Small-signal model analysis of droop-controlled modular multilevel converters with circulating current suppressing controller

Small-Signal Model Analysis of Droop-controlled Modular Multilevel Converters with Circulating Current Suppressing Controller

J. Freytes

, G. Bergna

, J.A. Suul

, S. D’Arco

, H. Saad

, X. Guillaud

1Université Lille, Centrale Lille, Arts et Métiers, HEI - EA 2697 - L2EP - Lille, France - e-mail: [email protected]

2Department of Electric Power Engineering, Norwegian University of Science and Technology - Trondheim, Norway - e-mail: [email protected]

3SINTEF Energy Research - Trondheim, Norway - e-mail: [email protected], [email protected]

4Réseau de Transport d’Électricité (RTE) - La Défense, France - e-mail: [email protected]

Keywords:HVDC Transmission, Modular Multilevel Conver- ter, State-Space Modelling, Small-Signal Stability Analysis

Abstract

This paper presents a small signal eigenvalue analysis applied to a droop-controlled HVDC terminal based on the Modular Multilevel Converter (MMC) topology. The applied linearised model is derived from previous modelling efforts recently propo- sed in the literature, which rely on the application of three Park transformations at different frequencies (ω,−2ωand 3ω) app- lied to associated variables defined within the MMC model. The investigated configuration is controlled under the well-known Circulating Current Suppression Controller (CCSC). The deve- loped small-signal model is utilized to evaluate two different approaches for calculating the insertion index for modulation of the MMC, and to reveal potential stability problems in the sy- stem. It is demonstrated by participation factor analysis that the potentially unstable modes of the system under the investigated control strategy are linked to the uncontrolled zero-sequence component of the common-mode current resulting from the CCSC.

1 Introduction

The Modular-Multilevel Converter (MMC) represents the recent development among the diverse available topologies of Voltage Source Converters (VSCs) and is allegedly the most suitable converter solution for HVDC transmission systems [1].

Modelling and control of the MMC can be considered in general as more challenging compared to two- or three-level VSCs, since it is characterized by additional internal dynamics related to the common-mode currents and the distributed capacitor voltages within each of its arms [2]. Furthermore, multiple frequency components naturally appear in the internal variables of the MMC [3]. Thus, it is not possible to obtain and further linearise a Steady-State Time Invariant (SSTI) state-space model in a single Synchronous Reference Frame (SRF), according to the modelling approaches commonly applied for two-level VSCs [4, 5].

In general, state-space models can ensure a flexible frame- work for controller design and system stability analysis. Most previously developed state-space models representing the inter- nal dynamics of MMCs result in Steady-State Time Periodic (SSTP) models, where the steady-state operation is represented by an orbit in state-space coordinates and not by a constant equilibrium point. Stability analysis of the MMC in SSTP representation was recently studied in [6] by the application of time-periodic system theory (Poincaré multipliers). Howe- ver, application of traditional eigenvalue-based techniques com- monly applied in the power system community depend on an SSTI model where states remain constant in steady state [7].

Thus, accurate SSTI representation and linearization of MMC models is necessary for detailed eigenvalue-based assessment of small-signal stability in AC and DC power networks.

A few SSTI modelling approaches for MMCs have been recently proposed in [8–14]. However, the models from [8–12]

rely on simplifications that prevent accurate representation of the internal dynamics characterizing the MMC. On the other hand, the more detailed models presented in [13, 14] are able to represent the particular dynamics of the MMC by following two different approaches. Nevertheless, these publications do not consider the DC side dynamics since they assume constant DC voltage sources.

This paper extends the previous modelling efforts presented in [14], which derived a dynamic nonlinear SSTI model of an MMC with Circulating Current Suppression Controller (CCSC) according to [15]. This extension is achieved, firstly, with the consideration of a simplified DC bus model. Since the DC voltage is assumed as a state-variable, a DC droop controller is considered. Secondly, the complete SSTI model (MMC, DC bus and controllers) is linearised for obtaining a small-signal representation. The stability of the resulting model is analyzed using traditional eigenvalue based methods. The SSTI model is linearized for two different cases, to evaluate the impact of two different approaches for calculating the insertion index for modulation of the MMC. The stability of the resulting models are analyzed by using traditional eigenvalue-based methods to reveal potential stability problems in the system.

