A Nonlinear Disturbance Observer Based Virtual Negative Inductor Stabilizing Strategy for DC Microgrid with Constant Power Loads

: For the dc microgrid system with constant power loads (CPLs), the dc bus voltage can easily cause high-frequency oscillation owing to the complicated impedance interactions. The large line inductance and the CPL-side capacitance will form an undamped LC circuit on the dc bus, which, together with the CPL, will make the system fall into the negative-damping region, thus causing the system instability. To address this problem, a virtual negative inductor (VNI) is built on the source side converter in this paper, which can effectively counteract the large line inductance, thus alleviating the instability problem. Moreover, a nonlinear disturbance observer (NDO) is proposed for estimating the converter output current, which relieves the strong dependence of the proposed VNI strategy on the output current measurement. And the proposed strategy is implemented in a totally decentralized manner, thus alleviating the single-point-failure problem in the central controller. For assuring the optimal parameter value for the proposed stabilizing strategy, a system root-locus diagram based parameter designing approach is adopted. And comparative Nyquist diagram based stability analyses are taken for studying the robustness of the proposed strategy to the system perturbations. Finally, detailed real-time simulations are conducted for validating the effectiveness of the proposed stabilizing strategy.


Introduction
Nowadays, distributed generations (DGs) from renewable sources and energy storage systems are reshaping the structure of the modern power system [1,2]. Due to incremental penetrations of the dc-based renewable sources (PVs, batteries, and supercapacitors) and loads (electric vehicles), the dc microgrid system is gaining people's increasing attentions. Compared with the traditional ac power system, the newly emerged dc microgrid system has the following advantages: a higher efficiency with less energy conversion stages, an improved reliability with highly flexible structures, and an enhanced controllability with no reactive power regulation or synchronization problem.
From the perspective of the system composition, the dc microgrid system is a typical heterogeneous system with various kinds of active loads [3][4][5][6][7][8]. Due to the high performance of the load converter, the active load may exhibit a constant power load (CPL) characteristics even when the bus voltage fluctuates. It is known that the CPL will introduce a negative incremental resistance to the dc bus, which lowers the system damping effect, thus deteriorating the system stability. The situation would be more serious for the system with a large CPL power, a small source-side capacitance, or a large line inductance [9][10][11][12][13].

Problem Formulation
The detailed equivalent circuit diagram of the studied dc microgrid system is shown in Figure  1, which consists of one battery as the energy source, one bidirectional dc/dc converter, one equivalent resistive load, and one equivalent active load. is the dc bus voltage, eq v is the CPL-side capacitance voltage, L i is the inductance current, o i is the converter output current, CPL P is the CPL power, in L and in R are the input inductance and resistance, respectively, o C is the converter output capacitance, e Z is the line impedance, dc R is the equivalent resistive load, and eq C is the CPL-side capacitance. The system parameter values for the studied dc microgrid have been listed in Table 1.  For ensuring the proportional power sharing control among multiple parallel-connected sources, droop control method is adopted here, which can be expressed as follows: As shown in Figure 1, v s is the battery source voltage, v o is the converter output voltage, v dc is the dc bus voltage, v eq is the CPL-side capacitance voltage, i L is the inductance current, i o is the converter output current, P CPL is the CPL power, L in and R in are the input inductance and resistance, respectively, C o is the converter output capacitance, Z e is the line impedance, R dc is the equivalent resistive load, and C eq is the CPL-side capacitance. The system parameter values for the studied dc microgrid have been listed in Table 1. For ensuring the proportional power sharing control among multiple parallel-connected sources, droop control method is adopted here, which can be expressed as follows: where, v * nom represents the nominal output voltage, R droop is the droop coefficient, which can be obtained by the maximum allowable voltage deviation and maximum output current as follows: If we neglect the source voltage dynamics, the studied DC microgrid system shown in Figure 1 can be simplified as shown in Figure 2.
where, * nom v represents the nominal output voltage, droop R is the droop coefficient, which can be obtained by the maximum allowable voltage deviation and maximum output current as follows: If we neglect the source voltage dynamics, the studied DC microgrid system shown in Figure 1 can be simplified as shown in Figure 2. Figure 2. Simplified circuit diagram of the studied dc microgrid system.
where, CPL R is the negative incremental resistance of the CPL, which can be expressed as follows: Based on the Thevenin's theorem, the dc bus voltage of the simplified equivalent circuit shown in Figure 2 can be derived as follows where, For ensuring the system stability, we expect that there is no right half plane poles in (4).
According to the Hurwitz stability condition ( 0 i a  ) and (5), the following inequality conditions should be fulfilled: As we can see from the second inequality of (6), a large line inductance e L will have an adverse effect on the maximum allowable CPL power. In other words, a large line inductance will decrease the system stability margin. For maintaining the stability of the studied dc microgrid system, a smaller line inductance e L would be preferred. However, as we know, the large line inductance is physically determined by the distributed nature of the dc microgrid system, which cannot be arbitrarily modified according to our needs. Therefore, traditional stabilizing methods always seek their ways by adding physical dampers or dc-link capacitors to the dc bus. Before [9], few stabilizing effort has ever been made from the sense of decreasing the line inductance. However, the where, R CPL is the negative incremental resistance of the CPL, which can be expressed as follows: Based on the Thevenin's theorem, the dc bus voltage of the simplified equivalent circuit shown in Figure 2 can be derived as follows where, For ensuring the system stability, we expect that there is no right half plane poles in (4). According to the Hurwitz stability condition (a i > 0) and (5), the following inequality conditions should be fulfilled: As we can see from the second inequality of (6), a large line inductance L e will have an adverse effect on the maximum allowable CPL power. In other words, a large line inductance will decrease the system stability margin. For maintaining the stability of the studied dc microgrid system, a smaller line inductance L e would be preferred. However, as we know, the large line inductance is physically determined by the distributed nature of the dc microgrid system, which cannot be arbitrarily modified according to our needs. Therefore, traditional stabilizing methods always seek their ways by adding physical dampers or dc-link capacitors to the dc bus. Before [9], few stabilizing effort has ever been made from the sense of decreasing the line inductance. However, the effectiveness of the stabilizing strategy proposed in [9] is strongly dependent on the converter output current measurement. If the current sensor fails, the strategy will be invalid. Inspired by this fact, we try a different new way to stabilize the dc microgrid system, namely, the NDO based virtual VNI stabilizing strategy proposed in this paper.

