Revisit Impedance-Based Stability Analysis of VSC-HVDC System

According to different partitioning styles, there exist different impedance-based stability analysis methods for complex systems such as voltage source converter (VSC)-based high-voltage direct current (HVDC) systems. Thus, the determination of the partitioning method is of importance in impedance-based stability analysis. However, how to partition a complex system is not paid much attention in the past. In this paper, the partitioning problems are revisited and a set of practical partitioning principles are proposed. Based on the principles, the VSC-HVDC system is partitioned into five parts, satisfying simplicity and integrity. In addition, the methods to form the equivalent circuits of the sub-systems are given and the equivalent circuits of the sub-systems are built, which is helpful to obtain the input-output models of the sub-systems. Furthermore, a stability criterion by directly applying the argument principle is used instead of the generalized Nyquist criterion (GNC), which could avoid the calculation of eigenvalue transfer functions, especially when a large number of sub-systems are involved. Finally, the correctness of the impedance-based stability analysis method is verified by control hardware-in-loop (CHIL) experimental results.


Subscript d, q
Direct-and quadrature-axis components in dq rotating frame.

I. INTRODUCTION
T HE point-to-point high-voltage direct current (HVDC)   transmission based on voltage source converters (VSCs) is a promising technology because it is a suitable solution to increase the capacity of AC networks and allow a greater international power transfer capacity [1], [2].The VSC-HVDC system is able to transmit bulk power in both directions and integrate large-scale renewable energy sources with the grid [3].In a VSC-HVDC system, abundant dynamic interactions exist between VSCs and AC networks, or VSCs and DC networks, which might result in instability [4], [5].Therefore, it is necessary to pre-assess the system stability before connecting to the existing networks.
Stability analysis of power converters is often carried out by small-signal methods.Generally, there are two kinds of smallsignal methods: the state-space method and the impedancebased method [6].The main drawback of the state-space method is that detailed information about the system is usually required.In fact, details of some commercial products are usually unavailable due to the protection of trade secrets.In addition, if the studied system is large and complex enough, an amount of computation is needed [7], [8], [9].In contrast, the impedancebased method is an effective and flexible tool.In this method, a power system is decomposed into the source and the load impedance equivalents.The stability is assessed by applying the Nyquist criterion to the impedance ratio of the source to the load.The impedance-based method is first presented to assess the stability of power electronics in [10].Afterward, it is extended from the stability criterion of the voltage source system to the current source system [11].The impedance modeling can be done under different frames, such as positive-negative sequence, DQ-frame, and reference frame [12], [13], [14], [15], [16].In sum, the impedance-based method has been widely applied in power system stability studies.
In VSC-HVDC systems, several impedance-based stability analysis methods according to different partitioning styles have been developed, including AC impedance, DC impedance, and hybrid AC-DC impedance/admittance methods.In order to investigate the influence of the AC grid impedance on the system stability, the HVDC system is partitioned from the AC grid to obtain the AC impedance [17].The DC impedance is obtained by partitioning the overall system into two sub-systems at the DC network.Different DC impedance models are developed for different partitioning positions of the DC network in [18], [19], [20], [21], [22].The hybrid AC-DC impedance/admittance is an extension of AC impedance and DC impedance [23], [24], [25], [26].In [25], the number of partitioned sub-systems is extended from two to multiple, which is a leap in the field of impedance-based stability analysis.The hybrid AC-DC impedance/admittance is defined based on the perturbations at the point of common coupling on the AC and DC sides, excluding the impacts of AC grid and DC network impedance.It is convenient to analyze the stability of complex systems and ensure the stability of sub-systems under certain assumptions.
However, all three methods mentioned above have some limitations.In both AC impedance and DC impedance methods, the systems are partitioned into two sub-systems.But the partitioned sub-systems are still complicated and it is difficult to guarantee that these sub-systems are stable when operating independently.Further partitioning of the HVDC system can reduce the complexity of the individual sub-systems.In the hybrid AC-DC impedance method, the system is partitioned into five sub-systems [25].As a result, it is easier to guarantee the sub-systems stability under certain assumptions.Proper partitioning principles are important for impedance-based methods.To partition a complex power converters-based system properly, partitioning boundaries and equivalent circuits of sub-systems should be determined.However, partitioning methods have not been paid much attention in the previous research works and some improper partitioning ways exist in [25] and [26].In addition, when a complex system is partitioned into multiple sub-systems, the generalized Nyquist criterion (GNC) is used for stability analysis, which increases the computation burden.
To address the problems mentioned above, the partitioning problems of the VSC-HVDC systems are revisited.A set of simple and practical partitioning principles are introduced and the impedance-based stability analysis methods are discussed.The main contributions of this paper are summarized as 1) A set of practical partitioning principles to partition complex systems is proposed, in which the assumptions for sub-systems are more reasonable and the detailed information of sub-systems is allowed to be unknown; 2) Stability criterion for the VSC-HVDC system by directly applying the argument principle is presented, which is easy to implement.The remainder of this article is organized as follows.Section II presents the partitioning principles and forms the equivalent circuits.In addition, the input-output models of the partitioned sub-systems are built and the stability analysis method of the overall system is introduced.Section III verifies the correctness of the VSC-i model and the impacts of outer loop controllers on the overall system stability through experimental results, thereby validating the correctness of the proposed stability analysis method.Finally, Section IV concludes this article.