2 Modeling of MMC

2.1 Arm Averaged Model (AAM) inabcframe

The Arm Averaged Model (AMM) of the MMC is recalled in Fig. 1. The model presents one leg for each phasej(j=a,b,c), each leg consisting of an upper and a lower arm. Each arm includes an inductanceL_arm, an equivalent resistanceR_armand an aggregated capacitorC_arm[16].

v^G_c v^G_b v^G_a Rf Lf

Rf Lf

Rf Lf i^Δ_c i^Δ_b

i^Δ_a

i^Ua i^Ub i^Uc

i^L_a i^L_b i^L_c idc

idc

v_ma^U

v_ma^L

i^Uma

Carm

v^U_Ca m^U_a m^U_b

m^L_a vdc

Cdc DC Bus

il

il=_v^Pl

dc AC source

Rarm

Larm

Rarm

Larm

Rarm

Larm

Larm

Rarm

Larm

Rarm

Larm

Rarm

i^U_mb

Carm

v^U_Cb

i^Umc

Carm

v_Cc^U m^U_c

i^Lmc

Carm

v_Cc^L m^L_c

i^Lmb

Carm

v^L_Cb m^L_b

i^Lma

Carm

v^L_Ca

Figure 1: MMC connected to a DC bus capacitor In the studied system, the simplified dynamics of a DC bus are considered. These dynamics are modeled with an equivalent capacitorC_dcwhich emulates the capacitance of the DC cables and potentially other converter stations connected to the grid.

Furthermore, in parallel withC_dcthere is a controlled current sourcei_lwhose output power isP_las an equivalent model of the power exchanged in the HVDC system.

Defining v^Δ_{m j}= (−^def v^U_{m j}+v^L_{m j})/2; v^Σ_{m j}= (v^{def U}_{m j}+v^L_{m j})/2 and i^Σ_j= (i^{def U}_j +i^L_j)/2, it is possible to derive the currents dynamics of the AAM as follows [14]:

L^ac_eqdi^Δ_j

dt =v^Δ_{m j}−v^G_j −R^ac_eqi^Δ_j (1) whereR^ac_eq = (Rarm+2R_f)/2 andL^ac_eq= (Larm+2L_f)/2. The currentsi^Δ_j are oscillating at the grid frequencyω. The common- mode arm currents dynamics are given by:

L_armdi^Σ_j dt =v_dc

2 −v^Σ_{m j}−Rarmi^Σ_j (2) The common-mode arm currenti^Σ_j is mainly composed of a DC component and an AC circulating current at−2ω[14].

The voltagesv^U_{m j}andv^L_{m j}, as well as the currentsi^U_{m j}andi^L_{m j} of each arm jare described by the following equations:

v^U_{m j}=m^U_jv^U_{C j}, v^L_{m j}=m^L_jv^L_{C j} (3)

i^U_{m j}=m^U_ji^U_j, i^L_{m j}=m^L_ji^L_j (4) wherev^U_{m j},v^L_{m j},i^U_{m j}andi^L_{m j} are respectively the voltages and currents of the upper and lower arm equivalent capacitors and m^U_j andm^L_j are the corresponding instantaneous duty cycles.

