Motor propulsion system fault-tolerant control method under ship power grid failure
Patent Information
- Application Number
- CN202610812811.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2026-06-08
- Publication Date
- 2026-08-28
- Estimated Expiration
- 2046-06-08
AI Technical Summary
传统以PI为主的矢量控制对参数漂移与外扰敏感,难以在强扰动下兼顾快速性与约束安全;单纯MPC虽能处理约束,但在模型失配、测量噪声和电网扰动强烈时性能下降
(1)将转子电流、转子侧输出电压与直流母线电压等安全约束显式纳入优化,降低过流与过压风险;
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Figure CN122339343B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of integrated ship power systems and motor drive control technology, specifically relating to a fault-tolerant model predictive control method for fused sliding mode control (SMC) of a doubly fed asynchronous induction motor (DFIM) rotor-side converter (RSC), which is used to improve the fault ride-through capability of the DFIM under sudden voltage drop / rise disturbances in the ship's power grid. Background Technology
[0002] Shipboard electrical networks have limited capacity, high equivalent impedance, and frequent load fluctuations. Switching between high-power propulsion loads, deck machinery, and pulse loads can easily cause sudden drops or rises in AC bus voltage. DFIM (Dual-Fault Induction Machine) relies on back-to-back converters to achieve wide-range speed regulation, with the rotor-side converter directly determining electromagnetic torque and current dynamics. During voltage surges, DFIM is prone to rotor-side overcurrent, DC bus overvoltage, torque surges, and control instability. Traditional PI-based vector control is sensitive to parameter drift and external disturbances, making it difficult to balance speed and constraint safety under strong disturbances. While simple MPC (Multi-Process Control) can handle constraints, its performance degrades under model mismatch, measurement noise, and strong grid disturbances. Therefore, a rotor-side fault-tolerant control method that combines explicit constraint handling with robust disturbance rejection is needed. Summary of the Invention
[0003] The objective of this invention is achieved through the following technical solutions.
[0004] To address the aforementioned issues, this invention proposes a fault-tolerant MPC fault ride-through control method integrating SMC. The core idea is that MPC is responsible for "optimal decision-making under constraints," while SMC is responsible for "robust compensation for model mismatch and external disturbances." Furthermore, fault classification triggers weight / constraint reconstruction to adapt to voltage sag / boost conditions. In this invention, x(k) represents the rotor current state variable. It is the optimal control input obtained by solving the rotor-side model predictive controller, which essentially corresponds to the rotor voltage vector or equivalent modulation amount that RSC should apply; on the other hand... It is a sliding mode compensation quantity constructed based on rotor current error, used to suppress model mismatch and external disturbances. The two are superimposed to form the fault period control quantity, which is then fused with the normal control quantity through a smooth switching mechanism to finally generate the synthetic control law u(k) applied to the rotor-side converter.
[0005] Specifically, this invention discloses a fault-tolerant control method for a motor propulsion system under ship electrical grid faults, comprising: firstly, acquiring stator voltage, frequency, DC bus voltage, and rotor three-phase current; obtaining rotor current components and speed information in a synchronous rotating coordinate system through coordinate transformation; and calculating voltage change and rate of change within a sliding window to achieve voltage sag / surge detection and fault classification. Secondly, constructing an MPC rolling optimization problem based on the DFIM discrete prediction model, explicitly writing constraints of rotor current, output voltage, DC bus voltage, and electromagnetic torque rate of change into the constraint set, and reconstructing the cost function weights and constraint upper limits online according to the fault level. Simultaneously, constructing a sliding mode surface with rotor current tracking error as the core and introducing a sliding mode compensation term in the form of a saturated function to form a synthetic control law.
[0006] Compared with the prior art, the present invention has at least the following beneficial effects: (1) Explicitly incorporate safety constraints such as rotor current, rotor-side output voltage and DC bus voltage into the optimization to reduce the risk of overcurrent and overvoltage; (2) Fault classification online reconfiguration weights and constraint upper limits to achieve coordinated optimization of reactive power support and active power limiting during voltage sag; (3) Sliding mode compensation suppresses parameter drift and external disturbances, improves control robustness and reduces secondary shocks during the recovery phase. Attached Figure Description
[0007] 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. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings: Figure 1 This is a schematic diagram of the overall flow of the control method for rotor-side RSC according to the present invention.
