**2. Methods**

#### *2.1. Parallel Scheme of the Multiple Rectification Module*

The studied high-current generation system is a new DC power generation system that realizes electromechanical energy conversion by using a multiple three-phase permanent magne<sup>t</sup> synchronous motor. The system directly generates low-ripple and high-intensity DC power through the combination of the modular rectification component and the high current confluence plate [15]. The advantages of this system are as the following:


Figure 1 illustrates the designed model of the multiple three-phase permanent magne<sup>t</sup> synchronous high-current generation system. It indicates that the designed system is mainly composed of a permanent magne<sup>t</sup> synchronous generator with a half-turn coil for each phase and two confluence plates. Figure 1a describes the minimum cell which is composed of each winding and the leg of the bridge, and Figure 1b describes the five MOSFETs on each leg which are fixed on the heat sink. Moreover, Figure 1c shows a confluence plate that fixes the heat sink to be used as the output bus. In order to improve the heat-dissipation efficiency, the inside of the confluence plate is machined with a water-cooled passage. Figure 1d shows the overall model of the system containing the permanent magne<sup>t</sup> in the synchronous generator.

**Figure 1.** Schematic models of the rectifier module and the power generation system: (**a**) A leg of bridge; (**b**) heat sink; (**c**) schematic of the confluence plate; (**d**) schematic model of a large DC power generation system.

The generator rotor contains 6 pairs of magnetic poles and the stator core is internally opened with 54 slots. Each slot is embedded with a half-turn coil, containing the insulation as one phase, while the three-phase stator windings are placed at an electrical angle of 120◦. It should be indicated that the two adjacent three-phase winding units differ from each other by an electrical angle of 2π/9. Moreover, each positive and negative confluence plate is composed of 54 MOSFET rectifier modules. The rectifier modules can be distributed along the circumference surface of the confluence plate as a stationary part of the synchronous generator. In order to simplify the analysis, only three rectifier modules were considered in the design and analysis. These modules correspond to the spaces of nine stator windings of the synchronous generator, which can form a 360◦ electrical angular space. The system is managed hierarchically, where the top layer is the managemen<sup>t</sup> layer and the controller area network (CAN) bus transmits the current sharing information. The second layer of each control module analyzes the collected module voltage and compares it with current signals to generate the PWM control information to drive the switching device of the rectifier. The bottom layer is a three-phase PWM rectifier module, which is fixed at the bottom of the generator together with the confluence plate to generate high currents. Figure 2 shows the block diagram of the synchronous DC power generation system with the multi-phase permanent magnet.

**Figure 2.** The block diagram of the synchronous DC power generation system with the multi-phase permanent magnet.

The digital current sharing by the CAN bus can avoid the defects of the conventional master–slave control mode. This reduces the number of controllers and avoids the failure of the main control module not achieving the appropriate current sharing [17]. The CAN bus realizes the current sharing by means of competition where the module which generates the maximum current is determined as the main module and the corresponding maximum current is used as the reference current of the submodule. Even if the main module fails and shuts down, the other modules can continue to qualify as the main module in a competitive manner.

A modular rectifier improves system performance. First of all, the modular parallel connection is adopted in this rectification system, and the ripple of output is small, so that a rapid voltage adjustment method can be adopted without causing distortion of the input current, and the dynamic performance is good. Secondly, when the modules are connected in parallel to supply power to the load, even if a module fails, as long as it is removed from the parallel system in time, the load power supply can be ensured without interruption, further improving the reliability and flexibility of the power supply system. Finally, when powered by a parallel system, the power of each parallel module will be reduced, while the low power module can operate at a higher switching frequency, which can reduce the volume of the filter capacitor and increase the power density of the parallel system, which is very suitable for low- voltage and high-power systems.