The voltage and current of the equivalent capacitor are related by the following equation:

C_armdv^U_{C j}

dt =i^U_{m j}, C_armdv^L_{C j}

dt =i^L_{m j}. (5) The addition and difference of the terms in (5) yields,

2C_armdv_{C j}^Σ

dt =m^Δ_ji^Δ_j

2 +m^Σ_ji^Σ_j (6) 2C_armdv_{C j}^Δ

dt =m^Σ_ji^Δ_j

2 +m^Δ_ji^Σ_j (7) wherem^Δ_j = (m^{def U}_j −m^L_j),m^Σ_j = (m^{def U}_j +m^L_j),v_{C j}^Δ = (v^{def U}_{C j}−v^L_{C j})/2 andv^Σ_{C j}= (v^{def U}_{C j}+v^L_{C j})/2. A frequency analysis can be performed by assuming thatm^U_j is phased shifted approximately 180° with respect tom^L_j (where m^U_j and m^L_j have a DC component of 0.5 and an oscillating component), resulting in m^Σ_j ≈1 and m^Δ_j ≈mcos(ωˆ t). By inspecting the right-side of (6), it can be seen that in steady-state, the first productm^Δ_ji^Δ_j oscillates at 2ω while the second productm^Σ_ji^Σ_j gives a DC value in the case CCSC is used or a 2ωsignal otherwise, resulting for both cases in 2ω oscillations inv_{C j}^Σ . Similarly forv^Δ_{C j}, the first product on the right-side of (7),m^Σ_ji^Δ_j, oscillates atω, while the second productm^Δ_ji^Σ_j oscillates atω in the case the CCSC is used or will result in a signal oscillating atω superimposed to one at 3ωotherwise.

2.2 State space modelling in rotating frame using voltage- based formulations inΣ-Δrepresentation

This section summarizes the time-invariant model of the MMC with voltage-based formulation as proposed in [14]. To achieve time invariance, first the MMC variables are classified into “Σ” and “Δ” variables. In steady state, the “Δ” variables are oscillating at the grid’s angular frequencyω; while “Σ” variables oscillate at−2ω. This is summarized in Table 1.

Variables oscillating atω Variables oscillating at−2ω i^Δ_j =i^U_j −i^L_j i^Σ_j = (i^U_j +i^L_j)/2 v^Δ_{m j}= (−v^U_{m j}+v^L_{m j})/2 v^Σ_{m j}= (v^U_{m j}+v^L_{m j})/2

m^Δ_j =m^U_j −m^L_j m^Σ_j =m^U_j +m^L_j

Table 1: MMC variables inΣ-Δrepresentation To achieve a SSTI model, it is necessary to refer the MMC variables to their corresponding SRFs, following the frequency classification shown in Table 1. For generic variablesx^Σand

x^Δ, time-invariant equivalents are obtained with the Park’s trans- formation defined in the Appendix A as (bold variables means matrix or vectors):

ω⁺⇒xxx^ΔΔΔ_dddqqqzzz=^def

x^Δ_d x^Δ_q x^Δ_z

=PPP_ω

x^Δ_a x^Δ_b x_c^Δ 2ω⁻⇒xxx^ΣΣΣ_dddqqqzzz=^def

x^Σ_d x^Σ_qx^Σ_z

=PPP−^2ω

x^Σ_a x^Σ_b x^Σ_c

The zero sequences of the vectors in “Δ” representation need additional post processing as they are the only variables that remain time-periodic in steady-state after applying the above transformations. This issue was solved in [14] by means of an auxiliary virtual variable, 90° shifted from the real one, and by using a Park transformation at+3ωto achieve time invariant signals.

3ω⁺⇒xxx^ΔΔΔ_ZZZ=^def

x^Δ_Z

d x^Δ_Zq

=PPP_3ω

x^Δ_z x^Δ90°_z

Using the above definitions, the MMC dynamics in their

“Σ−Δ” representation can be rewritten in a time-invariant form [14]. The resulting system is recalled in the following.

