**2. Stability Analysis of DC Power System for Electric Vehicles**

The DC power system of electric vehicles is mainly composed of a power generation unit, generator set, energy storage unit, AC/DC load, and power converter connected to each unit module. As shown in Figure 1, in a power generation unit, the energy flows in one direction, the battery is connected to the DC bus by a DC/DC converter, and the generator set provides energy to the bus by an AC/DC converter.

**Figure 1.** DC/DC converters with constant power characteristics in electrical systems.

All kinds of power electronic devices in the power system of electric vehicles are connected to the on-board high-voltage power supply system of electric vehicles in the form of a cascade, and most of these power electronic devices adopt closed-loop control; when the bus voltage changes, the output power can remain constant. When the input voltage changes, the input current changes in the opposite trend; constant power load has negative impedance characteristics. It is therefore said to have a negative impedance characteristic (ΔV/ΔI < 0) constant power load.

In a constant power load, P is constant. Thus, as shown in Figure 2, as the voltage at both ends of a constant power load increases/decreases, its current decreases/increases. Because the incremental impedance of CPL is negative (ΔV/ΔI < 0), in this case, the system will deviate from its stable region, resulting in CPL negative impedance instability [29]. Interaction with other devices may affect the dynamic characteristics and stability of the system.

**Figure 2.** Negative impedance characteristics of constant power load.

As shown in Figure 3a. The midpoint is the initial stable operating point of the system. When the system is subjected to external disturbances causing an increase in the input current of the CPL, as can be seen from Figure 3a, the voltage at both ends of the CPL is at this time, and according to KVL, the voltage at both ends of the filtering inductor is at this time. At this point, the inductance current will further increase, causing the system to move away from the stable operating point. On the contrary, when the input current of the CPL decreases due to external disturbances in the system, the voltage at both ends of the CPL is reduced. From KVL, it can be seen that the voltage at both ends of the filter inductor is reduced, and the inductor current will further decrease, thus keeping the system away from the stable operating point. The obtained volt ampere characteristic curve is shown in Figure 3b for when the load is a pure resistive load, where the point is the initial stable operating point of the system. When external disturbances increase the input current, there is a filter inductance voltage, which can be determined by KVL. At this time, the current will correspondingly decrease, so the system can return to the initial stable operating point; that is, the system is stable. CPL negative impedance was obtained via a small signal analysis [30]. In Reference [30], the equivalent model of CPL was extracted through a small signal analysis and a large signal analysis.

**Figure 3.** Changes in DC bus voltage during power fluctuations of different types of loads: (**a**) constant power load power fluctuation and (**b**) pure resistance load power fluctuation.

In order to analyze the three-phase interleaved parallel DC/DC converter system supplying CPL, the conjugated model and circuit were extracted for a small signal analysis. Based on the analysis of Reference [3], the destabilizing effect and limit of CPL negative impedance on converter were explained. Since the root of the characteristic equation is on the right-hand side, there is negative impedance instability in the output voltage of the system. In addition, the control performance is severely degraded due to the inevitable voltage fluctuations in the DC supply voltage [31].

According to the research of Reference [3], through the small signal analysis of the system equation of state, the transfer function of the system can be obtained as follows:

$$\mathbf{G}(\mathbf{s}) = \frac{(1-\mathbf{d})\mathbf{U}\_{\text{bus}} - \mathbf{I}\_{\text{L}}\mathbf{L}\mathbf{s}}{\mathbf{L}\mathbf{C}\mathbf{s}^{2} + \frac{\mathbf{L}}{\mathbf{R}\_{\text{L}}}\mathbf{s} + (1-\mathbf{d})^{2}},\tag{1}$$

From the transfer function, with the increase of CPL power, the negative incremental resistance characteristic of CPL becomes more obvious, and the root of the system characteristic equation begins to move to the right of the complex plane. As shown in Figure 4, once the power consumed by CPL exceeds the power consumed by resistive loads, that is, *PCPL* > *PR*, CPL plays a dominant role in the system. The damping coefficient of the corresponding system is less than 0, and the slope of the output characteristic curve is negative. In this case, the DC bus voltage will be in an oscillating state. When the power consumption of CPL is less than that of resistive load, that is, *PCPL* < *PR*, the resistive load plays a dominant role in the system, the damping coefficient of the corresponding system is greater than 0, and the slope of the output characteristic curve is positive. Under this condition, the DC bus voltage of the system is in a stable state.

**Figure 4.** Stable and unstable regions based on small signal theory.

In order to elucidate the impact of CPL power fluctuations on the stability of the DC bus voltage, we first introduced some common CPLs in special vehicles and preliminarily analyzed the dynamic characteristics of negative incremental resistance of constant power loads. Secondly, through a theoretical analysis, we found that the power imbalance between the generating and receiving ends is the fundamental cause of bus voltage fluctuations. Then, based on small signals, the reason for the low-frequency oscillation of the DC bus caused by CPL was obtained. Through a simplified circuit analysis of an ideal voltage source, filter inductor, and CPL in series, it was found that when CPL power fluctuates, it amplifies the power fluctuation, causing the system to move away from the initial operating equilibrium point.