DC/DC Converter Modeling and Inner Loop Controller
We first assume that the dc/dc source converter will be always operating in the continuous current mode. Then, the state-space average model of the dc/dc converter can be expressed as follows: where, d is the average duty ratio of dc/dc converter. The corresponding definition of variables in (7) have been declared above, under Figure 1.
Conventional dual loop controller is adopted here for regulating the output voltage of the dc/dc converter, as depicted in Figure 3.

DC/DC Converter Modeling and Inner Loop Controller
We first assume that the dc/dc source converter will be always operating in the continuous current mode. Then, the state-space average model of the dc/dc converter can be expressed as follows: where, d is the average duty ratio of dc/dc converter. The corresponding definition of variables in (7) have been declared above, under Figure 1.
Conventional dual loop controller is adopted here for regulating the output voltage of the dc/dc converter, as depicted in Figure 3. Figure 3. Basic structure of dc/dc converter dual loop controller.
From Figure 3, we can calculate the closed-loop transfer functions of the inner inductance current loop and the output voltage loop as shown in (8) and (9). It should be mentioned that (9) is obtained here by neglecting the inner dynamics of the inductance current loop.
where, pi k and ii k are the inductance current PI controller parameters, pv k and iv k are the output voltage PI controller parameters. From (8) and (9), we can see that both the inductance current loop and the output voltage loop can be taken as the typical second-order system. The inner loop controller parameters can be easily determined by designating the system damping ratio ξ and the natural frequency ω based on the classical control theory. Due to the space limitation, the detailed parameter designing process for the inner loop controller is just omitted here. The final designing results and the corresponding system parameters have been listed in Table 1.  (8) and (9). It should be mentioned that (9) is obtained here by neglecting the inner dynamics of the inductance current loop.
where, k pi and k ii are the inductance current PI controller parameters, k pv and k iv are the output voltage PI controller parameters. From (8) and (9), we can see that both the inductance current loop and the output voltage loop can be taken as the typical second-order system. The inner loop controller parameters can be easily determined by designating the system damping ratio ξ and the natural frequency ω based on the classical control theory. Due to the space limitation, the detailed parameter designing process for the inner loop controller is just omitted here. The final designing results and the corresponding system parameters have been listed in Table 1.

Virtual Negative Inductor Stabilizing Strategy
As stated in Section 2, a large line inductance may deteriorate the dc microgrid system stability margin. Because of the distributed nature of the dc microgrid system, the line inductance cannot be arbitrarily reduced according to our needs. For solving this problem, a virtual negative inductor (VNI) stabilizing strategy is proposed in this paper.
The proposed VNI stabilizing strategy is built on the source-side converter through the modified droop control method, which voltage reference value can be expressed as follows: where, s is the Laplacian operator, −L droop is the built VNI, v * o is the output voltage reference, v * nom is the nominal value of the bus voltage, and i o is the output current.
For clearly illustrating the basic principles of the proposed stabilizing strategy, a simplified circuit of the proposed VNI is shown in Figure 4.
where, s is the Laplacian operator, For clearly illustrating the basic principles of the proposed stabilizing strategy, a simplified circuit of the proposed VNI is shown in Figure 4. alleviating the large line inductance destabilizing effect. As shown in (10), there is a pure differentiating operator in the designated output voltage reference, which may bring undesired high-frequency noises to system. For solving this problem, a low-pass filter is introduced here. The modified output voltage reference of the proposed VNI stabilizing strategy can be then expressed as follows: where, τ is the time constant of the introduced low-pass filter.

Nonlinear Disturbance Observer
From (11), we can see that the implementation of the proposed VNI stabilizing strategy requires an accurate measurement of the dc/dc converter output current o i . Consequently, the effectiveness of the proposed stabilizing strategy will be strongly dependent on the output current sensor. If the sensor fails, the proposed VNI stabilizing strategy will be invalid. For solving this current sensor failure problem, a nonlinear disturbance observer (NDO) for the source-side dc/dc converter is proposed here for estimating the output current. The basic structure of the proposed NDO is shown in Figure 5. As shown in Figure 4, with the proposed VNI stabilizing strategy, a virtual negative inductor −L droop is built on the dc bus in series with the large line inductance L e . From the system point of view, the large line inductance L e will be counteracted by the built negative inductor −L droop , thus alleviating the large line inductance destabilizing effect.
As shown in (10), there is a pure differentiating operator in the designated output voltage reference, which may bring undesired high-frequency noises to system. For solving this problem, a low-pass filter is introduced here. The modified output voltage reference of the proposed VNI stabilizing strategy can be then expressed as follows: where, τ is the time constant of the introduced low-pass filter.