#### *2.2. Design of the Three-Phase Rectifier Controller, Based on the Lyapunov Algorithm*

2.2.1. Mathematical Model of the Three-Phase Voltage Source PWM Rectifier in the d-q Synchronous Rotating Coordinate System

The state equation of the three-phase voltage source PWM rectifier is described by the unipolar binary logic switching function. When the coordinate transformation of the constant power is carried out, the mathematical models in the three-phase rotating coordinate system can be written as the following [18]:

$$L\frac{di\_d}{dt} = \omega L i\_q - R i\_d + e\_d - \upsilon\_o p\_d \tag{1}$$

$$L\frac{d\dot{i}\_q}{dt} = -\omega L\dot{i}\_d - R\dot{i}\_q + \varepsilon\_q - \upsilon\_o p\_q \tag{2}$$

$$\mathbb{C}\frac{dv\_0}{dt} = \frac{3}{2}(P\_d i\_d + P\_q i\_q) - i\_o \tag{3}$$

where *vo*, *io*, *L* and C denote the output voltage, module current, AC side inductance, and the DC side filter capacitance, respectively. On the other hand, ω is the angular frequency. Furthermore, *pd* and *pq* are voltage modulation ratios on *d* and *q* axes, respectively. Finally, *id, iq, ed, eq* denote the currents and voltages of the network side in the synchronous rotating coordinate system, respectively.

#### 2.2.2. Lyapunov Control Algorithm

Studies showed that the hysteresis current control, the deadbeat control and the fuzzy control methods yield various advantages and disadvantages [19–21] from different points of view, including the circuit complexity, switching frequency, and the transient state. A common disadvantage of these methods is that system stability cannot be guaranteed under large signal interference. Since the system generates high current, the voltage produces large fluctuations when subjected to large signal interference, which affects the normal operation of the system. In [9], the controller is derived based on direct Lyapunov stability theory in order to calculate proper switching functions, Setting up the controller with this switch function, the results show the appropriate performance of the proposed controllers during both steady-state and transient dynamic conditions. Considering the abovementioned challenge, a nonlinear control strategy based on the Lyapunov direct method [22,23] is proposed to strictly guarantee the global asymptotic stability of the rectifier. The purpose of this control strategy is to make the output voltage close to the reference voltage (*Vr*) and provide a unity power factor with a nearly sinusoidal input current. When the system is controlled by the Lyapunov algorithm, it is necessary to construct an "energy-like" function for the system, and then design the controller under the premise of ensuring that the function is always negative. Finally, establishing the Lyapunov function based on the quantitative correlation of the energy storage of the inductor and the capacitor. Suppose that the Lyapunov function is a positive definite function, while its derivative is negative definite function. When *x* tends to infinity in any direction, *V*(*x*) also approaches infinity. The equilibrium point at the origin is globally asymptotically stable.

Letting the operating point of the system energy stability be the equilibrium point, we define a positive definite Lyapunov function as the following:

$$V(\overline{\mathbf{x}}) = \frac{3}{2} L \mathbf{x}\_1^2 + \frac{3}{2} L \mathbf{x}\_2^2 + \mathbb{C} \mathbf{x}\_3^2 \tag{4}$$

where *x1, x2* and *x3* are system state variables defined as the following:

$$\begin{cases} \mathbf{x}\_1 = \dot{\mathbf{i}}\_d - \dot{\mathbf{i}}\_{d0} \\ \mathbf{x}\_2 = \dot{\mathbf{i}}\_q \\ \mathbf{x}\_3 = V\_r - \mathbf{v}\_0 + k(I\_r - \dot{\mathbf{i}}\_0) \end{cases} \tag{5}$$

where *Vr*, *I*r, k and *id0* denote the reference voltage, reference current, proportional coefficient, and the steady value of *id*, respectively. It should be indicated that the reference current is maximum current in the multiple rectification. The derivative of the Lyapunov function is

$$V'(\overline{\mathbf{x}}) = 3\mathbf{x}\_1 \mathbf{L}\mathbf{x}\_1' + 3\mathbf{x}\_2 \mathbf{L}\mathbf{x}\_2' + 2\mathbf{x}\_3 \mathbf{C}\mathbf{x}\_3' \tag{6}$$

According to the first Lyapunov stability theorem, when the Lyapunov derivative is negative, the system is stable in the equilibrium point. Let *iq* = 0 to ensure unit power factor, when the system is stable, the parameter values of the equilibrium point are as follows:

$$\begin{cases} v\_o = V\_r \\ i\_o = I\_r \\ i\_d = i\_{d0} \\ \varepsilon\_d = E\_m \\ i\_q = 0 \\ p\_d = p\_{d0} \\ p\_q = p\_{q0} \end{cases} \tag{7}$$

where *pd0* and *pq0* are the control variables when the system is stable and *Em* is the amplitude of the phase voltage.