2.2.1 AC currents

Applying the Park transformation atωto (1), the time invari- ant dynamics of the AC side currentsiii^ΔΔΔ_dddqqqare given as follows:

L^ac_eqdiii^ΔΔΔ_dddqqq

dt =−vvv^G_dddqqq^GG +vvv^ΔΔΔ_mmmdddqqq−R^ac_eqiii^ΔΔΔ_dddqqq−JJJ_ωL^ac_eqiii^ΔΔΔ_dddqqq (8) with JJJ_ω being the cross-coupling matrix at the fundamental frequency as defined in (9),

JJJ_ω=^def

0 ω

−ω 0

(9) vvv^GG_dddqqq^G the grid voltage at the point of interconnection andvvv^ΔΔΔ_mmmdddqqqthe AC-side modulated voltage. This voltage is defined in (10) as a function of the modulation indexesmmm^ΔΔΔ_dddqqqandmmm^ΣΣΣ_dddqqqzzz,

vvv^ΔΔΔ_mmmdddqqq=1 4VVV^ΔΔΔ

mmm^ΔΔΔ_dddqqq,mmm^ΣΣΣ_dddqqq,m^Σ_z

, (10)

andVVV^ΔΔΔ, the following 2×5 time invariant voltage matrix:

VVV^ΔΔΔ=^def

⎡

⎣−2v^Σ_Cz−v^Σ_Cd; v^Σ_Cq; −v_Cd^Δ −v_CZ^Δ_d;v^Δ_Cq+v^Δ_CZq;−v^Δ_Cd v_Cq^Σ; v^Σ_Cd−2v^Σ_Cz; v^Δ_Cq−v_CZ^Δ_q; v_Cd^Δ −v^Δ_CZ

d; 2v_Cq^Δ

⎤

⎦. (11)

2.2.2 Common-mode arm currents

Similarly, applying the Park’s transformation at−2ωto (2), the dynamics of the common-mode arm currents in their time invariant representationiii^ΣΣΣ_dddqqqandi^Σ_z are obtained, shown in (12).

L_armdiii^ΣΣΣ_dddqqq

dt =−vvv^ΣΣΣ_mmmdddqqq−R_armiii^ΣΣΣ_dddqqq−2JJJ_ωL_armiii^ΣΣΣ_dddqqq L_armdi^Σ_z

dt =−v^Σ_mz−R_armi^Σ_z+v_dc

2 (12)

withv_dc representing the voltage at the MMC DC terminals.

The modulated voltages driving the currentsiii^ΣΣΣ_dddqqqandi^Σ_z arevvv^ΣΣΣ_mmmdddqqq andv^Σ_mz. These voltages are defined in (13), as a function of the modulation indexes:

vvv^ΣΣΣ_mmmdddqqqzzz=1 4VVV^ΣΣΣ

mmm^ΔΔΔ_dddqqq,mmm^ΣΣΣ_dddqqq,m^Σ_z

, (13)

andVVV^ΣΣΣ, the following 3×5 voltage matrix:

VV V^ΣΣΣ=^def

⎡

⎢⎢

⎢⎣

v^Δ_Cd+v_CZ^Δ

d −v^Δ_Cq+v^Δ_CZq 2v_Cz^Σ 0 v^Σ_Cd

−v_Cq^Δ −v^Δ_CZq −v_Cd^Δ +v^Δ_CZ

d 0 2v^Σ_Cz 2v^Σ_Cq v^Δ_Cd v_Cq^Δ v_Cd^Σ v^Σ_Cq 2v_Cz^Σ

⎤

⎥⎥

⎥⎦. (14)

2.2.3 Arm voltages sum

Applying the Park transformation at−2ω to (6), the time invariant dynamics of the voltage sum vectorvvv^ΣΣΣ_CCCdddqqqzzz result in (15).

C_armdvvv_CC^ΣΣΣ_Cdddqqqzzz

dt =iii^ΣΣΣ_mmmdddqqqzzz−

2JJJ_ω 000_2×1 000_1×2 0

C_armvvv_C^ΣΣΣ_CCdddqqqzzz (15)

withiii^ΣΣΣ_mmmdddqqqzzzrepresenting the modulated current as defined in (16), as a function of the modulation indexes,

iii^ΣΣΣ_mmmdddqqqzzz=1 8III^Σ

mmm^ΔΔΔ_dddqqq,mmm^ΣΣΣ_dddqqq,m^Σ_z

(16) andI^ΣΣΣ, the following 3×5 time invariant current matrix:

I^ΣΣΣ=^def

⎡

⎢⎢

⎢⎣

i^Δ_d −i^Δ_q 4i^Σ_z 0 4i^Σ_d i^Δ_q −i^Δ_d 0 −4i^Σ_z −4i^Σ_q i^Δ_d i^Δ_q 2i^Σ_d 2i^Σ_q 4i^Σ_z

⎤

⎥⎥

⎥⎦. (17)

2.2.4 Arm voltages difference

Finally, the steady-state time invariant dynamics of the voltage difference vectorsvvv^ΔΔΔ_CCCdddqqqandv_CZ^Δ are now recalled. Re- sults are obtained by applying the Park’s transformation atωand 3ωto (7). For the sake of compactness, the voltage difference vector is defined asvvv_C^ΔΔΔ_CCdddqqqZZZ=^def

v^Δ_Cd,v^Δ_Cq,v^Δ_CZ

d,v^Δ_CZq

.

C_armdvvv^ΔΔΔ_CCCdddqqqZZZ

dt =iii^ΔΔΔ_mmmdddqqqZZZ−

JJJ_ω 0002×2

0002×2 3JJJ_ω

C_armvvv^ΔΔΔ_CCCdddqqqZZZ (18) withiii^ΔΔΔ_mmmdddqqqZZZrepresenting the modulated current as defined in (19), as a function of the modulation indexes,

iii^ΔΔΔ_mmmdddqqqZZZ=1 8I^ΔΔΔ

m m

m^ΔΔΔ_dddqqq,mmm^ΣΣΣ_dddqqq,m^Σ_z

(19) andI^ΔΔΔ, the following 4×5 time-invariant current matrix:

I^ΔΔΔ=^def

⎡

⎢⎢

⎣

2i^Σ_d+4i^Σ_z −2i^Σ_q i^Δ_d −2i^Δ_q 2i^Δ_d

−2i^Σ_q −2i^Σ_d+4i^Σ_z −i^Δ_q −i^Δ_d 2i^Δ_q 2i^Σ_d 2i^Σ_q i^Δ_d i^Δ_q 0

−2i^Σ_q 2i^Σ_d i^Δ_q −i^Δ_d 0

⎤

⎥⎥

⎦. (20)

2.2.5 DC bus dynamics

The DC bus dynamics are modelled by (21), whereC_dcis the cable model terminal capacitance andP_lrepresents the power injection as seen from the MMC station.

C_dcdv_dc dt =−P_l

v_dc−3i^Σ_z (21)

An overview of the model structure corresponding to the MMC and DC bus equations is shown in Fig. 2.

Grid currents v_dq^G

vdc

i^Σ_dqz Eq. (8)

Eq. (12)

Calculations Calculations

Eq. (10) Calculations

v_mdq^Δ

v_mdqz^Σ Eq. (13)

Eq. (15) v^Σ_Cdqz

Calculations Eq. (18)

v^Δ_CdqZ m^Σ_dqz

m^Δ_dq Common-mode

currents i^Σdqz

i^Δ_dq

DC bus vdc

vdc i^Σz

Pl ÷il Eq. (21) Inputs

States Algebraic equations Differential equations

v^Σ_Cdqz

vCdqZ^Δ i^Δ_dq

v^Σ_Cdqz v^Δ_CdqZ

v^Σ_Cdqz v^Δ_CdqZ

v^Σ_mdqz v^Δ_mdq

Figure 2: MMC and DC bus equations resume

3 DC voltage droop control and circulating cur- rent suppressing controller

The MMC control structure is shown in Fig. 3. For the AC- side the MMC control strategy is based on a classical scheme with two cascaded loops. The outer loop controls the active powerP_acfollowing aP_ac—v_dcdroop characteristic with gain k_d[17].

