State Space Modeling and Stability Analysis of a VSC-HVDC System for Exchange of Energy

2.1 Study of two terminals VSC-HVDC controlled system

The converter is outlined as a fundamental part of the VSC-HVDC network. The fundamental ingredient of the converter is a permanent switch which is adapted to turn on or off a current [20]. The asymmetric structure of IGBT (Insulated Gate Bipolar Transistor) used in this VSC-HVDC system, enables lower on-state voltage droop and higher brazen voltage blocking capability at the expense of reverse voltage blocking capability efficiency reduced [21].

The VSC-HVDC scheme is shown in Figure 1. A DC cable is made as connection between the two VSC stations. On the DC side, capacitors permit to conserve the DC voltage and the phase reactors filter injected grid currents. On the AC side, the converter is connected to an AC network or a wind turbine with a transformer in order to insulate the station from the AC source and provides the appropriate AC voltage. The phase reactors associated with the transformer filter injected grid currents. Optional AC filters are exploited to supply the reactive power and also to filter Pulse-Width Modulation (PWM) harmonics.

1.png

Figure 1. VSC-HVDC transmission system

The various control methods developed in view of controlling the VSC-HVDC system can operate suitably in case of a communication failure. Specially the Master-Slave method and the Voltage Droop method [22]. The Voltage Droop method is typically used when more than one converter is controlling the DC voltage at the same time. On the other hand, the Master-Slave method is a simple control method adapted to command the VSC-HVDC system. Indeed, there is only one converter called Master one. It controls the DC bus voltage to keep it in a certain range. While the other converters which are called Slave, control their power by injecting or extracting the power to from the DC grid. For point to point HVDC structure the Master Slave control method is usually sufficient [22].

The dq rotating structure used for the modelling and controlling of an AC system is employed with the aim of attaining stationary electrical equations. To pathway the AC grid voltage frequency and the phase angle, a Phase Lock Loop (PLL) is applicated. In such a case, the q-axis voltage is null and the d-axis is synchronized with the grid voltage vector and has similar amplitude as the point of common coupling voltage vector. Thus, the active and reactive powers are separately controlled by current constituent [9].

As shown in Figure 2 which presents a VSC substation controlled in dq synchronous frame, the control system is divided into two parts: the inner control loop is used to control the currents (i_sd and i_sq) by adjusting d and q coordinates of modulated voltages (V_md and V_mq) in the Park frame, and the outer control loop is used to achieve precise functions: The d-axis current reference (i_sd-ref) can be generated by an active power or DC voltage controller. The q-axis current reference (i_sq-ref) can be obtained by a reactive power or AC voltage controller.

2.png

Figure 2. Point to-point controlled VSC-HVDC link

where:

R_s L_s: Resistance, Inductance of the phase reactors associated with the interface transformer filter.

C_s: DC capacitors.

L_sr: Smoothing reactors.

L_g C_g: Inductance, Capacitance of the AC filter.

(v_gi, i_gi): The couple voltage-current of an AC grid.

(u_si, i_ci): The couple voltage-current of the DC bus.

(v_mdi-ref, v_mqi-ref): Direct and quadrature control voltages of converter.

(v_mi, i_mi): Couple voltage-current delivered by the converter.

(P_gi, Q_gi): The active and reactive power through the system.

(U_si-ref): The DC Voltage reference.

(P_gi-ref, Q_gi-ref): The reference active and reactive power delivered by the transformer.

(P_mi, Q_mi): Active and reactive power delivered by the converter.

i_si: The phase reactor current.

i_li: The cable current.

u_Cci: The voltage cable.

The i index ϵ {1, 2}, it indicates the station number.

2.2 Stability study of the HVDC link via small signal stability method

2.2.1 Linearization of a VSC-HVDC link

The stability of a system is its capability, for an operating point to find equilibrium after a physical disturbance. Traditionally, the system stability is tested when it is submissive to small disturbance. Therefore, a linearized model of the system is created to analyse its stability [7-23]. In this part, we present a technique to model VSC-HVDC converter station for dynamic study.

In this work, the converter is deemed ideal and it is simulated by its average model in view of not taking into account harmonics effects [24], which are not our a matter of interest in this work. Indeed, the average-value model of the converter has the same dynamic performance of the detailed model, it is also more efficient for simulation time steps as explained in the study [25]. In this section, the linearization of the various components of a HVDC liaison is presented.