After substituting Equation (7) into Equations (1)–(3) and a simple manipulation, the following expressions are obtained:

$$
\sigma\_{d0} = (E\_m - R i\_{d0}) / V\_r \tag{8}
$$

$$p\_{q0} = -\omega L i\_{d0} / V\_r \tag{9}$$

$$\dot{a}\_o = \frac{3}{2V\_r} (E\_m \dot{i}\_{d0} - R \dot{i}\_{d0}^2) \tag{10}$$

$$i\_{d0} = \frac{1}{2} \left\{ \frac{E\_m}{R} \pm \left[ \left( \frac{E\_m}{R} \right)^2 - \left( \frac{8V\_r i\_o}{3R} \right) \right]^{\frac{1}{2}} \right\} \tag{11}$$

Assuming that the system is disturbed, the variations of voltage-space vector modulation ratio on *d* and *q* axes are Δ*pd* and <sup>Δ</sup>*pq*, respectively. Then the actual modulation ratios of output voltage can be obtained as the following:

$$p\_d = p\_{d0} + \Delta p\_d \tag{12}$$

$$p\_q = p\_{q0} + \Delta p\_q \tag{13}$$

Substitute Equations (5), (8) and (12) into Equation (1) results in the following equation:

$$\begin{array}{ll}L\mathbf{x}\_{1}^{\prime} = & \boldsymbol{\alpha}L\mathbf{x}\_{2} - V\_{r}\Delta p\_{d} - \frac{(E\_{\mathrm{nr}} - \mathrm{Ri}\_{d0})[\mathbf{x}\_{3} + k(I\_{r} - i\_{o})]}{V\_{r}}\\ & -[\mathbf{x}\_{3} + k(I\_{r} - i\_{o})]\Delta p\_{d} - \mathrm{Rx}\_{1} \end{array} \tag{14}$$

Similarly, Equations (5), (9) and (13) are applied to Equation (2) so that the following equation is obtained:

$$\begin{array}{ll}L\mathbf{x}'\_2 = & -\omega L\mathbf{x}\_1 - V\_r \Delta p\_q - \frac{\omega L i\_{d0}|\mathbf{x}\_3 + k(l\_r - i\_o)}{V\_r} \\ & -[\mathbf{x}\_3 + k(l\_r - i\_o)]\Delta p\_q - R\mathbf{x}\_2 \end{array} \tag{15}$$

Moreover, Equations (5), (10), (12) and (13) are implemented into Equation (3) to obtain the following equation:

$$\begin{array}{ll} \text{Cx}\_3' = & \frac{3}{2} \begin{bmatrix} \frac{(E\_m - Ri\_{d0})}{V\_r} x\_1 + i\_{d0} \Delta p\_d + \Delta p\_d x\_1\\ -\frac{\alpha Li\_{d0}}{V\_r} x\_2 + \Delta p\_q x\_2 \end{bmatrix} \end{array} \tag{16}$$

Substituting Equations (10) and (14)–(16) into Equation (6) yields the following equation:

$$\begin{array}{ll} V'(\overleftarrow{\mathbf{x}}) = & -3 \{ [V\_r + k(I\_r - i\_o)] \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3 \} \Delta p\_d\\ & -3 \mathbf{x}\_2 [V\_r + k(I\_r - i\_o)] \Delta p\_q - 3 \mathcal{R} \{ \mathbf{x}\_1^2 + \mathbf{x}\_2^2 \} \end{array} \tag{17}$$

When the following conditions are met, the Lyapunov derivative along any of the trajectories of the system is negative:

$$
\Delta p\_d = \gamma \{ |V\_r + k(I\_r - i\_o)| \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3 \}, \gamma > 0 \tag{18}
$$

$$
\Delta p\_q = \beta \mathbf{x}\_2 \left[ V\_r + k(I\_r - i\_o) \right] \rho > 0 \tag{19}
$$

where β and γ are arbitrary real constants.