The inner loops control the AC currents in SRF. The variables v^Δ∗_mdandv^Δ∗_mqare the output of the controllers regulating the grid side currentiii^ΔΔΔ_dddqqqto the desired referenceiii^{ΔΔΔ∗∗∗}_dddqqq by implementing standard SRF current controllers with decoupling feed-forward terms, as briefly recalled in (22).

v_mabc^Σ∗

ωL^ac_eq ωL^ac_eq

v_d^G

v_q^G i^Δ∗_d

i^Δ∗_q i^Δ_d i^Δ_q

+

−

− +

v^Δ∗_md

v^Δ∗_mq ++

−

++ PI + PI

P−1(θ)

θ

v_mabc^Δ∗

Coupling &

Linearization m^U_abc

m^L_abc Grid currents control

v^G_d

Uncompensated modulation

i^Σ_d +

−

− +

v^Σ∗_md

v^Σ∗_mq PI

PI

P−1(−2θ)

−2θ CCSC

i^Σ_q i^Σ∗_d = 0

i^Σ∗_q = 0

−

vx 2 + 2ωLarm

2ωLarm +

− + +

÷ +

− v^∗_dc vdc

÷ v^G_d DC voltage droop control

−1

kd +

P_ac0^∗ Q^∗ac P_ac^∗

vx

23

23

Figure 3: MMC droop-controlled with CCSC

dξξξ^ΔΔΔ_dddqqq dt =

iii^{ΔΔΔ∗∗∗}_dddqqq−iii^ΔΔΔ_dddqqq

(22) vvv^{ΔΔΔ∗∗∗}_mmmdddqqq=K^Δ_p

iii^{ΔΔΔ∗∗∗}_dddqqq−iii^ΔΔΔ_dddqqq

+ 1

T_i^Δξξξ^ΔΔΔ_dddqqq−JJJ_ωL^ac_eqiii^ΔΔΔ_dddqqq+vvv^GG_dddqqq^G Furthermore,v^Σ∗_mdandv^Σ∗_mqare the outputs of the CCSC regu- lators [15], as given in (23), forcing the circulating currentsiii^ΣΣΣ_dddqqq to zero (000_2×1). The output DC current of the converter is left uncontrolled and it is naturally adjusted to balance the AC and DC power [3].

dξξξ^ΣΣΣ_dddqqq dt =

000_2×1−iii^ΣΣΣ_dddqqq

(23) vvv^{ΣΣΣ∗∗∗}_mmmdddqqq=K^Σ_p

0

002×1−iii^ΣΣΣ_dddqqq

+ 1

T_i^Σξξξ^ΣΣΣ_dddqqq+2JJJ_ωL_armiii^ΣΣΣ_dddqqq The MMC insertion indexes are calculated directly from the output of the control loops for the ac-side and circulating current suppressing controller as shown in (24). This calculation of the insertion indexes will be referred to as the "Un-Compensated Modulation" (UCM) [14] since there is no compensation for neither the steady-state oscillations nor the dynamic dynamic variations in the equivalent capacitor voltages.

m^U_j =−v^Δ∗_{m j}−v^Σ∗_{m j} v_x +1

2, m^L_j =v^Δ∗_{m j}−v^Σ∗_{m j} v_x +1

2 (24)

With the definitions ofm^Σ_j andm^Δ_j from Section 2 and taking into account (24), the modulation indexes in SRF can be obtained as in (25),

m^Σ_d=−2v^Σ∗_md

v_x , m^Σ_q=−2v^Σ∗_mq

v_x , m^Σ_z =1 m^Δ_d=−2v^Δ∗_md

v_x , m^Δ_q=−2v^Δ∗_mq

v_x (25)

The variablev_xin (24) and (25) can be replaced by the measure ofv_dc, or its nominal valuev_dc,nom. It will be shown by the small-signal eigenvalue analysis that this choice can influence the dynamics and stability of the studied system

The control presented in this section will be referred asnon- energy based control since the internal stored energy is not being controlled explicitly. When thenon-energy basedcontrol strategy is applied, the average value of the equivalent arm capacitor voltagesv^U_{C j}andv_{C j}^L follows the DC bus voltagev_dc. Hence, for slow transients it can be considered that there is a capacitor connected to the DC bus with value 6×C_arm[3, 17].