[0008] Figure 2 This is a schematic diagram of the voltage sag / boost discrimination and fault classification, and weight / constraint reconstruction process.
[0009] Figure 3 This is a schematic diagram of the solution and synthesis control process for fault-tolerant MPC fused with SMC.
[0010] Figure 4 This diagram illustrates the relationship between current / voltage constraints and protection coordination during rotor-side fault ride-through.
[0011] Figure 5 This is the main circuit topology diagram of the DFIM doubly fed asynchronous induction motor back-to-back converter.
[0012] Figure 6This is a block diagram of the rotor-side fault-tolerant control of a doubly fed asynchronous induction motor (DFIM) based on model predictive control (MPC) and sliding mode compensation (SMC).
[0013] In Figure 7(a), the given current i on the d-axis of the pure PI rotor side is... dr * and feedback actual current i dr Follow the waveform diagram; Figure 7(b) shows the predicted control rotor-side d-axis current i using the fault-tolerant model with integrated SMC. dr * and feedback actual current i dr Follow the waveform.
[0014] In Figure 8(a), the q-axis current i on the pure PI rotor side is given. qr * and feedback actual current i qr Following the waveform diagram, Figure 8(b) shows the predicted control rotor-side q-axis current i using the fault-tolerant model with fused SMC. qr * and feedback actual current i qr Follow the waveform.
[0015] Figure 9 The demonstration shows the three-phase grid (stator) voltage V using a traditional pure PI algorithm on the rotor side and a fault-tolerant model predictive control fused with SMC. abc Waveform diagram of the change.
[0016] Figure 10 The display shows the grid-side active power P gc Grid-side reactive power Q gc Active power P on the machine side s Reactive power Q on the machine side s Waveform diagram of the change.
[0017] Figure 11(a) shows the total active power P on the rotor side using the traditional pure PI algorithm. t Total reactive power Q t The waveform diagram shows the total active power P on the rotor side predicted by the fault-tolerant model of the fused SMC. t Total reactive power Q t Waveform diagram of the change.
[0018] Figure 12(a) shows the mechanical torque T on the rotor side using the traditional pure PI algorithm. m and electromagnetic torque T e The waveform diagram, in Figure 12(b), shows the rotor-side mechanical torque T predicted by the fault-tolerant model fused with SMC. m and electromagnetic torque T e Waveform diagram of the change. Detailed Implementation
[0019] 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 so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.
[0020] Figure 1 This is a schematic diagram of the overall flow of the control method for rotor-side RSC of the present invention. The method specifically includes: 1. System and Rotor-Side Controlled Objects In one embodiment, the energy link of the DFIM propulsion system is: shipboard electrical grid → grid-side converter (GSC) → DC bus → rotor-side converter (RSC) → DFIM → propeller propulsion load. This invention focuses on fault ride-through control of the RSC under voltage sag / surge disturbances: using rotor current in a synchronous rotating coordinate system. , The core controlled variable is the RSC output voltage vector or its equivalent modulation quantity as the control input, and the gate drive signal is generated through PWM / SVPWM.
[0021] 2 Voltage sag / dip detection and classification (S1-S2) Figure 2 This is a schematic diagram of the voltage sag / boost discrimination and fault classification, and weight / constraint reconstruction process.
[0022] To quickly identify sudden voltage changes in the ship's power grid, this embodiment uses sliding window differential and differential derivative to achieve detection and debouncing: (1) Note: for Constant AC bus voltage (RMS or per-unit value can be used). Pushing forward from the current moment Historical AC bus voltage values for each sampling period This represents the voltage change during this period; This is the window length (number of sampling points).
[0023] (2) The voltage change rate (discrete difference approximation). The sampling period.
[0024] (3) Here, f(k) represents the frequency deviation, f0 represents the ship's electrical grid frequency, and f0 represents the rated frequency.