A. Schematic Diagram of The Studied System
A VSC-HVDC system with the proposed partitioning method is shown in Fig. 1(a).It consists of two power grids, one π-type DC cable, and two LC filter-type VSC-based converters named VSC-A and VSC-B.The HVDC system is used to illustrate the partitioning principles in impedance-based stability analysis methods.
Fig. 2 depicts the control block diagram of VSC-i of the VSC-HVDC system in the control synchronous reference frame.VSC-A works as an inverter and its current reference is given by the power controller.And VSC-B works as a rectifier and its current reference is given by the DC-link voltage controller.

B. Partitioning Principles
The key of the impedance-based stability analysis is to partition a complex system into several sub-systems whose stability can be guaranteed easily and then to judge the stability of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.overall system by the impedance ratio.One important reason for the popularity of the method is that lots of complex systems are formed by integrating multiple stable sub-systems.In the partitioning process, boundaries and equivalent circuits need to be determined.
In this study, new partitioning principles are proposed.The principles include: 1) simplicity: the stability of the partitioned sub-systems must be guaranteed easily; 2) integrity: the partitioned sub-systems are complete and could play independent roles in the overall system.The two principles ground the basis for partitioning.If the sub-systems are independent products from the market, they are easy to be identified and partitioned.It is worth noting that the integrity of the products must be ensured.The reason is that the impedance/admittance of the sub-systems can be easily measured when their parameters and structure are unavailable.In fact, the real challenge is to correctly identify the boundaries of sub-systems in a large-scale complex system.That is to say, it is not easy to ensure the integrity mentioned above.To identify the boundaries of the sub-systems correctly, the function and working principle of each sub-system should be understood deeply.
According to the principles above, the HVDC system is partitioned into five sub-systems according to their physical structures, as shown in Fig. 1(a).For ease of comparison, let Remark: The presented partitioning principles look like common sense, but they are very important because they make the impedance-based stability analysis method simpler and more practical.In fact, most existing methods do not follow these principles.For example, the VSC-HVDC systems in [20] and [21] are partitioned into two sub-systems and it is difficult to judge the stability of the sub-systems.Thus, they do not follow the first principle.Likewise, the improved partitioning methods in [25] and [26] break the second principle, complicating impedance/admittance measurement and leading to an improper aggregated model [27].

