrix. Indicates the sampling period.

4. The method according to claim 1, characterized in that, The extended state-space model is as follows: ， ， in, , , This represents the expanded state coefficient matrix, input coefficient matrix, and disturbance input matrix. This represents the expanded system state at time k. Indicates the corresponding control input, Represents the perturbation matrix. Indicates the output matrix. This is the system output.

5. The method according to claim 1, characterized in that, In the extended state-space model, ， ， in It includes the frequency that affects each arm. The vector of voltage perturbation in radians per second. This indicates the expanded system state. This represents a permutation of system states. This represents the voltage disturbance vector sequence, where k represents time and h represents the harmonic order. It is the fundamental frequency. It is its phase.

6. The method according to claim 1, characterized in that, The dynamic model of the state observer is as follows: ， ， in, Indicates the predicted estimated state. This represents the predicted output, where k represents time. It is the observer gain matrix.

7. The method according to claim 6, characterized in that, For the observer gain matrix Calculated using a steady-state Kalman filter, where: ， ， in, To estimate the covariance matrix in steady state, These are the covariance matrices of process noise and sensor noise, respectively. ， in, and This represents the estimated voltage disturbance value of the modeling motor and the disturbance coefficient of the current sensor.

8. The method according to claim 6, characterized in that, The control input of the optimal control strategy of the state observer dynamic model. for: ， Among them, control input compensation Where D represents the perturbation input matrix, This represents the estimated perturbation value in the αβ coordinate system. This indicates the required CHB output voltage to maintain the robotic arm current under the desired steady-state operating conditions.

9. A harmonic compensation system for a CHB converter based on Kalman filtering, characterized in that, Used to perform the method described in any one of claims 1-8.

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