[0025] Example of discrimination logic: Voltage drop: ≤U low And continue for N hold One sampling period (4) U low The lower threshold; N hold To maintain the points for jitter reduction.
[0026] Voltage surge: ≥U high And continue for N hold Each sampling period (5) U high This is the upper threshold.
[0027] Fault classification should include at least two levels: minor and severe, based on the drop / rise amplitude, duration, and other factors. The classification is performed using Δf(k); the classification result Level is used to trigger the online reconstruction of the subsequent MPC weight matrix and constraint set.
[0028] 3 Rotor-side fault-tolerant MPC modeling and rolling optimization (S3) Figure 3 This is a schematic diagram of the solution and synthesis control process for fault-tolerant MPC fused with SMC; this embodiment uses discrete state-space form to describe the rotor-side current dynamics, and introduces an equivalent disturbance term to represent model mismatch and external disturbances: (6) in for Time-state vector for Predicted state at time +1, i.e., rotor current state ( ); The control vector, i.e., the voltage vector output by the rotor-side converter. ); For the equivalent disturbance term, A is the state transition matrix, B is the control input matrix, and E is the disturbance input matrix; Acceptable (k), (k)]^T, where T represents time; For RSC control input (voltage vector or equivalent modulation amount), here referring to [ (k), (k)]^T; For equivalent disturbances, the MPC controller is based on numerous possible... Among the combinations, find one that will allow the current state to remain constant for a future period of time. The set of constraints that best fits the target value without violating the safety red line, where j represents the time increment.
[0029] To suppress control shocks while also considering fault-crossing targets, a quadratic cost function is constructed: (7) in N represents the current tracking state error penalty. p The prediction step size is the number of steps the controller predicts for the future. Represents the future at time k. The predicted values of the state at any given time, i.e., the d-axis and q-axis components of the rotor current. , ( ) is the future The state reference value at any given time, i.e., the given rotor current command. , Q is the state error weight matrix. The square of the weighted Euclidean norm is expressed as follows: ; It is a control increment penalty, N c To control the step size, N represents the number of steps the controller plans for future control actions. c N p , It controls the increment of the quantity. This refers to the voltage vector output by the rotor-side converter, and R is the control increment weight matrix, used to penalize drastic changes in the controller.
[0030] (8) The constraint set must contain at least: or | |≤I d,max , | |≤I q,max (9) This represents the upper limit of the rotor current, or a component-limited limit.
[0031] | (10) Indicates the rotor-side output voltage. This is the DC bus voltage.
[0032] (k)≤U dc,max (11) U dc,max This represents the maximum allowable limit / protection threshold for the DC bus.
[0033] (12) in, Indicates electromagnetic torque. It is the maximum permissible rate of torque change.
[0034] Limit the rate of torque change to suppress mechanical shock.
[0035] Fault classification triggers weight / constraint reconstruction: Q = Q(Level), R = R(Level), I max =I max (Level) (13) Minor fault: Q(Level) = Q light R(Level) = R light I max (Level)=I max,light Severe fault: Q (Level) = Q severe R(Level) = R severe Imax(Level) = I max,severe Q (Level) is used to adjust the degree of focus on control objectives such as current tracking, reactive power support, or DC bus stability; R (Level) is used to adjust the strength of suppression of control input amplitude or rate of change; I max (Level) indicates the maximum allowable rotor current constraint under the corresponding fault level.
[0036] Severe faults can increase the relevant weight of reactive power support and tighten [the system / mechanism]. or improve Constraint weights.
[0037] The controller employs a rolling time-domain optimization strategy: it solves for the optimal control sequence in each sampling period and applies only the first step of the control input as the current output.
[0038] 4. Sliding mode compensation design (S4) To improve robustness to parameter drift and grid disturbances, a sliding mode surface based on rotor current error is constructed and a continuous compensation law is formed: (14) This is the rotor current error vector. This indicates the d-axis current on the rotor side. This indicates the actual current fed back from the d-axis on the rotor side. ; This indicates the q-axis current on the rotor side. This indicates the actual current fed back to the q-axis on the rotor side.
[0039] (15) It is a sliding surface coefficient matrix or vector.