Nonlinear Disturbance Observer
From (11), we can see that the implementation of the proposed VNI stabilizing strategy requires an accurate measurement of the dc/dc converter output current i o . Consequently, the effectiveness of the proposed stabilizing strategy will be strongly dependent on the output current sensor. If the sensor fails, the proposed VNI stabilizing strategy will be invalid. For solving this current sensor failure problem, a nonlinear disturbance observer (NDO) for the source-side dc/dc converter is proposed here for estimating the output current. The basic structure of the proposed NDO is shown in Figure 5. As shown in Figure 5, the NDO receives the output voltage o v , the inductance current L i , and the average duty ratio d as the input signals. For the continuity of description, the corresponding designing process of the NDO have been specified in the Appendix A. Only the final designing results of the NDO are presented here, which can be expressed as follows: where, z is the dummy variable for the proposed NDO, 2 l is the observer gain to be designed, and ˆo i is the estimated output current.
With the NDO expressed in (12), we can relieve the strong dependence of the proposed VNI stabilizing strategy on the output current sensor. Before proceeding to the next combination of the proposed NDO with the VNI stabilizing strategy, we would like to analyze the effect of the proposed NDO here first.
As depicted in Appendix A, we find out that the effect of the proposed NDO shown in Figure 5  As shown in Figure 5, the NDO receives the output voltage v o , the inductance current i L , and the average duty ratio d as the input signals. For the continuity of description, the corresponding designing process of the NDO have been specified in the Appendix A. Only the final designing results of the NDO are presented here, which can be expressed as follows: where, z is the dummy variable for the proposed NDO, l 2 is the observer gain to be designed, andî o is the estimated output current.
With the NDO expressed in (12), we can relieve the strong dependence of the proposed VNI stabilizing strategy on the output current sensor. Before proceeding to the next combination of the proposed NDO with the VNI stabilizing strategy, we would like to analyze the effect of the proposed NDO here first.
As depicted in Appendix A, we find out that the effect of the proposed NDO shown in Figure 5 is equivalent to a simple first-order low-pass filter, which can be expressed as follows: where, T NDO is the time constant of the equivalent low-pass filter, which can be expressed as Based on (11) and (12), the detailed structure of the NDO based VNI stabilizing strategy for the studied dc microgrid system can be depicted as shown in Figure 6.
where, z is the dummy variable for the proposed NDO, 2 l is the observer gain to be designed, and ˆo i is the estimated output current.
With the NDO expressed in (12), we can relieve the strong dependence of the proposed VNI stabilizing strategy on the output current sensor. Before proceeding to the next combination of the proposed NDO with the VNI stabilizing strategy, we would like to analyze the effect of the proposed NDO here first.
As depicted in Appendix A, we find out that the effect of the proposed NDO shown in Figure 5 is equivalent to a simple first-order low-pass filter, which can be expressed as follows: where, NDO T is the time constant of the equivalent low-pass filter, which can be expressed as Based on (11) and (12), the detailed structure of the NDO based VNI stabilizing strategy for the studied dc microgrid system can be depicted as shown in Figure 6. Figure 6. Detailed structure of NDO based VNI stabilizing strategy. Figure 6. Detailed structure of NDO based VNI stabilizing strategy.
As shown in Figure 6, the estimated output currentî o from the NDO is directly sent to the VNI stabilizing controller, which modifies the output voltage reference v * o for the dc/dc converter. As shown in (11)- (14), there are three parameters remaining to be designed in the proposed NDO based VNI stabilizing strategy, namely, the virtual negative inductor −L droop , the time constant of the low-pass filter τ, and the time constant of the nonlinear disturbance observer T NDO .

Parameter Designing
For maximizing the stability margins of the studied dc microgrid system, a system root-locus based approach is adopted here for the proposed NDO based VNI stabilizing strategy. In this parameter designing approach, the system root-locus are plotted with the three parameters to be designed varying from a smaller value to a larger one. The optimal parameter value will be the one that corresponds the point that is farthest away from the right-half-plane, which is also the most stable point for the system.
The small-signal model of the studied dc microgrid system is required for plotting the system root-locus diagram, which has been presented in the Appendix B. Based on the small-signal model, we can now plot the system root locus diagram.
As shown in Figure 7, the root locus of the studied dc microgrid system is plotted by varying the NDO time constant T NDO from 0.7 ms to 2.6 ms. As we can see from Figure 7, the allowable range for the nonlinear disturbance observer time constant T NDO is from 0.9 ms to 1.9 ms. If T NDO is too large or too small, the system root-locus will enter into the right half plane (RHP), making the system unstable. The optimal value for the NDO time constant T NDO is 1.2 ms, which makes the system dominant poles farthest away from the RHP. for the nonlinear disturbance observer time constant NDO T is from 0.9 ms to 1.9 ms. If is too large or too small, the system root-locus will enter into the right half plane (RHP), making the system unstable. The optimal value for the NDO time constant NDO T is 1.2 ms, which makes the system dominant poles farthest away from the RHP. As shown in Figure 8, with the time constant of the NDO set to be 1.2 ms, the system root-locus diagram is plotted with the virtual negative inductor L droop varying from 0.02 mH to 0.4 mH. It can be observed that there is an inflection point (0.1 mH) in the system root-locus diagram, which is also the dominant poles farthest away from the RHP. Therefore, the optimal value for the L droop can be confirmed to be 0.1 mH from Figure 8. With the NDO time constant set to be 1.2 ms and the virtual negative inductor set to be 0.1 mH, we vary the low-pass filter time constant from 0.02 ms to 0.4 ms. The corresponding system root locus diagram for this case is shown in Figure 9. With the NDO time constant set to be 1.2 ms and the virtual negative inductor set to be 0.1 mH, we vary the low-pass filter time constant from 0.02 ms to 0.4 ms. The corresponding system root locus diagram for this case is shown in Figure 9. With the NDO time constant set to be 1.2 ms and the virtual negative inductor set to be 0.1 mH, we vary the low-pass filter time constant from 0.02 ms to 0.4 ms. The corresponding system root locus diagram for this case is shown in Figure 9. As shown in Figure 9, as long as the low-pass filter time constant τ is lower than 0.3 ms, the system dominant poles will always be the pole-pairs with the negative real parts nearly −97.7. Hence, the system stability characteristics will not be affected with the low-pass filter time constant varying from 0.02 ms to 0.3 ms. However, as shown in Figure 9, there are two branches for the sub-dominant poles when increasing the low-pass filter time constant, one from the left to the right, and the other from the right to the left. For ensuring the fast convergence of the system, we select the most negative sub-dominant poles for the low-pass filter time constant here, namely, 0.08 ms.
The detailed implementation procedure of the proposed root locus diagram based parameter designing approach is summarized as follows: Step 1: Start; Step 2: Construct the small signal model of the studied dc microgrid system; Step 3: Calculate the system eigenvalues based on the small-signal model; As shown in Figure 9, as long as the low-pass filter time constant τ is lower than 0.3 ms, the system dominant poles will always be the pole-pairs with the negative real parts nearly −97.7. Hence, the system stability characteristics will not be affected with the low-pass filter time constant varying from 0.02 ms to 0.3 ms. However, as shown in Figure 9, there are two branches for the sub-dominant poles when increasing the low-pass filter time constant, one from the left to the right, and the other from the right to the left. For ensuring the fast convergence of the system, we select the most negative sub-dominant poles for the low-pass filter time constant here, namely, 0.08 ms.
The detailed implementation procedure of the proposed root locus diagram based parameter designing approach is summarized as follows: Step 1: Start; Step 2: Construct the small signal model of the studied dc microgrid system; Step 3: Calculate the system eigenvalues based on the small-signal model; Step 4: Determine the optimal value of the nonlinear disturbance observer gain T NDO from the root locus diagram shown in Figure 7; Step 5: Determine the optimal value of the virtual negative inductor L droop from the root locus diagram shown in Figure 8; Step 6: Determine the optimal value of the low-pass filter time constant τ from the root locus diagram shown in Figure 9; Step 7: End. For the clearance of description, the stabilizer parameter designing results have been summarized in Table 2.