The general state space of a linearized system is expressed by [26]:

$\left\{\begin{array}{l}\Delta \dot{X}=A \Delta \dot{X}+B \Delta U \\ \Delta Y=C \Delta X+D \Delta U\end{array}\right.$     (1)

where, for each considered subsystem: ΔX is the state vector, ∆U is the input vector and the output vector is presented by ∆Y, A is the state matrix, the input matrix is B, the output matrix is presented by C, and the feed-forward matrix is presented by D.

We notice that the linearized block diagram of each subsystem presented in Figures 3, 4, 5 and 6 contains a proportional integral (PI) regulator. Therefore, to facilitate the creation of a global state space model, we choose to change the model defined by Eq. (1) to an increased one by adding a new state auxiliary vector "∆Z" to the main state vector "∆X" in order to eliminate the use of the controller [9]. Thus, the new state space system is obtained:

$\left\{ \begin{array}{l}\Delta \mathop {X'}\limits^.  = A'\Delta \mathop {X'}\limits^.  + B'\Delta U'\\\Delta Y' = C'\Delta X' + D'\Delta U'\end{array} \right.$     (2)

With $\Delta X' = \left[ \begin{array}{l}\Delta X\\\Delta Z\end{array} \right]$     (3)

where, ∆X’ is the new state vector, ∆U’ is the new input vector, the new output vector is presented by ∆Y’, A’ is the new state matrix, B’ is the new input matrix, new output matrix is given by C’ and finally, D’ is the new feed-forward matrix. However, when the state matrix A’ is. null, we fix the controlled state vector ∆U’ as: ∆U’=k∆X’. With: k is the gain matrix assuring the stability of the system.

A current controller linear model

The model of the VSC-HVDC system associated to its current control loop as given in Figure 2 is the basic structure for such VSC system connected to grid. Disregarding the switching losses, power at both sides of the converter is the same: Assumptions are made with the aim of simplifying the complexity of the system and making the AC side of the VSC not to impact the DC system. With this assumption and referring to Figure 2, we can assume that:

$V_{m d i} i_{s d i}+V_{m d i} i_{s q i}=U_{s i} i_{m i}=P_{m i}$     (4)

Eq. (4) is not linear. So, with the aim of evaluating its stability, we should linearize the system around an operating point. Each quantity is prescribed by an operating point denoted by capital letter and the subscript 0, and a small variation denoted by the Greek symbol. After linearization, Eq. (4) becomes:

$\Delta {V_{mdi}}{I_{sd0}} + \Delta {i_{sdi}}{V_{md0}} + \Delta {V_{mqi}}{I_{sq0}} + \Delta {i_{sqi}}{V_{mq0}} = \Delta {U_{si}}{I_{m0}} + \Delta {i_{mi}}{U_{s0}}$     (5)

The i index refers to the station number which is in this case ϵ{1, 2}.

Thus, the linearized current controller of a VSC is presented in Figure 3.

The steady state error between the reference and the output current ought to be avoided by the integration of a PI controller in the current controller. The corrector gain of the current controller PI corrector is defined by:

$P I=\frac{K_{p i} p+K_{i i}}{p}$      (6)

With: p presents Laplace Coefficient. The transfer function (TF) of the current control is the same for the two axes. The expression of the d-axis current control loop transfer function is expressed by:

$\frac{{{i_{sd}}}}{{{i_{sd - ref}}}}(p) = \frac{{1 + \frac{{{K_{pi}}}}{{{K_{ii}}}}p}}{{1 + \frac{{{K_{pi}} + {R_s}}}{{{K_{ii}}}}p + \frac{{{L_s}}}{{{K_{ii}}}}{p^2}}}$     (7)

The formulations of K_pi and K_ii deduced from the pole placement method, are:

$K_{p v}=2 \zeta w_{n} C_{s}$ and $K_{i i}=L_{s} w_{n}^{2}$     (8)

where: $\zeta$ is the damping ratio.

$w_{n} \frac{3}{T_{r}}$ is the natural frequency [rad/s].

To create the new state space current controller, we choose new state variables $\Delta Z_{1 i}$ and $\Delta Z_{2 i}$  such as:

$\left\{ \begin{array}{l}\frac{{d(\Delta {Z_{1i}})}}{{dt}} = \Delta {i_{sdi - ref}} - \Delta {i_{sdi}}\\\frac{{d(\Delta {Z_{2i}})}}{{dt}} = \Delta {i_{sqi - ref}} - \Delta {i_{sqi}}\end{array} \right.$     (9)

The vectors and the following matrices represent the new linearized current controller state space representation according to the system (2).