[0040] Slippage compensation amount: (16) K is the sliding mode gain matrix; Boundary layer thickness; (·) is a saturation function to reduce chattering.
[0041] The saturation function is defined as: sat(z) = 1 (z > 1), sat(z) = z (|z| ≤ 1), sat(z) = -1 (z < -1).
[0042] 5. Synthetic Control Law and Fault Traceability Strategy (S5) Figure 4 This diagram illustrates the relationship between current / voltage constraints and protection coordination during rotor-side fault ride-through. The optimal MPC control quantity is superimposed with sliding mode compensation to form a synthetic control law. u(k)=U MPC (k)+U SMC (k) (17) U MPC For the optimal control quantity, U SMC This is the sliding mode compensation amount.
[0043] During voltage dips, to meet the ship's electrical grid voltage support requirements and protect rotor-side components, the controller increases the reactive power support weight and enhances... Prioritize and limit active power-related components and current limits; during voltage surges, the controller increases... Constrain weights and limit energy injection, triggering braking resistor activation or energy feedback strategy suppression when necessary. Rising upwards.
[0044] 6. Protect coordination and smooth exit (S5) To avoid secondary impacts when the fault is cleared, a smoothing factor is used. To achieve a smooth transition between fault and normal control, its function is to ensure a smooth transition of voltage or current commands to the motor, thereby guaranteeing the stability and safety of the control system when switching out of a fault state. (18) Δγ is the amplitude limiting function, and Δγ is the ramp step.
[0045] (19) For fault mode control variables, This is the control value for normal mode.
[0046] By setting the ramp step value ,control It changes at a fixed rate between 0 and 1. The function ensures it does not go out of bounds. Assume... =1, = The system is entirely controlled by fault logic; When =0, = The smooth transition ends, and the normal PI speed / current dual closed-loop fully takes over the motor. Before execution... Implement amplitude limiting and coordinate with overcurrent and overvoltage protection logic; when the recovery criteria are met, gradually restore the weights and constraints to normal values.
[0047] Figure 5 This is the main circuit topology diagram of the DFIM doubly fed asynchronous induction motor back-to-back converter. It includes the GSC grid-side converter, the RSC rotor-side converter (controlled object), and the DFIM.
[0048] Figure 6 This is a block diagram of the rotor-side fault-tolerant control of a doubly fed asynchronous induction motor (DFIM) based on model predictive control (MPC) and sliding mode compensation (SMC) (or an architecture diagram of the MPC-SMC composite fault-tolerant control system on the RSC side).
[0049] In the simulation, the external power grid U s A three-phase programmable voltage source was used for modeling, with the rated line voltage RMS value set to 100V and the frequency set to 50Hz. To simulate the grid voltage disturbance process, the positive sequence voltage amplitude of the power source varied in segments according to a preset time series. Specifically, in During this period, the grid voltage remains at its rated value, i.e., the amplitude is ;when At that time, the grid voltage dropped sharply, with the amplitude from Mutation and in Maintain this low voltage state within the range; when At that time, the grid voltage returned to its rated value. And in the subsequent It maintains stable operation within the specified range.
[0050] Figure 7(a) shows the d-axis current on the pure PI rotor side. and feedback actual current Figure 7(b) shows the fault-tolerant model predictive control of the rotor-side d-axis given current using the SMC-integrated method of this invention. and feedback actual current Figure 8(a) shows the q-axis current on the pure PI rotor side. and feedback actual current Figure 8(b) shows the fault-tolerant model predictive control rotor-side q-axis given current using the SMC-integrated method of this invention. and feedback actual current The graph shows that under PI control, Significant sudden changes and oscillations can easily occur during voltage dips and recoveries in the power grid, under MPC control. The fluctuation amplitude is smaller, the peak value is more controllable when a fault occurs, and the recovery process is smoother, demonstrating better d-axis current constraint capability and dynamic adjustment performance; under PI control, Significant overshoot and oscillation are prone to occur both before and after a fault, with particularly pronounced fluctuations during voltage dips. Oscillations also exhibit a certain degree of attenuation during the recovery phase. In contrast, MPC control can predict the current evolution at the next moment based on the system model and select the optimal control variable from candidate control actions, thus... It maintains good tracking capability even under fault disturbances.
