Active Disturbance rejection Hopf bifurcation suppression strategy based on DC microgrid DC-DC converter

To solve the problem that the output voltage of the Buck converter is easily affected by the change of system parameters, which reduces the energy transmission quality, a linear active disturbance rejection control strategy is proposed in this paper. Firstly, a discrete mathematical model is established to analyze that Hopf bifurcation disturbance is the cause of low-frequency oscillation of output voltage, and the existence of Hopf bifurcation disturbance is verified by simulation. Secondly, based on the theoretical analysis, the suppression effect of improving the error feedback control rate on Hopf bifurcation disturbance is designed. Moreover, the stability of the control strategy under Hopf bifurcation disturbance is analyzed by using frequency-domain theory. Finally, the suppression of Hopf bifurcation disturbance by the proposed control strategy and the traditional PI control strategy under different working conditions is compared by simulation analysis. The results show that the proposed linear active disturbance rejection control strategy can effectively suppress Hopf bifurcation disturbance and guarantee the performance of the Buck converter.


Introduction
In recent years, due to the development of new power systems, new energy has been widely connected to DC microgrids.As the basic unit of the DC microgrid, the DC-DC converter not only significantly improves the operational performance and efficiency of the DC microgrid but also has nonlinear behavior (such as bifurcation, chaos, etc.) when parameters change due to its nonlinear structural characteristics.
Nonlinear dynamic behavior will reduce the output power quality, affect the structural stability of the converter, and bring many stability problems to the DC microgrid and the converter itself [1].Therefore, the effective suppression of the bifurcation phenomenon can avoid the system output instability, ensure the converter performance, and improve the power quality.Zhong et al. [2] have studied the phenomenon of low-frequency oscillation of the Buck circuit and its stability conditions, but no controller is used to suppress the disturbance.In [3], two control methods were used to suppress the chaotic bifurcation phenomenon.The portability of the control methods is high but the robustness of the two control methods is insufficient.Tse and Bernardo [4] analyzed the evolution process of the nonlinear behavior of the voltage Buck converter and proved that the low-frequency oscillation phenomenon was caused by Hopf bifurcation.The stability conditions were studied, but the nonlinear behavior was not considered a kind of dynamic interference.The control method was highly dependent on the precise information of the model information In this paper, a discrete model is established to analyze the existence of Hopf bifurcation and its bifurcation phenomenon, and then a linear active disturbance rejection controller is designed.Firstly, nonlinear behaviors such as Hopf bifurcation are summarized as disturbances that affect the stable output of the system, and a disturbance rejection strategy of linear active disturbance rejection control is proposed.The core idea of this method is to take the internal uncertainty of the system (steady or time-varying, linear or nonlinear) and external uncertainty (external disturbance) together as "total disturbance".The extended state observer is constructed to estimate and compensate the "total disturbance" in real-time so as to obtain strong anti-disturbance ability.This method does not require direct measurement of the disturbance, nor does it require prior knowledge of the disturbance action law so that the relevant controller design can be carried out.Therefore, the method can be used to observe, estimate and compensate for Hopf bifurcation disturbance.Finally, the effectiveness and correctness of active disturbance rejection control (LADRC) are verified by theoretical analysis and simulation.

Hopf bifurcation in voltage-type Buck converter
When where where c ( ) y t is the input of the controller,

 
,( 1) 3) is a transcendental formula, the solution n d is difficult to solve, so it is generally solved by numerical method.

Discrete model
Taking the clock cycle T as the sampling interval, the following discrete model can be obtained according to Formulas ( 1) and ( 2) : where can be inversely solved, so the fixed point of period 1 can be obtained by solving Formula (4).
Further, the Jacobian matrix of the discrete mapping Formula (4) at s1 x can be written as: The fixed point s1 x can be obtained from Formulas (4) to (6), and the Jacobian matrix and its can be used to judge the stability of the system.When the system is unstable, the existence proof of Hopf bifurcation can be used to know that Hopf bifurcation occurs in its state variables.

Simulation analysis of bifurcation phenomenon of Hopf bifurcation existence proof
The system parameters are: L=0.5 mH, R=5 Ω, C=300 μF, L r =0.02 Ω, C r =0.05 Ω, T=100 μs, ramp   Figure 3 is the time-domain waveform of the output voltage obtained at IW 186 k  .Figure 4 shows the bifurcation diagram of the output voltage when the integral coefficient changes IW k obtained by using the discrete model.At that time, IW 186 k  system voltage was unstable, which was caused by low-frequency oscillation caused by Hopf bifurcation disturbance in system state variables.In this paper, the Hopf bifurcation disturbance occurred is classified as the total disturbance of the system.LADRC is used to suppress and eliminate Hopf bifurcation disturbance.The compensation effect of LESO on interference can avoid the side effects of delay and easy oscillation caused by differentiation, effectively suppress noise signal and compensate for the static difference.Figure 5 shows the structure diagram of second order LADRC, in which LESO is a new state variable formed by extending the disturbance affecting the output of the controlled object, and an observer capable of observing the extended state established by using a special feedback mechanism [7].The second-order system with single input and single output is taken as the research object, and the controlled object is:

Principle of second-order linear active disturbance rejection control
1 ( , , ) y bu f y t     (7) in Formula ( 7), y, u, and  are the output, input, and external disturbance of the system; t is the time- varying state of the system, b is the control quantity, and the gain is the inherent parameter of the system.However, the real value of b cannot be accurately estimated by the actual system, so the estimate of b is b 0 , and the system can be written as: (8) The linear expansion observer (LESO) corresponding to the design is: in Formula (9),   , L is the error feedback control gain matrix of the observer.The design of realize the tracking of all variables in the system.The observer gain matrix is obtained by pole assignment:  is observer gain.The compensation link LSEF adopts the controller of linear PD combination, and its form is: where r is the given input value, to reduce the system oscillation caused by the rapid change of the given value.Then the design control law is: ) Then the closed-loop transfer function of the whole system is transformed into a second-order system without zero: The gain matrix of compensator obtained by pole assignment is and c  is the bandwidth of the compensator.

Stability analysis
The controlled system is: Control quantity: Then Formula (13) can be expressed as: , z y r z y     : The characteristic formula is 2 . If the characteristic formula is guaranteed to be a Hurwitz polynomial, then the system is asymptotically stable. 1 2 lim 0 lim 0 That is, LADRC has good stability.

Simulation verification analysis
A Buck circuit is built on the digital simulation software for simulation verification.This verifies the effectiveness and correctness of the LADRC control strategy proposed in this paper.
1) Working condition 1: Voltage mutation on the input side Considering the forward flow of energy, 50% input voltage surges and decreases are added at 0.12 s, respectively, as disturbances and the circuit output voltage waveform is shown in Figure 9. Figure 7.The input voltage drops by 50%.
2) Working condition 2: Load mutation on the output side Considering the influence of load mutation on the output voltage, 50% load surge and drop are applied at 0.12 s respectively, and the output voltage waveform is observed as follows: Figure 9. Exit load drops by 50%.
3) Performance index analysis  1, as well as linear active disturbance rejection stability analysis, it can be shown that LADRC control has a more obvious inhibition effect on Hopf bifurcation disturbance, while PI controller's inhibition ability on disturbance decreases due to the lack of differential link.However, when the input voltage drops by 50%, the nonlinear structure of the system changes and enters the stability domain.The results of Hopf bifurcation disturbance suppression are similar.In general, PI control and LADRC controller adjust time in the face of disturbance basically the same.
As can be seen from the above, under different working conditions LADRC control is better than PI control to effectively compensate Hopf bifurcation disturbance and make the system voltage output stable.

Conclusion
In order to improve the power quality and solve the problem of stable output of DC microgrid DC-DC converter, this paper uses mathematical modeling and simulation analysis to summarize the Hopf bifurcation phenomenon as a kind of self-excited Hopf bifurcation disturbance, aiming at which a LADRC disturbance suppression strategy is proposed.This control strategy can compensate the nonlinear behavior (Hopf bifurcation) by reducing the disturbance to the total disturbance without directly measuring the external disturbance or knowing the disturbance action law in advance.The theoretical analysis and simulation results show that the LADRC control strategy has better dynamic and steady performance than the traditional PI control for nonlinear Hopf bifurcation disturbance under different working conditions.

Figure 1
Figure 1 and figure 2 show the structure diagram of DC microgrid and Buck converter.According to the study of Xu et al. [5], when the converter is completely in discontinuous on-off mode, its output voltage does not have low-frequency fluctuations, namely Hopf bifurcations.The differential formula model of a closed-loop controlled Buck converter in continuous current mode can be obtained from Saublet et al.'s work [6].When n nT t nT d T    has:


becomes unstable by obtaining the eigenvalues of IW 185 k  Jacobian matrix.According to Lyapunov's second stability criterion, when IW 185 k  the system becomes unstable, a pair of complex eigenvalues move out of the unit circle, indicating that Hopf bifurcation occurs in the system.The system voltage time-domain response output diagram and system bifurcation diagram are as follows.

(Figure 3 .
Figure 3.Time domain response diagram of the output voltage.

Figure 4 .
Figure 4. Bifurcation diagram with k IW as the bifurcation parameter.

Table 1 .
Comparison of system performance indicators.