#### 2.2.3. Saturation Constraint and Decoupling Control Variables

The switching state of the rectifier is determined by the space vector pulse width modulation method (SVPWM) [24]. In order to ensure that the rectifier is in sinusoidal steady-state operation and the switching function is not saturated, the following conditions must be met:

$$(p\_{d0} + \Delta p\_d)^2 + \left(p\_{q0} + \Delta p\_q\right)^2 \le \frac{4}{3} \tag{20}$$

The control variables, which satisfy the SVPWM are obtained from Equation (20) as the following:

$$(\Delta p\_d)\_{m1} = \frac{2(p\_{d0} + \Delta p\_d)}{\sqrt{3\left[ (p\_{d0} + \Delta p\_d)^2 + \left(p\_{q0} + \Delta p\_q\right)^2 \right]}} - p\_{d0} \tag{21}$$

$$\left(\Delta p\_{\eta}\right)\_{m1} = \frac{2\left(p\_{q0} + \Delta p\_{q}\right)}{\sqrt{3\left[\left(p\_{d0} + \Delta p\_{d}\right)^{2} + \left(p\_{q0} + \Delta p\_{q}\right)^{2}\right]}} - p\_{q0} \tag{22}$$

The two control variables ( Δ*pd)m1* and ( <sup>Δ</sup>*pq)m1* are mutual coupling and non-linear variables, which cannot guarantee a negative Lyapunov derivative. Therefore, it is necessary to decouple the control variables. Assuming that the system is controlled by *pq*, (i.e., *pd* = *0*), then the range of <sup>Δ</sup>*pq* can be written as

$$-\left(p\_{q0m} + p\_{q0}\right) \le \Delta p\_q \le p\_{q0m} - p\_{q0} \tag{23}$$

*Pq0m* represents the maximum possible steady-state value that *pq* can take for the maximum DC load current. Similarly, the range of Δ*pd* can be written as

$$- \left( p\_{d0m} + p\_{d0} \right) \le \Delta p\_d \le p\_{d0m} - p\_{d0} \tag{24}$$

*Pd0m* is the maximum steady-state value of *pd*, which can be calculated by Equation (25) as the following:

$$p\_{d0m} = \sqrt{\frac{4}{3} - p^2\_{q0m}}\tag{25}$$

Therefore, the control variables are rewritten as the following:

$$(\Delta p\_d)\_{n2} = \begin{cases} -(p\_{d0m} + p\_{d0}) & \gamma \{ [V\_r + k(I\_r - i\_0)] \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3 \} < -(p\_{d0m} + p\_{d0})\\ \gamma \{ [V\_r + k(I\_r - i\_0) \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3] \} & -(p\_{d0m} + p\_{d0}) \le \gamma \{ [V\_r + k(I\_r - i\_0)] \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3 \} \le (p\_{d0m} - p\_{d0})\\ p\_{d0m} - p\_{d0} & \gamma \{ [V\_r + k(I\_r - i\_0) \mathbf{x}\_1 - i\_{d0} \mathbf{x}\_3] \} > (p\_{d0m} - p\_{d0}) \end{cases} \tag{26}$$

$$\begin{pmatrix} \left(\Delta p\_q\right)\_{m\_2} = \begin{cases} -\left(p\_{q0m} + p\_{q0}\right) & \beta \ge \left[V\_r + k(I\_r - i\_o)\right] < -\left(p\_{q0m} + p\_{q0}\right) \\ \beta \ge \left[V\_r + k(I\_r - i\_o)\right] & -\left(p\_{q0m} + p\_{q0}\right) \le \beta \ge \left[V\_r + k(I\_r - i\_o)\right] \le p\_{q0m} - p\_{q0} \\ p\_{q0m} - p\_{q0} & \beta \ge \left[V\_r + k(I\_r - i\_o)\right] > p\_{q0m} - p\_{q0} \end{cases} \tag{27}$$

Figure 3 illustrates that the voltage vector of the rotation is in the rectangular area of the dotted round line. Under the control rules of Equations (26) and (27), the Lyapunov derivative is a negative definite function. Meanwhile, the stability of the system is independent of the circuit parameters.

**Figure 3.** Modified control vector area.

2.2.4. SVPWM Algorithm

> The final voltage modulation ratio *pd, pq* is as follows:

$$p\_d = p\_{d0} + \left(\Delta p\_d\right)\_{m2} \tag{28}$$

$$p\_d = p\_{d0} + (\Delta p\_d)\_{m2} \tag{29}$$

Then the inverse Park transformation on *pd, pq*, is as shown in Figure 4.