[0051] Figure 9 The demonstration shows the three-phase grid (stator) voltage V using a rotor-side pure PI conventional algorithm and a fault-tolerant model predictive control (SMC) fused with SMC (i.e., the method of this invention). abc The waveform diagram shows the three-phase grid voltage V under two control strategies. abc The changes are basically consistent, both reflecting the voltage drop and recovery process caused by external power grid fault settings. Specifically, they are manifested as follows: During this period, the three-phase voltage remains at its rated amplitude; During this period, the three-phase voltage amplitude decreased significantly to half of the rated value; It later returned to normal levels. Due to V abc Since the voltage is directly set by an external programmable grid voltage source, this waveform is mainly used to illustrate the consistency between the two sets of simulation conditions. That is, the PI control and MPC control are compared under the same grid disturbance conditions, thus ensuring the fairness and effectiveness of the comparison results.
[0052] Figure 10 The display shows the grid-side active power P gc Grid-side reactive power Q gc Active power P on the machine side s Reactive power Q on the machine side sThe waveform diagram illustrates the dynamic response of grid-side power and generator-side power under external grid voltage disturbances. The upper part represents the grid-side active power P. gc With grid-side reactive power Q gc The curve shows the change in active power P on the machine side, with the lower part representing the active power P on the machine side. s With the reactive power Q on the machine side s The curves show the changes. As can be seen from the figure, during normal operation, the power on both the grid side and the generator side remains at a relatively stable level. When the grid voltage changes abruptly, the active and reactive power on both the grid side and the generator side show a significant dip. After the fault is cleared, the power quantities can recover to near the level before the disturbance, indicating that the proposed control method can maintain the system power regulation capability under grid disturbance conditions and has good dynamic recovery performance.
[0053] The total active power P in Figure 11(a) is obtained by superimposing the power from the machine side and the grid side respectively. t Total reactive power Q t The waveform diagram shows the total active power P on the rotor side predicted by the fault-tolerant model of the fused SMC. t Total reactive power Q t The waveform diagram shows that under PI control, active power experiences a significant dip at the moment of fault occurrence, deviates considerably from the steady-state value during the fault period, and exhibits overshoot and oscillation during the recovery phase. Reactive power also experiences significant fluctuations before and after the fault, indicating that traditional PI control has limited power regulation capability under voltage disturbances. In contrast, under MPC control, the total active power drop is smaller, fluctuations are weaker during the fault duration, and the recovery to steady state is faster. The total reactive power can respond more quickly and establish support, thus demonstrating superior dynamic power regulation capability. These results demonstrate that the MPC control method proposed in this invention can more effectively maintain system active power output and enhance reactive power regulation capability during grid voltage changes, thereby improving the overall power support performance during fault ride-through.
[0054] Figure 12(a) shows the mechanical torque T on the rotor side using the traditional pure PI algorithm. m and electromagnetic torque T e The waveform diagram, in Figure 12(b), shows the rotor-side mechanical torque T predicted and controlled by the fault-tolerant model integrating SMC according to the present invention. m and electromagnetic torque T eWaveform diagrams. As shown in Figures 12(a) and 11(b), there are significant differences in the electromagnetic torque response characteristics under PI control and MPC control under grid voltage disturbance conditions. Under PI control, the electromagnetic torque ripple is smaller during normal operation, and the overshoot and oscillation after fault recovery are relatively weaker. Under MPC control, the electromagnetic torque exhibits a faster dynamic reconstruction trend after a fault, but is accompanied by larger start-up overshoot, steady-state pulsation, and fault recovery shock. This indicates that current MPC control has certain potential in terms of dynamic support, but there is still room for further optimization of torque smoothness.