Stability Analysis
In this section, the stability of the proposed NDOB-VNI stabilizing strategy is studied with the impedance-based stability analysis method. The output impedance model of the dc/dc converter is first presented. And the Nyquist stability criterion for the studied dc microgrid system is carefully derived. With the help of the system Nyquist diagram, three cases are analyzed for studying the robustness of the proposed NDOB-VNI stabilizing strategy to the system perturbations. These three cases include the change of the CPL power, the variations of the droop coefficient, and the perturbations of the CPL-side capacitance.

Output Impedance Model of the DC/DC Converter
Based on the small-signal model obtained in (B1), we can plot the small-signal model diagram of the dc/dc converter, as shown in Figure 10.

Stability Analysis
In this section, the stability of the proposed NDOB-VNI stabilizing strategy is studied with the impedance-based stability analysis method. The output impedance model of the dc/dc converter is first presented. And the Nyquist stability criterion for the studied dc microgrid system is carefully derived. With the help of the system Nyquist diagram, three cases are analyzed for studying the robustness of the proposed NDOB-VNI stabilizing strategy to the system perturbations. These three cases include the change of the CPL power, the variations of the droop coefficient, and the perturbations of the CPL-side capacitance.

Output Impedance Model of the DC/DC Converter
Based on the small-signal model obtained in (B1), we can plot the small-signal model diagram of the dc/dc converter, as shown in Figure 10. where, where, The output impedance model of the dc/dc converter can be then obtained from Figure 10 as follows:

Nyquist Stability Criterion
The equivalent circuit diagram of the studied dc microgrid system considering the source dynamics is illustrated in Figure 11, where, Z o is the output impedance of the dc/dc converter, Z e is the line impedance, C eq is the equivalent CPL-side capacitance, and R dc is the equivalent resistive load.  Figure 11. Equivalent circuit diagram of the studied dc microgrid system.
As shown in Figure 11, the dc bus voltage dc v can be expressed as The stability of the studied dc microgrid system can be ensured if there is no right-half-plane poles in (28). For the clearance of description, we can reorganize the denominator in (28) as follows where, S Z is the source output impedance, L Y is the load input admittance, and M T is the system minor loop gain, which can be expressed as follows , According to the Argument Principle, the number of RHZ in (29) can be calculated by the number of times that Nyquist trajectory encircles the point (−1,0): where, (0,0) N and ( 1,0) N − are the number of times that Nyquist trajectory encircles the point (0,0) and the point (−1,0) in clockwise direction, respectively. It should be mentioned that as the source and CPL are designed to be stable alone, the (1 ) would be just zero in (33). Therefore, the system stability analysis problem is then transformed into judging whether the Nyquist trajectory of M T encircles the critical point (−1,0): Figure 11. Equivalent circuit diagram of the studied dc microgrid system.
As shown in Figure 11, the dc bus voltage v dc can be expressed as The stability of the studied dc microgrid system can be ensured if there is no right-half-plane poles in (28). For the clearance of description, we can reorganize the denominator in (28) as follows where, Z S is the source output impedance, Y L is the load input admittance, and T M is the system minor loop gain, which can be expressed as follows According to the Argument Principle, the number of RHZ in (29) can be calculated by the number of times that Nyquist trajectory encircles the point (−1,0): where, N (0,0) and N (−1,0) are the number of times that Nyquist trajectory encircles the point (0,0) and the point (−1,0) in clockwise direction, respectively. It should be mentioned that as the source and CPL are designed to be stable alone, the RHP(1 + Z S · Y L ) would be just zero in (33). Therefore, the system stability analysis problem is then transformed into judging whether the Nyquist trajectory of T M encircles the critical point (−1,0): (a) If N (−1,0) (Z S · Y L ) equals to zero, the system will be stable; (b) Or else, the system will be unstable.