$\Delta X^{\prime}=\left[\begin{array}{c}\Delta i_{s d i} \\ \Delta i_{s q i} \\ \Delta U_{s i} \\ \Delta Z_{1 i} \\ \Delta Z_{2 i}\end{array}\right] \Delta U^{\prime}=\left[\begin{array}{c}\Delta i_{s d i-r e f} \\ \Delta i_{s q i-r e f}\end{array}\right] ; \Delta Y^{\prime}=\left[\begin{array}{c}\Delta i_{s d i} \\ \Delta i_{s q i} \\ \Delta U_{s i}\end{array}\right]$     (10)

3.png

Figure 3. Linearized block diagram of the current controller VSC

$A' = \left[ {\begin{array}{*{20}{c}}{\frac{{ - {R_s}}}{{{L_s}}}}&0&0&{\frac{1}{{{L_s}}}}&0\\0&{\frac{{ - {R_s}}}{{{L_s}}}}&0&0&{\frac{1}{{{L_s}}}}\\{(\frac{{ - {V_{md0}} - {I_{sq0}}{L_s}w}}{{{C_s}{U_{s0}}}} +\frac{{z{'_1}}}{{{C_s}}})}&{(\frac{{ - {V_{mq0}} + {I_{sd0}}{L_s}w}}{{{C_s}{U_{s0}}}} + \frac{{z{'_2}}}{{{C_s}}})}&{(\frac{{{I_{m0}}}}{{{C_s}{U_{s0}}}} + \frac{{z{'_3}}}{{{C_s}}})}&{(\frac{{ - {I_{sd0}}}}{{{C_s}{U_{s0}}}} + \frac{{z{'_4}}}{{{C_s}}})}&{(\frac{{ - {I_{sq0}}}}{{{C_s}{U_{s0}}}} + \frac{{z{'_5}}}{{{C_s}}})}\\

{ - 1}&0&0&0&0\\0&{ - 1}&0&0&0\end{array}} \right]$

$B^{\prime}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & 1\end{array}\right]$; $C^{\prime}=\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0\end{array}\right]$; $D^{\prime}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0 \\ 0 & 0\end{array}\right]$     (11)

where, The i index ϵ {1, 2}, it means the station number and $Z_{1}^{\prime}, Z_{2}^{\prime}, Z_{3}^{\prime}, Z_{4}^{\prime},$ and $Z_{5}^{\prime}$, are constant values.

A Power controller Linear model

4.png

Figure 4. Linearized block diagram of the active power controller

The linearized model of the active power controller is illustrated in Figure 4.

After linearization, the expressions of the active power $P_{g i}$ and reactive power $Q_{g i}$ presented in Figure 2 become:

$\Delta P_{g i}=\Delta V_{g d i} I_{s d 0}+\Delta i_{s d i} V_{g d 0}$     (12)

$\Delta Q_{g i}=-\left(\Delta V_{g d i} I_{s q 0}+\Delta i_{s q i} V_{g d 0}\right)$     (13)

The PI power controller is expressed by:

$P I=\frac{K_{p p} p+K_{i p}}{p}$     (14)

The power control transfer function has the following form:

$\frac{{{P_g}}}{{{P_{g - ref}}}}(p) = \frac{{1 + \frac{{{K_{pp}}}}{{{K_{ip}}}}p}}{{1 + \frac{{1 + {K_{pp}}}}{{{K_{ip}}}}p}}$     (15)

This TF is assimilated at a first order TF with a time constant $\Gamma_{p}=\frac{T_{r}}{3}$. In order to reach a required response time, the parameters of the power controller are:

$K_{p p}=0$ and $K_{i p}=\frac{1}{\Gamma_{p}}$     (16)

According to the active power controller linearized block diagram defined in Figure 4, we define the new state variables “$\Delta Z_{a i}$” and “$\Delta Z_{r i}$” respectively related to the active and reactive power controller:

$\left\{ \begin{array}{l}\frac{{d\Delta {Z_{ai}}}}{{dt}} = \Delta {P_{gi - ref}} - \Delta {i_{sdi}}{V_{gd0}} - \Delta {V_{gdi}}{I_{sd0}}\\\frac{{d\Delta {Z_{ri}}}}{{dt}} = \Delta {Q_{gi - ref}} - \Delta {i_{sqi}}{V_{gd0}} - \Delta {V_{gdi}}{I_{sq0}}\end{array} \right.$     (17)

