2.3.3. Parameter Adjustment Criterion and Case

Usually, an increase in the transformer capacity and the addition of other types of equipment such as static synchronous reactive power compensators on the source side are considered in locomotive network systems to address LFO caused by impedance mismatch. Such problems also exist in systems under emergency power supply schemes. However, when using PV and battery locomotive traction, based on the stability law of the control parameters revealed in Section 2.3.2, the goal can be achieved by adjusting the converter parameters accordingly. The specific parameter adjustment criteria are as follows: Firstly, if the system experiences instability, consider lowering the PV boost converter *k*PVup, battery DC/DC converter *k*Bup, and RPC single-phase inverter *k*invp. Secondly, consider slightly lowering the *k*invi during PV and battery co-traction or battery traction, and slightly increasing the *k*invi during PV traction. It is worth noting that the RPC outerloop proportional gain seriously affects the control accuracy (i.e., voltage amplitude); therefore, prioritize adjusting the proportional gain of the PV and battery DC converters unless faced with a situation that has special requirements. Taking the ratio of PV modules to battery modules as 1:1 as an example, the following explains the methods of parameter adjustment and governance in three cases. The Nyquist curve of Case 1 is shown in Figure 11a. When using the PV and battery locomotive co-traction, the battery system's energy storage inductance deteriorates from 1 mH to 1.23 mH, and the curve surrounds (−1, j0); at this time, the adjustment *k*Bup is reduced from 15 to 5, and the Nyquist criterion shows that the system will return to stability. The theoretical curve of Case 2 is shown in Figure 11b. When the *k*PVup of the PV locomotive traction deteriorates, it causes a critical oscillation in the system. The *k*Bup is adjusted and connected to the battery system; at this time, the PV and battery combined locomotive traction are stable. The test validation waveforms of the parameter adjustment criteria and related conclusions described in this section will be presented in Section 3.2.1.

**Figure 11.** (**a**) Nyquist curve for Case 1; (**b**) Nyquist curve for Case 2.

2.3.4. The Influence of Mixed Proportional Parallel Numbers of PV and Battery Modules on System Stability

The number of PV and battery hybrid proportional parallel connections will also change the source-side impedance model to weaken or improve the stability of the system. Therefore, the original parameters were substituted into the hybrid parallel connection system. Based on the generalized Nyquist criterion, the bar chart shown in Figure 12a was obtained. The results show that the system operated stably under different PV and battery module proportions. To reveal its stability mechanism, the number of PV and battery modules were introduced as system variables, and because all the modules are connected in parallel, thus, expanding the impedance ratio expression *Z*0/*Z*inv in Section 2.3.1, we achieve the following formula:

$$L\_{\rm hui} = \left(\frac{\frac{Z\_{\rm pv} Z\_{\rm Bat}}{n\_{\rm PV} n\_{\rm bat}}}{\frac{Z\_{\rm PV}}{n\_{\rm PV}} + \frac{Z\_{\rm bat}}{n\_{\rm bat}}}\right) / Z\_{\rm inv} \tag{17}$$

where *n*pv represents the number of PV modules, and *n*bat represents the number of battery modules.

**Figure 12.** (**a**) The proportion of stable system operation when the PV and battery modules were connected in parallel; (**b**) analysis results of the passive criterion when the ratio of modules was 1:1.

If *n*pv = *n*bat = *n*, the expression is rewritten as *L*hui = (*Z*0/*Z*inv)/*n*, which is equivalent to proportionally reducing the Nyquist function curve (*L*hui) by *n* times. Systems that do not originally include (−1, j0) will be more stable. In the previous work, it was proven and verified that the system was stable when the PV traction module ratio was 1:0, the battery traction module ratio was 0:1, and the PV and battery co-traction module ratio was 1:1 under the original parameters; therefore, the stability of the system with the PV and battery module ratios of 4:0, 0:4, and 3:3 in Figure 12a is explained. However, it is complex and difficult to give an answer for the stability mechanism of non 1:1, 1:0 systems using the impedance ratio criterion, since it changes a variable to directly judge the stability of the overall system. Therefore, this paper introduces a passive criterion applicable to the stability analysis of variables, subsystems, and overall systems. The judgment criterion is as follows: if the sum of the real parts of the total admittance of all parallel systems in the bus is guaranteed to be constantly greater than 0, all subsystems can operate stably at the same time. This passive criterion was proposed and verified by Riccobono A. in 2012 [34] but compared to the impedance ratio criterion, it is less used. Based on the "PV–battery locomotive network" coupling system, this article additionally provides the proof process of the passive criterion from the perspective of dissipative system stability, as shown in Appendix B. The criterion combines Equations (5), (6), and (14) and Figure 7a, resulting in the following expression:

$$\begin{aligned} \operatorname{Re}\left[Y\_{\text{tot}}(s)\right] &= \operatorname{Re}\left[n\_{\text{PV}}Y\_{\text{PV}}(s) + n\_{\text{bat}}Y\_{\text{BAt}}(s) + Y\_{\text{inv}}(s)\right] \ge 0\\ &= \operatorname{Re}\left[\frac{n\_{\text{PV}}}{Z\text{pV}(s)} + \frac{n\_{\text{bat}}}{Z\_{\text{BAt}}(s)} + \frac{1}{Z\_{\text{inv}}(s)}\right] \ge 0 \end{aligned} \tag{18}$$