**Figure 4.** Coordinate transformation.

Here, *pd* and *pq* are inversely transformed into *U*α and *<sup>U</sup>*β, and the electrical angle θ is taken as the angle. Then, *pd* and *pq* are respectively vector-decomposed, and the voltage components on the α-axis and the β-axis are calculated as follows:

$$\mathcal{U}\_{\alpha} = p\_d \ast \cos \theta - p\_q \ast \sin \theta \tag{30}$$

$$\mathcal{U}\_{\beta} = p\_q \ast \cos \theta - p\_d \ast \sin \theta \tag{31}$$

Then, we apply the inverse Park transformation on *U*<sup>α</sup>*, <sup>U</sup>*β, converting the target voltage vector *Uout* into a balanced three-phase voltage vector *Ua, Ub, Uc,* as follows:

$$\begin{cases} \begin{aligned} \mathcal{U}\_{\mathsf{a}} &= \mathcal{U}\_{\beta} \\ \mathcal{U}\_{b} &= \frac{-\mathcal{U}\_{\beta} + \sqrt{3}\mathcal{U}\_{\mathsf{a}}}{2} \\ \mathcal{U}\_{c} &= \frac{-\mathcal{U}\_{\beta} - \sqrt{3}\mathcal{U}\_{\mathsf{a}}}{2} \end{aligned} \tag{32}$$

We can determine the sector of *Uout* by defining variables *a*, *b*, and *c* according to Equation (33).

$$N = 4a + 2b + c \tag{33}$$

The correspondence between the value of N and the sector is as Table 1:

**Table 1.** The sector.


In Figure 5, the target voltage vector *Uout* is analyzed in the I sector, TS is the PWM carrier period, *T*0, *T*7 is zero voltage vector time, *T*4 is the *U*4 time, *T*6 is the *U*6 time, and θ is electrical angle of the *Uout*. The following relationships can be obtained from Figure 5.

$$\begin{aligned} \left[\begin{array}{c} \mathcal{U}\_{a} \\ \mathcal{U}\_{\beta} \end{array}\right] T\_{s} &= \mathcal{U}\_{out} \left[\begin{array}{c} \cos\theta \\ \sin\theta \end{array}\right] T\_{s} = \frac{2}{3} \mathcal{U}\_{dc} \left[\begin{array}{c} 1 \\ 0 \end{array}\right] T\_{4} + \frac{2}{3} \mathcal{U}\_{dc} \left[\begin{array}{c} \cos\frac{\pi}{3} \\ \sin\frac{\pi}{3} \end{array}\right] T\_{6} \end{aligned} \tag{34}$$

**Figure 5.** I sector.

From Equation (34), there are

$$\begin{cases} T\_4 = \frac{\sqrt{3}T\_s}{\underline{U}\_{dc}} (\frac{\sqrt{3}}{2} \underline{U}\_a - \frac{\underline{U}\_\beta}{2}) \\\ T\_6 = \frac{\sqrt{3}T\_s}{\underline{U}\_{dc}} \underline{U}\_\beta \\\ T\_0 = T\_7 = \frac{1}{2} (T\_s - T\_4 - T\_6) \end{cases} \tag{35}$$

In the same way, the time of each vector of *Uout* in other sectors is obtained. The result is shown in Table 2, where the *K* = √ 3*Ts*/*Udc*.


**Table 2.** The time of the basic space vector of each sector.

#### **3. Results and Discussion**

In order to verify the efficiency of the proposed control strategy compared to that of conventional methods, the direct power control algorithm [25,26], dead-Beat control algorithm, which are widely used in three-phase PWM rectifier systems, were selected in the present study. The studied parameters included the dynamic performance, reliability and independency of the large signal interference for the control of the new synchronous generation system with a multiple three-phase permanent magnet. Figure 6 shows the block diagram of the control system. Moreover, Figure 7 illustrates the three-module parallel simulation model. The simulation parameters of the three-phase PWM rectifier are as Table 3:

**Table 3.** Electrical parameters power circuit.


**Figure 6.** System diagram of the single module control of the space vector pulse width modulation method (SVPWM) rectifier.

**Figure 7.** The three-module parallel simulation model: (**a**) The main circuit model of a three-module system during the parallel simulation; (**b**) control loop model of a three-module system during the parallel simulation.