For the active power controller, we obtain:

$\Delta X^{\prime}=\left[\begin{array}{l}\Delta i_{\text {sdi-ref}} \\ \Delta Z_{a i}\end{array}\right] ; \Delta U^{\prime}=\left[\begin{array}{l}\Delta P_{\text {gi-ref}} \\ \Delta i_{\text {sdi}} \\ \Delta V_{\text {gdi}}\end{array}\right] ; \Delta Y^{\prime}=\Delta i_{\text {sdi-ref}}$     (18)

$A^{\prime}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] ; B^{\prime}=\left[\begin{array}{ccc}0 & 0 & 0 \\ 1 & -V_{g d 0} & -I_{s d 0}\end{array}\right] ; \quad C^{\prime}=\left[\begin{array}{ll}0 & \frac{1}{V_{g d 0}}\end{array}\right]$ and $D^{\prime}=\left[\frac{1}{V_{g d 0}} 0-\frac{I_{s d 0}}{V_{g d 0}}\right]$     (19)

And for the reactive power controller, we obtain:

$\Delta X^{\prime}=\left[\begin{array}{c}\Delta i_{s q i-r e f} \\ \Delta Z_{r i}\end{array}\right] ; \Delta U^{\prime}=\left[\begin{array}{l}\Delta Q_{g i-r e f} \\ \Delta V_{g d i} \\ \Delta i_{s q i}\end{array}\right] ; \Delta Y^{\prime}=\Delta i_{s q i-r e f}$     (20)

$A^{\prime}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] ; B^{\prime}=\left[\begin{array}{ccc}0 & 0 & 0 \\ 1 & I_{s q 0} & V_{g d 0}\end{array}\right] ; C^{\prime}=\left[\begin{array}{ll}0 & -\frac{1}{V_{g d 0}}\end{array}\right]$ and $D^{\prime}=\left[-\frac{1}{V_{g d 0}}-\frac{I_{s q 0}}{V_{g d 0}} 0\right]$     (21)

In this case the gain K defined in the respective feedback power controller, is fixed assuring the final form of the matrix A’₁=N’K [27].

B. DC voltage controller Linear Model

The balance of powers proves that the DC power is converted to real power in the AC part when we neglect the phase reactor and the shunt conductance capacitance losses. Indeed, the conclusions of our analysis will be only due to the interaction between converter controllers and the DC grid. So the relation between AC and DC side can be written as follows:

${{i}_{m}}{{U}_{s}}={{P}_{g}}={{i}_{sd}}{{V}_{gd}}$     (22)

Thus, the linearization of the Eq. (22) is developed as follows:

${{I}_{m0}}\Delta {{U}_{si}}+{{U}_{s0}}\Delta {{i}_{mi}}={{I}_{sd0}}\Delta {{V}_{gdi}}+\Delta {{i}_{sdi-ref}}{{V}_{gd0}}$      (23)

The PI used in the DC voltage controller is expressed by:

$PI=\frac{{{K}_{pv}}p+{{K}_{iv}}}{p}$     (24)

The linearized model of the DC voltage controller is illustrated in Figure 5.

5.png

Figure 5. Block diagram of the linearized DC voltage controller

The voltage control loop transfer function is:

$\frac{{{U}_{si}}}{{{U}_{si-ref}}}(p)=\frac{1}{1+\frac{{{K}_{pv}}}{{{K}_{iv}}}p+\frac{{{C}_{s}}}{{{K}_{iv}}}{{p}^{2}}}$     (25)

The parameters of the controller are deduced by identifying the polynomial characteristic of the voltage TF to the desired second order polynomial.

${{K}_{iv}}={{C}_{s}}w_{n}^{2}$and${{K}_{pv}}=2\zeta {{w}_{n}}{{C}_{s}}$     (26)

where, $\zeta$ is the damping ratio and w_n is the natural frequency.