Comparative Stability Analysis
With the output impedance model of the dc/dc converter and the Nyquist based stability criterion stated above, we will conduct the comparative stability analyses for the studied dc microgrid system in this section.

Case 1: CPL Power Change
In this case, three same CPL power conditions are considered for these two comparative systems, which are 0.8 kW, 1.8 kW, and 2.8 kW, respectively. The parameter settings for this comparative case have been listed in Tables 1 and 2. The comparative stability analysis results are shown in Figure 12. In this case, three same CPL power conditions are considered for these two comparative systems, which are 0.8 kW, 1.8 kW, and 2.8 kW, respectively. The parameter settings for this comparative case have been listed in Tables 1 and 2. The comparative stability analysis results are shown in Figure 12. As shown in Figure 12a, all of the Nyquist trajectories of the NDOB-VNI controlled dc microgrid system do not encircle the critical point (−1,0), which means the system remains stable for the CPL power change under the proposed NDOB-VNI stabilizing strategy. However, as shown in Figure 12b, the Nyquist trajectories for the conventional droop controlled system with 1.8 kW and 2.8 kW CPL power encircle the point (−1,0), which means that the conventional droop controlled system are unstable with the CPL power change. The proposed NDOB-VNI stabilizing strategy shows a better robustness to the CPL power change.

Case 2: Droop Coefficient Variation
During the system operation, the droop coefficient may be adjusted online for balance of the battery state of charge. The varying droop coefficient may impact the system stability. Therefore, in this case, comparative stability analysis are conducted for the dc microgrid system with a varying droop coefficient, as shown in Figure 13. Three droop coefficients are considered, namely, 0.4, 0.6, and 0.8, respectively.
From Figure 13, we can see that a large droop coefficient may have an adverse effect on the system stability. As shown in Figure 13a, all of the three Nyquist trajectories for the NDO based VNI controlled dc microgrid system do not encircle the critical point (−1,0), which means the NOD based VNI strategy is robust to the droop coefficient variation. However, as shown in Figure 13b, for the conventional droop controlled system with droop coefficients of 0.6 and 0.8, the system Nyquist trajectories do encircle the critical point (−1,0), which means that the conventional droop controlled system is unstable with the droop coefficient variation. Imaginary Axis Figure 12. Comparative stability analysis result of the studied dc microgrid system with CPL power changes. (a) Proposed NDOB-VNI stabilizing strategy controlled system; (b) Conventional droop controlled system. Figure 12a, all of the Nyquist trajectories of the NDOB-VNI controlled dc microgrid system do not encircle the critical point (−1,0), which means the system remains stable for the CPL power change under the proposed NDOB-VNI stabilizing strategy. However, as shown in Figure 12b, the Nyquist trajectories for the conventional droop controlled system with 1.8 kW and 2.8 kW CPL power encircle the point (−1,0), which means that the conventional droop controlled system are unstable with the CPL power change. The proposed NDOB-VNI stabilizing strategy shows a better robustness to the CPL power change.

Case 2: Droop Coefficient Variation
During the system operation, the droop coefficient may be adjusted online for balance of the battery state of charge. The varying droop coefficient may impact the system stability. Therefore, in this case, comparative stability analysis are conducted for the dc microgrid system with a varying droop coefficient, as shown in Figure 13. Three droop coefficients are considered, namely, 0.4, 0.6, and 0.8, respectively.
From Figure 13, we can see that a large droop coefficient may have an adverse effect on the system stability. As shown in Figure 13a, all of the three Nyquist trajectories for the NDO based VNI controlled dc microgrid system do not encircle the critical point (−1,0), which means the NOD based VNI strategy is robust to the droop coefficient variation. However, as shown in Figure 13b, for the conventional droop controlled system with droop coefficients of 0.6 and 0.8, the system Nyquist trajectories do encircle the critical point (−1,0), which means that the conventional droop controlled system is unstable with the droop coefficient variation.

Case 3: CPL-side Capacitance Perturbation
During the system operation, the CPL-side capacitance eq C may be varying due to the connection or disconnection of the CPL. For illustrating this impact on the system stability, three CPLside capacitances are considered in this case, which are 2000 μF , 1100 μF, and 470 μF , respectively. The comparative stability analysis results of the studied dc microgrid system with a varying CPLside capacitance are shown in Figure 14. As shown in Figure 14a, all of the three Nyquist trajectories for the NDO based VNI controlled system do not encircle the critical point (−1,0), which means that the proposed NDO based VNI stabilizing strategy is robust to the CPL-side capacitance perturbation. However, for the conventional droop controlled system, the case with capacitance of 2200 μF and 1100 μF are unstable.
Different from the conventional view, we can see from Figure 14b that a large dc-link capacitance, on the contrary, will destabilize the studied dc microgrid system. In the studied dc microgrid system,

Case 3: CPL-side Capacitance Perturbation
During the system operation, the CPL-side capacitance C eq may be varying due to the connection or disconnection of the CPL. For illustrating this impact on the system stability, three CPL-side capacitances are considered in this case, which are 2000 µF, 1100 µF, and 470 µF, respectively. The comparative stability analysis results of the studied dc microgrid system with a varying CPL-side capacitance are shown in Figure 14.