#### *3.1. The Control Based on the Lyapunov Algorithm*

Figure 8 shows the output voltage, current and AC side A-phase voltage and current response curves of the system during the load is 32 Ω. Figure 8a indicates that during the power-on process, the system voltage overshoot is 10%, the system reaches the steady state in 0.02 s and the output voltage is 5 V. Moreover, Figure 8b shows that the output current is 150A. Figure 8c shows the waves of the input. It is observed that the current fluctuates before 0.02 s and there is a short overshoot. Meanwhile, the system is stable after 0.02 s and achieves unit power factor. Figure 8d shows the spectrum of the A-phase current. It is found that the A-phase current contains low harmonics, where the total harmonic distortion (THD) rate is THD = 1.02%.

**Figure 8.** *Cont*.

**Figure 8.** Simulation waves of the Lyapunov control: (**a**) Steady-state voltage wave; (**b**) steady-state current wave; (**c**) A-phase voltage and current waves; (**d**) A-phase current spectrum.

Figure 9 shows the output voltage, current and AC side of the A-phase voltage and current response curves of the system when the load suddenly changes from 32 to 16 Ω. Figure 9a illustrates that the system is stable in the 0.02 s, the load is halved in 0.03 s and the voltage changes to about 0.2 V. Moreover, Figure 9b shows that the system reaches the steady state after 0.02 s and the output current is doubled to 300 A. Figure 9c shows that the phase voltage and current briefly fluctuate and they re-implement the unit power factor.

**Figure 9.** Simulation waves of the disturbed Lyapunov control: (**a**) Voltage wave; (**b**) Current wave; (**c**) A-phase voltage and current waves. (**d**) three-phase AC current waveform.

#### *3.2. The Control Based on the Direct Power Algorithm*

Figure 10 shows the voltage, current and AC side of the A-phase voltage and current response curves of the system during the control of the direct power algorithm. Figure 10a shows that the system is stable at 0.02 s, while the voltage drops when the load is halved at 0.04 s. The system adjusts to a new steady state after 0.03 s and the output voltage drops to 4.8 V, which is less than the reference voltage. Figure 10b shows the output current wave. It is observed that the current increases rapidly and reaches 148 A. Furthermore, Figure 10c shows the AC side of the A-phase voltage and current waveforms. It is found that the current increases after 0.04 s, while the unit power factor is still not guaranteed.

**Figure 10.** Simulation waves of the disturbed direct power control: (**a**) Voltage wave; (**b**) current wave; (**c**) A-phase voltage and current waves.

#### *3.3. The Control Based on the No Beat Control Algorithm*

Figure 11 is the waveform with no beat control. It can be seen that the voltage overshoot is 20%, the system reaches the steady state in 0.03 s when the load resistance suddenly changes from 32 Ω to 16 Ω at 0.06 s, the voltage change is about 0.8 V, and the steady state is restored after 0.03 s. The output current is doubled to 300 A, and the phase A voltage and current achieve a unit power factor after a short period of fluctuation.

**Figure 11.** Simulation waves of the no beat control algorithm: (**a**) Voltage wave; (**b**) current wave; (**c**) A-phase voltage and current waves.

Figure 12 shows output current waveforms of the three-module system in parallel mode. Each rectifier module is controlled by the autonomous current sharing. In all of the parallel modules, the module with the highest output current automatically becomes the main module through the current sampling and the current sharing control bus sends reference current information to other modules to achieve the current sharing. The simulation results indicate that after the current sharing control, each module can generate a stable current of 300 A.

**Figure 12.** Output current waveform of the three modules in parallel mode.

It is concluded from the abovementioned comparisons that the voltage overshoot is 10%, 14%, and 20% when the system is controlled by the Lyapunov algorithm, direct power algorithm, and dead-beat algorithm, respectively. Moreover, when the load varies, the Lyapunov algorithm restores the steady state after 0.02 s adjustment, while the output voltage is 5 V. On the other hand, the direct power algorithm reaches a new steady state after 0.03 s, while the output voltage is 4.8 V. In other words, the steady state voltage of the Lyapunov algorithm is slightly higher than that of the direct power algorithm. Meanwhile, the voltage drops and produces 4% error. When the dead-beat algorithm is used as control, the load changes and the system reaches a new steady state after 0.03 s. Here, the restoring time of steady state is longer than that of Lyapunov algorithm. On the AC side, the Lyapunov control algorithm can realize the unit power factor and the overshoot of the current before and after the load mutation is low. However, when the direct power control is adopted, the phase difference of the single-phase voltage and current exists all the time. The unit power factor cannot be realized by the direct power control algorithm because it needs to estimate the instantaneous reactive power and then estimate the voltage vector of the AC side. Since the AC current is large, when the current transient tracking index is satisfied, the AC side inductance is limited. The inductance affects the accuracy of the estimation of instantaneous reactive power, which leads to the phase difference between the voltage and current on the AC side. Thus, the unit power factor cannot be realized.