Applying Equation (18), the function curve is plotted under the original parameters when the locomotive traction has a PV and battery module ratio of 1:1, as shown in Figure 12b. At this time, the real parts of the admittance of the PV and battery modules are both greater than 0 in the entire frequency band. If the subsystem is paralleled in any proportion, it will increase the distance between Re[*Y*tot(*s*)] and the horizontal axis, and the system will obtain an additional stability margin to maintain stability. This reveals the reason why the system always maintains stability after the PV and battery module mixing ratio is paralleled. Based on the above analysis, the problem of LFO in PV and battery locomotive co-traction under specific working conditions can also be solved by using the law of the influence of the number of module parallel connections on stability to reshape the system impedance from the main circuit structure through the passive criterion. In Section 3.2.2 Case 1, we designed experiments and demonstrated the validity of the theoretical analysis.

Increasing the number of PV battery parallel connections in a targeted manner can address instability caused by multiple locomotives running together or unreasonable RPC parameter settings but it is difficult to use this method to solve instability caused by the deterioration of the system. The instability process of this phenomenon can be revealed using passive criteria, as shown in Figure 13, which shows the criterion curve after the 1:1 parameters of the PV and battery modules deteriorate. Increasing *n*pv and *n*bat will cause Re[*Y*PV(*s*)], Re[*Y*bat(*s*)], and Re[*Y*tot(*s*)] to extend towards negative infinity, thereby exacerbating the system instability, and this was validated in Section 3.2.2 Case 2.

**Figure 13.** Passive criterion analysis results after deterioration of the system parameters.

Based on the research in this section, it is recommended to ensure that the real part of the admittance of the PV and battery subsystem is always greater than 0 during the design.

#### 2.3.5. Passive Criterion Is Used to Reveal the Influence Law of Parameters on Stability

As shown in Section 2.3.4, the passive criterion based on the DC bus has clear boundary conditions and judgment criteria, which makes it easier to explore the law of the influence of parameters on the stability of the system, in comparison with the Nyquist criterion. This is reflected in the bivariate function curve, which directly shows the influence of a parameter of the system on its stability. This means that the law of stability near the original parameter will no longer be experimentally searched as before, and the three-dimensional plotting of the function can immediately reveal the influence of arbitrary values of a parameter on the stability of the original system. In order to verify the above analysis, passive criteria are used to reveal the stability influence laws of *k*PVup and *k*Bup, which have been verified before. By taking *k*PVup and *k*Bup as variables, we can obtain functions similar to Equation (19) but because the expressions are verbose, they are presented in the form of graphs, as shown in Figures 14 and 15, where the z axis is Re[*Y*tot(*s*)].

$$\operatorname{Re}[Y\_{\text{tot}}(s)] = F(k, \omega) \tag{19}$$

**Figure 14.** The influence of *k*PVup on the system stability: (**a**) global diagram; (**b**) local diagram of the system instability.

 

**Figure 15.** The influence of *k*Bup on the system stability: (**a**) global diagram; (**b**) local diagram of the system instability.

Figures 14a and 15a show that with the increase in *k*PVup and *k*Bup, the surface gradually extends to the negative plane; in other words, the system gradually becomes unstable, which is consistent with the results obtained using Nyquist's criterion and in the semiphysical simulation, however, we are more concerned about the LFO phenomenon that often occurs in actual railway systems, that is, the critical instability of the system. Accordingly, the local amplification when the corresponding parameter is unstable is shown in Figures 14b and 15b, and the values of the LFO parameters given by the criterion are *k*PVup = 5.679 or *k*Bup = 31.8, which are obviously lower than the values given by the generalized Nyquist criterion, as shown in Figure 10b. The system ran stably in the simulation experiment under these parameters. It is also necessary to continue to increase the parameter values to cause LFO. This is caused by the large conservatism of the passive criterion. Assuming that the law revealed by the criterion is used to control the LFO, if the stability of a parameter affecting the law is not monotonic and there are lots of extreme points, this situation may lead to a counterproductive parameter adjustment. Here, the law revealed by the criterion is more suitable when parametric influences are monotonic over a large scale. However, in the system design, the conservativeness of the criterion is favorable, and it will leave a certain stability margin.

In addition, by applying passive criteria to the AC bus, the designed virtual impedance compensator successfully suppresses LFO [17] but the specific proof process in the AC system and its conservative improvement are the next research foci concerning this criterion.