4 Model linearization and time domain valida- tion

The non-linear SSTI model presented in the Section 2 with the control from Section 3 can be linearised around a steady-state operating point. This linearised model is used for evaluating small-signal dynamics and stability by eigenvalue analysis but the detail model is not shown due to lack of space.

To validate the developed small-signal model of the MMC withNon-energy basedcontrol, results from simulation of three different models will be shown and discussed in the following:

1. EMT: The system from Fig. 1 implemented in EMTP-RV.

The MMC is modeled with the so-called “Model # 3:Arm Switching Function” from [16]. The considered controller is from Fig. 3.

2. Simp: Non-linear time-invariant model implemented in Matlab/Simulink. This model represents the equations resumed in Fig. 2 with the controller in SRF from Fig. 3 and the modulation indexes from (25).

3. StSp: linearised time-invariant model from theSimpmodel implemented in Matlab/Simulink.

The main system parameters are listed in Table 2.

U_1n 320[kV] R_f 0.521[Ω] T_i^Δ 4.7×10⁻³ f_n 50[Hz] L_f 58.7[mH] K^Δ_p 34.9 N 400[-] R_arm 1.024[Ω] T_i^Σ 2.3×10⁻³ C_arm 32.55[μF] L_arm 48[mH] K_p^Σ 41.07 C_dc 195.3[μF] v_dcn 640[kV] k_d 0.1[pu]

Table 2: Parameters for the time domain simulation Starting with a DC power transfer of 1pu (from DC to AC), a step is applied onP_lof−0.1[pu]. The reactive power is control- led to zero during the event. Simulation results are gathered in Fig. 4.

Figure 4(a) shows the results of the DC powers. Since the DC power is a non-linear relation between states, the results for P_dcof the linear model is post-processed asP_dc=3i^Σ_zv_dc.

The step applied on P_l produces power imbalance in the DC bus, and the MMC reacts with the droop controller and

its internal energy to achieve the new equilibrium point. The DC voltage resulting from the different models are shown in Fig. 4(c). The DC bus voltage deviation is defined by the droop parameterk_d. The internal energy of the MMC participates in the dynamics of the DC voltage regulation by discharging its internal capacitors into the DC bus during the transients, as seen in the voltagev^Σ_Czfrom Fig. 4(d). The behavior ofv^Σ_Czis similar to the DC bus voltage. This is the reason that it can be considered that the equivalent DC bus capacitanceC_dc is reinforced with 6×C_armas discussed in [17].

Finally, the results of the differential currents are shown in Fig. 4(b) (onlydqcomponents). The EMT model presents os- cillations at 6ωin steady state. This oscillations were neglected during the development of the time-invariant model [14]. As seen in the comparisons from Fig. 4, the model captures the average dynamics with reasonable accuracy even if the 6^thhar- monic components are ignored. For all other variables, there are negligible differences between the different models.

5 Small signal stability analysis of a droop- controlled MMC connected to a DC bus

In this section, the small-signal dynamics and stability of the linearised model are studied. The impact of two main parame- ters influencing the DC voltage dynamics are evaluated: the DC capacitor value and the droop parameters [17]. Moreover, the effect of the choice in the modulation index calculation from (24) and (25) is evaluated, i.e. division by the measured value ofv_dc, or a constant valuev_dc,nom.

5.1 Influence of the DC capacitor

The electrostatic constantH_dcis defined as, H_dc=1

2C_dcv²_dcn

P_n . (26)

The value ofH_dcis varied from 40ms down to 5ms. This last value represents a small equivalent capacitance of the DC bus (24,4μF<<(6×C_arm)).

5.1.1 Active power from DC to AC sides

The first results considers a power direction from DC to AC side of 1GW of power transfer. Results are shown in Fig. 5. The eigenvalues trajectories for the parametric sweep ofH_dcin the case of the modulation with measuredv_dcis shown in Fig. 5(a).