C. Equivalent Circuits
After determining the partitioning boundaries, how to form the equivalent circuits of the partitioned sub-systems is another important issue.Based on the partitioning diagram in Fig. 1(a), these equivalent circuits are formed by adding proper controlled current or voltage sources, marked in red, at the ports of the sub-systems as shown in Fig. 3.
Since only sub-systems VSC-A and VSC-B are related to control, more attention is paid to them.According to the first principle of the partitioning method, VSC-A and VSC-B subsystems should be stable under certain conditions.In this study, the products VSC-A and VSC-B can work properly under ideal test conditions.According to the control structures shown in Fig. 2, VSC-A is a typical AC/DC converter with power control, its ideal test condition should be configured as shown in Fig. 3 Once the equivalent circuits of VSC-A and VSC-B subsystems have been determined, it is easy to get the equivalent circuits of the remaining sub-systems.To form the remaining equivalent circuits, the principle that the same port in different sub-systems has to be connected to different types of sources needs to be followed.For example, as the left port of the sub-system shown in Fig. 3(b) is a voltage source, the right port of the sub-system shown in Fig. 3(a) is a current source.Similarly, as the left port of the sub-system shown in Fig. 3(d) is a current source, the right port of the sub-system shown in Fig. 3(c) is a voltage source.
Remark: The roles of the equivalent circuits include: 1) specifying the working environments (ideal test conditions) of the sub-systems; 2) helping to build the input-output models of the sub-systems, especially telling us which variables should be selected as the inputs (excitations) of the input-output models.

D. Input-Output Models of VSC-A and VSC-B Subsystems
According to Fig. 3 where voltage sources v A o and u A dc are the inputs (excitations) of the VSC-A sub-system; voltage source v B o and current source i B dcm are the inputs (excitations) of the VSC-B sub-system.Detailed derivations of the above mathematical models are given in Appendix-A.
Fig. 3(b) and (d) are commonly-used inverter and rectifier circuits respectively, and it is not difficult to guarantee their stability in the design.Assuming that the inverter and rectifier circuits are stable under ideal test conditions, it can be concluded that Y A and H B are stable transfer function matrices, i.e., Y A and H B do not contain right half-plane (RHP) poles.Thus, the RHP pole issue in stability analysis is avoided.
Remark: In model aggregation, the determination of the inputs is crucial.The assumption that the sub-system is stable only determines the stability of a specific aggregated model (specific input-output mapping).If the built aggregated model disagrees with the equivalent circuit, the RHP issue cannot be avoided by reasonable assumptions, thus hampering the stability analysis.

E. Input-Output Models of AC and DC Network Subsystems
The circuit structures of the sub-systems shown in Fig. 3(a) and (e) are the same, so the mathematical models of them in the control frame have the same form.The model is given in matrix form as ( According to Fig. 3(c), the model of the DC network subsystem is expressed as Detailed derivations of the mathematical models of the AC grid and DC network sub-systems are given in Appendix-B.Because both AC grid and DC network sub-systems are passive circuits, E i ac and K dc are stable transfer function matrices.