Case 3: CPL-side Capacitance Perturbation
During the system operation, the CPL-side capacitance eq C may be varying due to the connection or disconnection of the CPL. For illustrating this impact on the system stability, three CPLside capacitances are considered in this case, which are 2000 μF , 1100 μF, and 470 μF , respectively. The comparative stability analysis results of the studied dc microgrid system with a varying CPLside capacitance are shown in Figure 14. As shown in Figure 14a, all of the three Nyquist trajectories for the NDO based VNI controlled system do not encircle the critical point (−1,0), which means that the proposed NDO based VNI stabilizing strategy is robust to the CPL-side capacitance perturbation. However, for the conventional droop controlled system, the case with capacitance of 2200 μF and 1100 μF are unstable.
Different from the conventional view, we can see from Figure 14b that a large dc-link capacitance, on the contrary, will destabilize the studied dc microgrid system. In the studied dc microgrid system, Imaginary Axis

Nyquist Diagram
Real Axis Imaginary Axis Figure 14. Comparative stability analysis result of the studied dc microgrid system with CPL-side capacitance perturbations. (a) Proposed NDOB-VNI stabilizing strategy controlled system; (b) Conventional droop controlled system.
As shown in Figure 14a, all of the three Nyquist trajectories for the NDO based VNI controlled system do not encircle the critical point (−1,0), which means that the proposed NDO based VNI stabilizing strategy is robust to the CPL-side capacitance perturbation. However, for the conventional droop controlled system, the case with capacitance of 2200 µF and 1100 µF are unstable. Different from the conventional view, we can see from Figure 14b that a large dc-link capacitance, on the contrary, will destabilize the studied dc microgrid system. In the studied dc microgrid system, the large line inductance and the CPL-side capacitance form an undamped LC circuit, which has a resonance frequency similar to the control bandwidth of the converter inner inductance current loop. The resonance frequency overlap is the key point that causes the system high-frequency oscillations. Particularly, for the studied dc microgrid system, the decrease of the CPL-side capacitance is the right direction to eliminate this frequency overlap, coincidently. Therefore, we cannot draw a general conclusion about how the CPL-side capacitance impacts the system stability. But what we can confirm is that the proposed NDO based VNI stabilizing strategy shows a better robustness to the CPL-side capacitance perturbation.

Numerical Simulations
In this section, detailed real-time simulations are conducted for validating the effectiveness of the proposed NDO based VNI stabilizing strategy. The experiment setup is shown in Figure 15, which consists of an AppSIM real-time simulator enabling a detailed simulation of the studied dc microgrid system. Three comparative simulating cases are studied here, which aim at validating the effectiveness and robustness of the proposed stabilizing strategy. The system parameters used for simulation have been listed in Tables 1 and 2. Energies 2018, 11, x FOR PEER REVIEW 15 of 23 the large line inductance and the CPL-side capacitance form an undamped LC circuit, which has a resonance frequency similar to the control bandwidth of the converter inner inductance current loop. The resonance frequency overlap is the key point that causes the system high-frequency oscillations. Particularly, for the studied dc microgrid system, the decrease of the CPL-side capacitance is the right direction to eliminate this frequency overlap, coincidently. Therefore, we cannot draw a general conclusion about how the CPL-side capacitance impacts the system stability. But what we can confirm is that the proposed NDO based VNI stabilizing strategy shows a better robustness to the CPL-side capacitance perturbation.

Numerical Simulations
In this section, detailed real-time simulations are conducted for validating the effectiveness of the proposed NDO based VNI stabilizing strategy. The experiment setup is shown in Figure 15, which consists of an AppSIM real-time simulator enabling a detailed simulation of the studied dc microgrid system. Three comparative simulating cases are studied here, which aim at validating the effectiveness and robustness of the proposed stabilizing strategy. The system parameters used for simulation have been listed in Tables 1 and 2.

Effectiveness Validation
In the first case, transient numerical simulations are conducted for showing the effectiveness of the proposed stabilizing strategy. The system parameters set for this case have been listed in Tables 1 and 2. The CPL power for two comparative systems steps up from 0.8 kW to 1.8 kW at the time of 2.5 s, as shown in Figure 16.
As shown in Figure 16a, when the CPL power steps up from 0.8 kW to 1.8 kW, the conventional droop controlled system starts to oscillate with a frequency nearly 2244 rad/s. The oscillation amplitudes of the dc/dc converter inductance current, output voltage, and output current are 3.2 A, 0.7 V, and 2.36 A, respectively. The peak values of the corresponding variables are 28.98 A, 191.8 V, and 14.47 A, respectively.
As shown in Figure 16b, the proposed NDOB-VNI stabilizing strategy controlled system remains stable with the step change of the CPL power. Moreover, the estimated output current tracks the real output current well. With the proposed NODB-VNI stabilizing strategy, the perturbed dc microgrid system is restored to a new stable state within 50 ms, which shows an excellent dynamics. Moreover, the peak value of the dc/dc converter inductance current, output voltage and output current are 27.26 A, 192.4 V, and 13.27 A respectively, which are lower than that of the conventional droop controlled system listed above. From the simulation results shown in Figure 11, we can see