In this case, for the selected values the system remains stable.

When the modulation indexes are calculated with a constant value (e.g. v_dc,nom), the results varies as depicted in Fig. 5(b).

The trajectories of the eigenvalues are similar as in Fig. 5(a), but in this case, an instability is observed when the value ofH_dc is near 20ms, since the the pair of eigenvaluesλ1,2shift to the right hand plane.

For validating the results, a new time-domain simulation is performed with a DC capacitor of 97,65μF. The simulated

Pdc−StSp Pdc−Simp.

Pdc−EMT Pl

Pdc[pu]

Time [s]

1.15 1.2 1.25 1.3 1.35

0.85 0.9 0.95 1 1.05

(a)P_dc[pu]

i^Σ_q −StSp i^Σd−StSp i^Σ_q −Simp.

i^Σd−Simp.

i^Σq−EMT i^Σd−EMT

iΣ dq[pu]

Time [s]

1.15 1.2 1.25 1.3 1.35

- 0.01 - 0.005 0 0.005 0.01

(b)i^Σ_dq[pu]

vdc−StSt vdc−Simp.

vdc−EMT

vdc[pu]

Time [s]

1.15 1.2 1.25 1.3 1.35

0.98 0.985 0.99 0.995 1

(c)v_dc[pu]

vCz^Σ −StSp vCz^Σ −Simp.

vCz^Σ −EMT

vΣ Cz[pu]

Time [s]

1.15 1.2 1.25 1.3 1.35

0.98 0.985 0.99 0.995 1

(d)v_Cz^Σ[pu]

Figure 4: Time domain validation – Step applied onP_lof 0.1pu –EMT: EMTP-RV simulation,Simp.: Non-linear time-invariant model in Simulink,StSp: Linear time-invariant state-space model in Simulink

DCCapacitorHdc[s]

Imaginary

Real

- 250 - 200 - 150 - 100 - 50 0 0.005

0.01 0.015 0.02 0.025 0.03 0.035 0.04

- 1500 - 1000 - 500 0 500 1000 1500

λ1,2

(a) Modulation index calculated with measuredv_dc

DCCapacitorHdc[ms]

Imaginary

Real

- 250 - 200 - 150 - 100 - 50 0 0.005

0.01 0.015 0.02 0.025 0.03 0.035 0.04

- 1500 - 1000 - 500 0 500 1000 1500

λ1,2

(b) Modulation index calculated with nominal valuev_dc,nom

Figure 5: Parametric sweep of DC capacitorH_dc— DC Opera- ting pointv_dc0=1[pu],P_dc0=1[pu] —k_d=0.1pu

event is similar to the previous section. Results are shown in Fig. 6, where the instability is clearly highlighted. For theEMT model, the simulation is started with two capacitors in parallel on the DC side, one of 97,65μF and one with value 195μF.

Att=1.15s the larger capacitor is disconnected, resulting in a DC bus capacitance of 97,65μF, i.e. the unstable value. The frequency of the oscillations is approximately 110Hz.

5.1.2 Active power from AC to DC sides

It is known that the converters dynamics depend on the ope- rating point [18]. The same parametric sweep as the previous section is performed with the opposite power transfer direction (i.e. from AC to DC side). Results are shown in Fig. 7(a) and 7(b) for the modulation index calculated with measuredv_dcor v_dc,nom respectively. For both cases, when the DC capacitor decreases the system is unstable. When the measured voltage is used for the modulation index, the system can support lower values of DC capacitor than for the constant value.

5.2 Influence of the droop parameter

In this case, the droop parameterk_dis varied from 0.2pu down to 0.05pu. The considered power direction is from AC to DC sides since it is the worst case from previous section. Results are shown in 8. For both cases when lower values of droop are used, the eigenvaluesλ1,2shift to the right-hand plane resulting in unstable behavior. Nevertheless, it is observed that in Fig. 8(a) the droop parameterk_dthat makes the system unstable is lower than for Fig. 8(b).