F. Overall System Model and Stability Analysis
Combining (1)−( 4), the model of the overall system is obtained.For simplicity, we keep the matrices Y A and H B unsplit, then the state variables are arranged as X = [ṽ A od , ṽA oq , ũA dc , ṽB od , ṽB oq , i B dcm , ĩA od , ĩA oq , ĩA dcm , ĩB od , ĩB oq , ũB dc ] T .After some simple manipulations, the closed-loop system is expressed as where I is the identity matrix, H all is given in Appendix-C, and O is the zero matrix.
Stability Criterion: If all the sub-systems (e.g., Y A , H B , E A ac , E B ac , and K dc ) are stable under ideal test conditions, the closedloop system of ( 5) is stable if and only if namely, the Nyquist diagram of G(s) does not encircle the origin.In which, G(s) = det(I-H all ), Δ N∞ argG(s) represents the change in arg G(s) after one cycle of s moving along the positive direction of the Nyquist path N∞ [29], and it is an integer multiple of 2π.The detailed proof is given in Appendix-D.
If we cannot guarantee all the elements of H all are stable transfer functions, it will not easy to determine the system stability just by the Nyquist diagram due to the possible RHP poles of H all .In fact, to build the mathematical models according to the steps listed in this paper, the condition that all the elements of H all are stable transfer functions will be guaranteed automatically.
Though only a point-to-point VSC-HVDC system is taken as an example to illustrate the stability analysis, it is easy to expand the analysis method to multi-terminal VSC-HVDC systems or other complex power electronics-dominated power systems with advanced control structures by the steps listed before.
Remark: In our method, two assumptions have been made: 1) All sub-systems are stable: This assumption is reasonable because the stability of each sub-system from the market is usually tested before shipment.It is worth noting that this assumption is not a necessary condition.In reality, two unstable sub-systems can still be stable when they are inter-connected but it rarely happens.2) No unstable zero-pole cancellations exist inside each sub-system: This is a common problem in measurementbased stability analysis because it only provides the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.specific input-output relationship without inner dynamic information.However, in practical situations, internally unstable equipment cannot work properly.Consequently, the possible zero-pole cancellation issue can also be avoided by testing the equipment before shipment.In summary, reasonable assumptions greatly simplify the stability analysis, meanwhile, it also leads the method to be conservative in terms of stability assessment.

III. EXPERIMENT AND VERIFICATION
In order to verify the correctness of the partitioning principles, experiments based on control-hardware-in-loop (CHIL) are carried out.The CHIL platform is shown in Fig. 4. The VSC-HVDC system is emulated in the OPAL-RT4510 simulator.The controller is implemented in the DSP-TMS320F28335 control board.The experimental data is recorded by the host machine of OPAL-RT4510 and can be displayed through an oscilloscope.

A. Model Verification of VSC-A and VSC-B
In Figs. 5 and 6, the curves are drawn by the elements of Y A and H B , and the red dots are measurement results.It can be seen that the theoretical results agree well with the measurement results, verifying the correctness of VSC-A and VSC-B subsystem models.

B. Impact of Power Controller Bandwidth on Stability
The premise that all sub-systems are stable must be satisfied before stability analysis of the overall system so that some constraints about the control parameters should be imposed, which are listed in Table I  stability.In Case 1, the system is near the stable boundary of the power controller and its active power flows from VSC-A to VSC-B.Case 2 is at the unstable boundary.In Nyquist diagrams, there are two ways for the arrow to cross the negative real axis, i.e., positive and negative crossings [31].The system stability can be judged by the difference between positive and negative crossing.Fig. 7 shows the Nyquist diagrams of G(s) in Case 1 and Case 2. In Fig. 7(a), there are four positive and negative crossings respectively, so the overall system is stable in Case 1.In Fig. 7(b), there are two unstable closed-loop poles, thus the system is unstable in Case 2. Fig. 8 shows the current waveforms of the DC network and the AC-side current waveforms of VSC-A before and after the change from Case 1 to Case 2. Experiments verify the correctness of the partitioning principles and the stability criterion.In addition, Case 1 and Case 2 also indicate that excessive power controller bandwidth can lead to overall system instability.For comparisons, the GNC diagrams in Case 1 and Case 2 are drawn in Fig. 9.In which, there are 6 eigenvalue curves marked with different colors.The stability conclusions obtained by the GNC are consistent with the one given by the proposed stability criterion, indicating that the proposed stability criterion is identical to the GNC in terms of the stability assessment.
From Figs. 7 and 9, the proposed method only needs to draw one Nyquist curve while the GNC gives multiple eigenvalue loci.The curve crossing is easy to be identified in the proposed method.Due to the numerical nature, especially for a largedimensional model, the identification of crossing may not be an easy task in the GNC plots.All these arguments point to the simplicity of the proposed method and its adaptability to complex systems.
Figs. 10 and 11 show the Nyquist diagram of G(s) and experimental waveforms of VSC-A in Case 3, respectively.According to the crossing difference depicted in Fig. 10 and waveforms in  Fig. 11, the system is stable in Case 3. As can be seen from the position of the DC network current waveform, the direction of power flow is different from Case 1 and Case 2. It shows that the system is less prone to instability in this direction of power transmission.