[0055] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A fault-tolerant control method for a ship's electric propulsion system under electrical grid failure, characterized in that, This method, implemented using a rotor-side converter (RSC), includes the following steps: S1, Collects AC bus voltage of the ship's electrical grid With frequency f and DC bus voltage Rotor current dq component , Rotation speed With rotor electrical angle And filter and synchronize the sampled signal; S2, calculate the voltage change within the sliding time window. With rate of change It also determines voltage drops or rises based on preset thresholds and debouncing logic and completes fault classification. S3. Based on the fault classification results, the cost function weights and constraint set of the rotor-side fault-tolerant MPC are updated online. A rolling optimization problem with rotor current tracking and fault ride-through objectives as the core is constructed and solved to obtain the optimal control quantity U. MPC ; S4, construct the sliding surface s(k) with rotor current tracking error as the core, and use a continuous saturation function to form the sliding compensation amount U. SMC To suppress parameter uncertainties and external disturbances; S5, place U MPC with U SMC The control quantity during the fault period is superimposed and then merged with the normal control quantity through smooth switching to generate the synthetic control rate u(k) applied to the rotor-side converter. After limiting and protection coordination, the PWM drive signal of RSC is generated, enabling the doubly fed asynchronous induction motor (DFIM) to achieve stable operation under reactive power support, active power limiting and rotor current constraint during voltage sag / boost, and smoothly return to normal control after the fault is cleared.
2. The method according to claim 1, characterized in that: In step S2, the voltage change and rate of change within the sliding time window respectively satisfy the following conditions: ,as well as ,in The length of the sliding window. Indicates the current time AC bus voltage value, Indicates the current time Push forward Historical AC bus voltage value for each sampling period, T s The sampling period; Voltage sag detection ≤U low And continue N hold Each sampling period, voltage surge is determined as follows: ≥U high And continue N hold Each sampling period; the fault classification includes at least two levels: mild and severe, and frequency deviation is introduced. As an auxiliary criterion; where N hold It is the fault persistence determination threshold, i.e., the number of debouncing hold points. For shipboard power grid frequency, For the rated frequency, U high U is the upper threshold. low The lower threshold is [value].
3. The method according to claim 1, characterized in that: In step S3, the rotor-side MPC adopts a discrete prediction model. , in for Time-state vector for Predicted state at time +1, i.e., rotor current state ; The control vector, i.e., the voltage vector output by the rotor-side converter. ; Let U be the equivalent disturbance term, A be the state transition matrix, B be the control input matrix, and E be the disturbance input matrix. The optimal control input vector obtained by solving the rotor-side model predictive controller is U. MPC The MPC cost function includes at least a current tracking error term and a control increment term: , in N represents the current tracking state error penalty. p The prediction step size is the number of steps the controller predicts for the future. Represents the future at time k. The predicted values of the state at any given time, i.e., the d-axis and q-axis components of the rotor current. , ( ) is the future The state reference value at any given time, i.e., the given rotor current command. , Q is the state error weight matrix. The square of the weighted Euclidean norm is expressed as follows: ; This is the control increment penalty, where Nc is the control step size, representing the number of steps the controller plans for future control actions. c N p , It is the increment of the control quantity, that is , This refers to the voltage vector output by the rotor-side converter, and R is the control increment weight matrix, used to penalize drastic changes in the controller.
4. The method according to claim 1, characterized in that: The constraint set in step S3 includes at least rotor current constraints. or | |≤I d,max 、| |≤I q,max , The upper limit of the rotor current and the output voltage constraint on the rotor side. DC bus voltage constraint U dc (k)≤U dc,max and electromagnetic torque change rate constraint , It is the maximum allowable rate of torque change, and the weights and constraints are reconfigured online based on the fault classification. The sampling period.
5. The method according to claim 1, characterized in that: In step S4, the sliding surface is defined as ,in For the sliding surface matrix, =[C1 Cn], Let the current error vector be... For the sliding surface variable, the unit is C and Decision; slip compensation amount is Where K is the sliding mode gain matrix, denoted as boundary layer thickness, and sat(·) as saturation function.
6. The method according to claim 1, characterized in that: During voltage sags, the reactive power support target weight is increased and the priority of rotor current commands is raised, while active power-related components and rotor current limits are restricted. Increase the DC bus voltage constraint weight and limit energy injection during voltage surges; A smoothing factor is used during the fault recovery phase to achieve a smooth switching of the synthetic control law.
Citation Information
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