Effectiveness Validation
In the first case, transient numerical simulations are conducted for showing the effectiveness of the proposed stabilizing strategy. The system parameters set for this case have been listed in Tables 1 and 2. The CPL power for two comparative systems steps up from 0.8 kW to 1.8 kW at the time of 2.5 s, as shown in Figure 16.
As shown in Figure 16a, when the CPL power steps up from 0.8 kW to 1.8 kW, the conventional droop controlled system starts to oscillate with a frequency nearly 2244 rad/s. The oscillation amplitudes of the dc/dc converter inductance current, output voltage, and output current are 3.2 A, 0.7 V, and 2.36 A, respectively. The peak values of the corresponding variables are 28.98 A, 191.8 V, and 14.47 A, respectively.
As shown in Figure 16b, the proposed NDOB-VNI stabilizing strategy controlled system remains stable with the step change of the CPL power. Moreover, the estimated output current tracks the real output current well. With the proposed NODB-VNI stabilizing strategy, the perturbed dc microgrid system is restored to a new stable state within 50 ms, which shows an excellent dynamics. Moreover, the peak value of the dc/dc converter inductance current, output voltage and output current are 27.26 A, 192.4 V, and 13.27 A respectively, which are lower than that of the conventional droop controlled system listed above. From the simulation results shown in Figure 11, we can see that the proposed NDOB-VNI stabilizing strategy shows a better robustness to the CPL power variations. The effectiveness of the proposed NDOB-VNI stabilizing strategy is validated. It should be noted that in Figure 16b, the NDO estimated value is different from the real output current during the system transient stage. This dynamic difference is indeed caused by the limited convergence rate of the NDO. We have pointed out that the estimating effect of NDO is equivalent to a first order low-pass filter in Section 3. The dynamic difference between ˆo i and o i shown in Figure 16b is in fact caused by the low-pass filtering effect of the NDO. Similarly, observable differences can be also found in the following simulating cases. Moreover, we have considered this dynamic error during the designing process. Therefore, this slight difference will not affect the effectiveness of the proposed stabilizing strategy.

Test of Droop Coefficient Variation
For validating the robustness of the proposed stabilizing strategy to the droop coefficient variation, a comparative simulation case is studied here, as shown in Figure 17. The droop coefficient varies from 0.4 to 0.6 at the time of 2.5 s for both two comparative systems, with the CPL power equal to 1 kW. The system parameters for this case have been listed in Tables 1 and 2. As shown in Figure 17a, when the droop coefficient varies from 0.4 to 0.6, the conventional droop controlled system starts to oscillate with a frequency nearly 2244 rad/s. The studied dc microgrid system becomes unstable, which is also consistent with the system stability analysis results shown in Figure 12.
On the contrary, as shown in Figure 17b, the proposed NDOB-VNI stabilizing strategy controlled system remains stable. The estimated output current also follows the real output current well. The system is restored to a new stable state within 50 ms, which also shows a great dynamic characteristics with acceptable instant peak values of system variables. From Figure 17, we can see that a large droop coefficient may deteriorate the stability of the conventional droop controlled system. But with the proposed NDOB-VNI stabilizing strategy, the studied dc microgrid system could be more robust to the droop coefficient variation. It should be noted that in Figure 16b, the NDO estimated value is different from the real output current during the system transient stage. This dynamic difference is indeed caused by the limited convergence rate of the NDO. We have pointed out that the estimating effect of NDO is equivalent to a first order low-pass filter in Section 3. The dynamic difference betweenî o and i o shown in Figure 16b is in fact caused by the low-pass filtering effect of the NDO. Similarly, observable differences can be also found in the following simulating cases. Moreover, we have considered this dynamic error during the designing process. Therefore, this slight difference will not affect the effectiveness of the proposed stabilizing strategy.

Test of Droop Coefficient Variation
For validating the robustness of the proposed stabilizing strategy to the droop coefficient variation, a comparative simulation case is studied here, as shown in Figure 17. The droop coefficient varies from 0.4 to 0.6 at the time of 2.5 s for both two comparative systems, with the CPL power equal to 1 kW. The system parameters for this case have been listed in Tables 1 and 2. As shown in Figure 17a, when the droop coefficient varies from 0.4 to 0.6, the conventional droop controlled system starts to oscillate with a frequency nearly 2244 rad/s. The studied dc microgrid system becomes unstable, which is also consistent with the system stability analysis results shown in Figure 12.
On the contrary, as shown in Figure 17b, the proposed NDOB-VNI stabilizing strategy controlled system remains stable. The estimated output current also follows the real output current well. The system is restored to a new stable state within 50 ms, which also shows a great dynamic characteristics with acceptable instant peak values of system variables. From Figure 17, we can see that a large droop coefficient may deteriorate the stability of the conventional droop controlled system. But with the proposed NDOB-VNI stabilizing strategy, the studied dc microgrid system could be more robust to the droop coefficient variation.

Test of CPL-Side Capacitance Variations
In this simulation case, the robustness of the proposed NDOB-VNI stabilizing strategy to the variations of the CPL-side capacitance is tested, as shown in Figure 18. The CPL-side capacitance varies from 470 μF to 1100 μF at the time of 2.5 s, with the CPL power of 2.9 kW. As shown in Figure 18a, the conventional droop controlled system is initially operating in a stable state with the 470 μF CPL-side capacitance. At the time of 2.5 s with the variations of CPLside capacitance from 470 μF to 1100 μF , the conventional droop controlled system starts to oscillate with a frequency of 2244 rad/s. Different from the conventional view that a large dc capacitance may help to stabilize the system, we can see that a large CPL-side capacitance will indeed, on the other way, destabilize the system. However, as shown in Figure 18b, the NDOB-VNI stabilizing strategy controlled dc microgrid system remains stable with the variations of the CPL-side capacitance. And the new steady-state

Test of CPL-Side Capacitance Variations
In this simulation case, the robustness of the proposed NDOB-VNI stabilizing strategy to the variations of the CPL-side capacitance is tested, as shown in Figure 18. The CPL-side capacitance varies from 470 µF to 1100 µF at the time of 2.5 s, with the CPL power of 2.9 kW.