C. Impact of DC Voltage Controller Bandwidth on Stability
The remaining cases in Table III are used to analyze the impact of DC-link voltage controller bandwidth on stability.Fig. 12 shows the Nyquist diagrams of G(s) in Case 4 and Case 5. Fig. 13 shows the current waveforms of the DC network and  AC-side current waveforms of VSC-B.From Figs. 12 and 13, it is found that the system is stable in Case 4 but unstable in Case 5.The two cases indicate that excessive DC-link voltage controller bandwidth tends to cause system instability.Fig. 14 shows the Nyquist diagrams of G(s) in Case 6 and Case 7. In the two cases, the direction of power flow is different from that in Case 4 and Case 5. Fig. 15 shows the current waveforms of the DC network and the AC-side of VSC-B, and Fig. 16 shows the AC-side current waveforms of VSC-A.Comparing the AC-side current waveforms of VSC-A and VSC-B, the oscillation in VSC-B is more obvious than that in VSC-A.Both the stability analysis and experiments give the same result that the overall system is stable in Case 6 and unstable in Case 7.

IV. CONCLUSION
This article proposes a set of partitioning principles in impedance-based stability analysis of VSC-HVDC systems.Different from the existing partitioning styles, the proposed method ensures the simplicity and integrity of the sub-systems.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Even if the sub-systems are commercial products and their parameters and structure are unavailable, the impedance-based stability analysis method still works.Based on the partitioning principles, the equivalent circuits of the sub-systems are formed by adding controlled current or voltage sources at the ports of the sub-systems.According to the equivalent circuits, the input-output models of sub-systems are built easily.The system stability is assessed by directly applying the argument principle.Experimental results verify the correctness of the impedancebased stability analysis method.

A. Modeling of VSC-A and VSC-B Subsystems
The model of the used phase-locked loop (PLL) is expressed in frequency-domain as The current and voltage vector dynamic equations of the AC filter in the control frame are expressed as After linearization of (A2) and (A3) around the operating point, we have where Y i cf , Z i c , and Y i o are defined as follows Substituting (A4) into (A5) gives The DC-link capacitance dynamic equation of VSC-i is defined as follows Then the model of VSC-B is described by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Modeling of AC Grid and DC Network Subsystems
The dynamic equation of the AC network sub-system in the control frame is as follows where v i gdq = V i gd + jV i gq cos Δθ i − j sin Δθ i (B2) And the model of the phase difference Δθ i between the control frame and the grid frame is expressed as Linearizing and rearranging (B1), the model of the AC network sub-system in the control frame can be given in matrix form as where The dynamic equations of the DC network are modeled as follows

D. Proof of the Stability Criterion
The characteristic polynomial of ( 5) is According to the classical control theory, the closed-loop system of ( 5) is stable if and only if where Z [G (s)] denotes the zeros of G (s), C denotes the field of complex numbers, and (s) denotes the real part of s.Since G (s) is analytic on C, except at the poles of H all , based on the argument principle [28], the stability condition of (D2) can be equivalently transformed to where N∞ is the Nyquist path [29] including the points +jÝ and -jÝ, Z and P respectively denote the number of zeros and poles of G(s) belonging to the set {s ∈ C| (s) ≥ 0 + }.It is not hard to find that H all is stable because it is comprised of stable sub-system transfer functions Y A , H B , E A ac , E B ac , and K dc .Therefore, according to the calculation law of determinant [30], G(s) does not contain RHP poles (i.e., P = 0).
As a result, the closed-loop system is stable if and only if namely, the Nyquist diagram of G(s) does not encircle the origin.