#### *3.4. Verification of the Experimental Result*

The main electrical parameters of the power circuit and control data, used in the implementation tests, are given in Table 3. The development of control algorithms was performed and simulated with Matlab/Simulink and the real-time implementation with a Texas Instrument digital signal processor (DSP) board (TMS320F28335). Each three-phase winding was used as an independent module. Each module, using a 32-bit DSP28335 processor, operated at a frequency of 150 MHz, with a bandwidth of 600 Mbps and a single precision floating-point-unit (FPU). The current sharing control was realized by CAN bus among the controllers, which improved the independence and reduced the interference of the modules, so the traditional fixed-point MCU would affect the result of algorithm processing. The control strategy can be used in the mainstream 32-bit floating-point microprocessor and 10-bit or more AD sampling. Figure 13 shows the experimental platform, which is the setup for experimental verification. This setup was used to verify the validity of the proposed Lyapunov control algorithm. When carrying out a load test, the setup can adjust the resistor box terminal to provide 10–1000 <sup>m</sup>Ω.

**Figure 13.** Large current load experiment of the integrated DC output system: (**a**) Experimental platform; (**b**) load experiment; (**c**) the schematic of the synchronous system.

Figure 14 shows the experiment waves of the output current and the three-phase AC current waveforms when a single rectifier module is in operation. When the load suddenly changed from 32 to 16 Ω at 0.04 s, as can be seen, the system reached the steady state in less than 0.02 s, their steady-state performances were both quite good. The proposed control can highly improve the dynamic performance of the DC output current with a shorter transition time. It is consistent with the simulation results.

**Figure 14.** The experiment waves of a single rectifier module: (**a**) Output current wave; (**b**) three-phase AC current waveform.

Figure 15 shows the experiment waves of the output voltage and the corresponding frequency spectrum when a single rectifier module is in operation. The logarithmic uniform distribution is used to facilitate the comparison of the longitudinal axis. The diagram shows that the output voltage of the rectifier module is stable. The amplitude of the voltage is 5 V when the main frequency is 0 Hz, and the

maximum value of the ripple of each frequency does not exceed 0.06 V, which meets the requirements of the DC output.

**Figure 15.** The experiment voltage waves of the DC power generation: (**a**) Output voltage wave; (**b**) output voltage spectrum.

Figure 16 shows the experiment waves of the output current and the corresponding spectrum when a single rectifier module is in operation. The vertical axis is logarithmically and uniformly distributed. Moreover, the unit ratio is 100 A. When the main wave of the output current is stable, the main frequency is still 0 Hz, which conforms to DC characteristics. Although there is a certain amplitude in other frequency bands, none of them exceeds 3 A.

**Figure 16.** The experiment current waves of the DC power generation: (**a**) Output current wave; (**b**) output current spectrum.

Figure 17 shows the waveforms' output voltage when the system load increases and decreases suddenly. The diagram illustrates that when the load changes, the output external characteristics of the rectifier module of the generation system changes, under the condition of the same duty cycle. This may be attributed to the internal resistance inside the device. The output voltage in the diagram can be quickly restored to a stable state without overshooting, which proves that the dynamic control performance of the Lyapunov algorithm is superior over the conventional methods.

**Figure 17.** Output voltage waves of resistance mutation: (**a**) Load increases; (**b**) load decreases.

Figures 18 and 19 present the current waveforms of the paralleling system when the light load switches to the heavy load, and vice versa. They indicate that the current variation and adjustment times are different between module 1 and 2. This originates from the two modules within the inherent relationship between different parameters and their corresponding master–slave structure, in which the regulating speed of the submodule is slower than that of the main module. Meanwhile, the control process contains the vibration. However, the two modules can quickly achieve stable output and current sharing, which proves the better dynamic response of digital current sharing, compared to the conventional methods.

**Figure 18.** Switching the light load to the heavy load.

**Figure 19.** Switching the heavy load to the light load.