Test of CPL-Side Capacitance Variations
In this simulation case, the robustness of the proposed NDOB-VNI stabilizing strategy to the variations of the CPL-side capacitance is tested, as shown in Figure 18. The CPL-side capacitance varies from 470 μF to 1100 μF at the time of 2.5 s, with the CPL power of 2.9 kW. As shown in Figure 18a, the conventional droop controlled system is initially operating in a stable state with the 470 μF CPL-side capacitance. At the time of 2.5 s with the variations of CPLside capacitance from 470 μF to 1100 μF , the conventional droop controlled system starts to oscillate with a frequency of 2244 rad/s. Different from the conventional view that a large dc capacitance may help to stabilize the system, we can see that a large CPL-side capacitance will indeed, on the other way, destabilize the system. However, as shown in Figure 18b, the NDOB-VNI stabilizing strategy controlled dc microgrid system remains stable with the variations of the CPL-side capacitance. And the new steady-state As shown in Figure 18a, the conventional droop controlled system is initially operating in a stable state with the 470 µF CPL-side capacitance. At the time of 2.5 s with the variations of CPL-side capacitance from 470 µF to 1100 µF, the conventional droop controlled system starts to oscillate with a frequency of 2244 rad/s. Different from the conventional view that a large dc capacitance may help to stabilize the system, we can see that a large CPL-side capacitance will indeed, on the other way, destabilize the system. However, as shown in Figure 18b, the NDOB-VNI stabilizing strategy controlled dc microgrid system remains stable with the variations of the CPL-side capacitance. And the new steady-state value of the system variables remains almost unchanged as shown in Figure 18b. We can easily find out from the simulation results that the proposed NDOB-VNI stabilizing strategy shows a better robustness to the CPL-side capacitance variations than that of the conventional droop control method.
As shown in Figures 16-18, the proposed NDOB-VNI stabilizing strategy controlled dc microgrid system shows a better stability to the variations of CPL power, droop coefficient and CPL-side capacitance. Therefore, the effectiveness and robustness of the proposed NDOB-VNI stabilizing strategy for the perturbed dc microgrid system can be validated.

Conclusions
In this paper, a NDO based VNI stabilizing strategy is proposed for the dc microgrid system with CPLs. The proposed strategy builds a VNI on the source-side converter through the modified droop control method, which counteracts the large line inductance, hence improves the system stability. A NDO is constructed on the source side converter for estimating the output current, thus alleviating the sensor failure problem. Moreover, we theoretically prove that the effect of the built nonlinear disturbance observer is equivalent to a first order low-pass filter, which illustrates the impact of the proposed NDO on the system dynamics. Small-signal models of the studied dc microgrid system are carefully derived. And a root-locus based parameter designing approach is adopted for obtaining the optimal controller parameter value. An explicit Nyquist stability criterion is established and with the help of the system Nyquist diagram, robustness to the CPL power change, the droop coefficient variation and the CPL-side capacitance perturbation are comparatively studied. From the analysis result, we prove that the proposed NDOB-VNI stabilizing strategy shows a better robustness these system perturbations. Detailed real-time simulations are also conducted for validating the effectiveness of the proposed NDO based VNI stabilizing strategy. As the proposed NDO based VNI stabilizing strategy is implemented in a totally decentralized manner, the single-point-failure problem in the central controller is then alleviated. Only the local measurement of the source-side converter is required for implementing the proposed strategy, which saves the high-bandwidth communications between multiple energy sources. As the implementation process of the proposed stabilizing strategy does not require any global system information, the proposed stabilizing strategy can be used regardless of the system topology. However, the proposed NDO based VNI stabilizing strategy is only applicable for the dc/dc converter operating as the voltage terminal under the droop control method. In future, an improved version of the proposed stabilizing strategy applicable for more different operational conditions would be expected.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
We first reorganize the dc/dc converter model (7) as the standard nonlinear affine form as follows: where, x is the state variable including the converter inductance current and output voltage, which can be expressed as u is the input signal, namely the duty ratio for the dc/dc converter w is the disturbance signal to be estimated, namely the converter output current and f (x) is the system equation, g 1 (x) is the input channel function, and g 2 (x) is the disturbance channel function. The standard NDO functions for (A1) can be expressed as follows: where,ŵ is the estimated value, z is the internal state of the NDO, p(x) is the nonlinear function of NDO to be designed, and l(x) is the NDO gain, which can be expressed as Here, we define the NDO estimation error as follows: The dynamics of the NDO estimation error can be calculated from (A1) and (A5)-(A7) as follows: . e w = As indicated in (A8), l 1 (x) has no impact on the dynamics of the NDO estimation error, which can be set to be zero for simplicity. Moreover, we can see that if: (a) the NDO observer gain satisfies l 2 (x) < 0, (b) and the disturbance signal tends to be constant as the time goes to the infinity the NDO estimation error shown in (A8) will have a ultimate asymptotically stable point at e w = 0, which means that the estimated currentî o will eventually converge to the real one i o .
Here, we take the nonlinear disturbance observer gain l 2 (x) as a negative constant for simplicity, namely, Then, from (A1), (A5), (A6), and (A11), the nonlinear disturbance observer designed for the dc/dc converter output current can be expressed as follows: where,î o is the estimated output current. The proposed NDO can be equivalent to a first-order low-pass filter. Substitute the physical variables definition in (A1)-(A4) and (A7) into (A8), we can obtain Replacing the differentiator in (A13) with the Laplacian operator s, we can reorganize the equation in (A13) as follows:î where, T NDO is the time constant of the NDO, which can be expressed as follows:

Appendix B
The small-signal model of the studied dc microgrid system under the proposed NDO based VNI stabilizing strategy can be expressed as follows: i eq = ∆v dc − R e ∆i eq − ∆v eq /L e ∆ . v eq = ∆i eq − ∆i CPL /C eq ∆ .
where, ∆ represents the small perturbation of the variable, and the uppercase letters V and I represent the steady state values of voltage and current, respectively, i oest is the estimated output current. And x vr , x ir , z and x V N I represent the inner states of the voltage controller, current controller, NDO, and VNI stabilizing controller, respectively.