Fig. 2 .
Fig. 2. Control schematic of VSC-i of the HVDC system.

Fig. 1 (
a).As observed, the proposed partitioning method differs from the existing partitioning ones shown in Fig.1(b) and (c).The main difference lies in the partitioning of DC-link capacitors (C i dc and C i d ).In Fig.1(b) and (c), the partitioning breaks the respective integrity of VSC-i and π-type DC cable because C i dc and C i d belong to two separate sub-systems in practical situations.C i dc is a part of the power converters VSC-i from the market, but it needs to be stripped when measuring the VSC-i impedance/admittance in Fig.1(b).While in Fig.1(c), the equivalent capacitor C i d of π-type DC cable is partitioned into the sub-system of VSC-i, therefore an additional capacitor C i d need to be installed in measurement, making it complicated.Moreover, the sub-system VSC-B in Fig.1(b) is incomplete and cannot work properly because its DC-link voltage controller fails to work.
Fig.2, VSC-A is a typical AC/DC converter with power control, its ideal test condition should be configured as shown in Fig.3(b), where v A o represents an ideal AC voltage source and u A dc represents an ideal DC voltage source.On the other hand, VSC-B is a commonly-used AC/DC converter with DC-link voltage control, its ideal test condition should be configured as shown in Fig.3(d), where v B o represents an ideal AC voltage source and i B dcm represents an ideal DC current source.Though VSC-A and VSC-B have the same topology, their equivalent circuits are different due to different control structures.Once the equivalent circuits of VSC-A and VSC-B subsystems have been determined, it is easy to get the equivalent circuits of the remaining sub-systems.To form the remaining equivalent circuits, the principle that the same port in different sub-systems has to be connected to different types of sources needs to be followed.For example, as the left port of the sub-system shown in Fig.3(b) is a voltage source, the right port of the sub-system shown in Fig.3(a) is a current source.Similarly, as the left port of the sub-system shown in Fig.3(d) is a current source, the right port of the sub-system shown in Fig.3(c) is a voltage source.Remark: The roles of the equivalent circuits include: 1) specifying the working environments (ideal test conditions) of the sub-systems; 2) helping to build the input-output models of the sub-systems, especially telling us which variables should be selected as the inputs (excitations) of the input-output models.
(b) and (d), the input-output models of VSC-A and VSC-B in the control frame are obtained and expressed as

Fig. 5 .Fig. 6 .
Fig. 5. Bode diagrams of the elemental curves of Y A and measurement results of the VSC-A sub-system.

Fig. 8 .
Fig. 8. Experimental waveforms of i dcl , i A oa , i A ob and i A oc from Case 1 to Case 2.

Fig. 11 .Fig. 12 .
Fig. 11.Experimental waveforms of dcl , i A oa , i A ob and i A oc in Case 3.

Fig. 13 .
Fig. 13.Experimental waveforms of i dcl , i B oa , i B ob and i B oc from Case 4 to Case 5.

Fig. 15 .
Fig. 15.Experimental waveforms of i dcl , i B oa , i B ob and i B oc changing from Case 6 to Case 7.

Fig. 16 .
Fig. 16.Experimental waveforms of i A oa , i A ob and i A oc in Case 7.

From Fig. 2 ,
the control input of VSC-i can be expressed as ṽi cd where T s is the control period, Y A k , Y B k , Z A L , Z B L , m A , m B , D elay , and T d are ) into (A7) and (A10), the sub-system VSCi can be represented in matrix form.The model of VSC-A is expressed as d s(L dc s + R dc ) .C. Matrix of the Closed-Loop System (C1) shown at the top of this page.
. And the remaining parameters are given in Table II.Case 1, Case 2, and Case 3 as illustrated in Table III are selected to analyze the impact of power controller bandwidth on Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.