In accordance with the linearized block diagram of the voltage controller defined in Figure 5, a new state variable “$\Delta Z_{v i}$” ought to be created.

where,

$\frac{d\Delta {{Z}_{vi}}}{dt}=\Delta {{U}_{si-ref}}-\Delta {{U}_{si}}$     (27)

Thus, we obtain:

$\Delta X^{\prime}=\left[\begin{array}{l}\Delta i_{\text {sdi-ref}} \\ \Delta Z_{v i}\end{array}\right]$;

$\Delta {{U}^{'}}=\left[ \begin{align}  & \Delta \mathop{U}_{si-ref} \\ & \Delta \mathop{V}_{vgdi} \\ & \Delta \mathop{U}_{si} \\ & \Delta \mathop{i}_{li} \\\end{align} \right]$;     (28)

$\Delta Y^{\prime}=\Delta i_{\text {sdi-ref}}$

$A^{\prime}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] ; B^{\prime}=\left[\begin{array}{llll}0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0\end{array}\right] ; C^{\prime}=\left[\begin{array}{ll}0 & -\frac{U_{s 0}}{V_{g d 0}}\end{array}\right]$ and $D^{\prime}=\left[\begin{array}{llll}0 & -\frac{I_{s d 0}}{V_{g d 0}} & \frac{I_{m 0}}{V_{g d 0}} & \frac{U_{s 0}}{V_{g d 0}}\end{array}\right]$     (29)

2.2.2 Studied VSC-HVDC model association

To find a liaison between two subsystems which are constituents of a total system ⅀, two systems ⅀A and ⅀B are designed.

The index 'a' is an auxiliary subscribe identifying ⅀A. The state space model of ⅀A is defined by:

$\Sigma A=\left\{\begin{array}{l}\frac{d}{d t}\left[\begin{array}{l}X_{a 1} \\ X_{a 2}\end{array}\right]=A_{a}\left[\begin{array}{l}X_{a 1} \\ X_{a 2}\end{array}\right]+B_{a}\left[\begin{array}{l}U_{a 1} \\ U_{a 2}\end{array}\right] \\ {\left[\begin{array}{l}Y_{a 1} \\ Y_{a 2}\end{array}\right]=C_{a}\left[\begin{array}{l}X_{a 1} \\ X_{a 2}\end{array}\right]+D_{a}\left[\begin{array}{l}U_{a 1} \\ U_{a 2}\end{array}\right]}\end{array}\right.$     (30)

The index 'b' is an auxiliary subscribe identifying ⅀B. The state space representation of ⅀B is defined by:

$\Sigma B=\left\{\begin{array}{l}\frac{d}{d t}\left[\begin{array}{l}X_{b 1} \\ X_{b 2}\end{array}\right]=A_{b}\left[\begin{array}{l}X_{b 1} \\ X_{b 2}\end{array}\right]+B_{a}\left[\begin{array}{l}U_{b 1} \\ U_{b 2}\end{array}\right] \\ {\left[\begin{array}{l}Y_{b 1} \\ Y_{b 2}\end{array}\right]=C_{b}\left[\begin{array}{l}X_{b 1} \\ X_{b 2}\end{array}\right]+D_{b}\left[\begin{array}{l}U_{b 1} \\ U_{b 2}\end{array}\right]}\end{array}\right.$     (31)

In the whole system, the transfer matrix $T_{a b}$ links the inputs of subsystem $\sum A$ to the outputs of subsystem $\sum B$ :

$\left[\begin{array}{l}U_{a 1} \\ U_{a 2}\end{array}\right]=T_{a b}\left[\begin{array}{l}Y_{b 1} \\ Y_{b 2}\end{array}\right]$     (32)

In the same manner, the transfer matrix $T_{b a}$ links the inputs of subsystem $\sum B$ to the outputs of subsystem $\sum A,$ in the global system:

$\left[\begin{array}{l}U_{b 1} \\ U_{b 2}\end{array}\right]=T_{b a}\left[\begin{array}{l}Y_{a 1} \\ Y_{a 2}\end{array}\right]$     (33)

Hence, the state space representation of the global system is:

$\Sigma=\left\{\begin{array}{l}\frac{d}{d t}\left[\begin{array}{l}X_{a 1} \\ X_{a 2} \\ X_{b 1} \\ X_{b 2}\end{array}\right]=A\left[\begin{array}{l}X_{a 1} \\ X_{a 2} \\ X_{b 1} \\ X_{b 2}\end{array}\right]+B\left[\begin{array}{l}U_{a 1} \\ U_{a 2} \\ U_{b 1} \\ U_{b 2}\end{array}\right] \\ {\left[\begin{array}{l}Y_{a 1} \\ Y_{a 2} \\ Y_{b 1} \\ Y_{b 2}\end{array}\right]=C\left[\begin{array}{l}X_{a 1} \\ X_{a 2} \\ X_{b 1} \\ X_{b 2}\end{array}\right]+D\left[\begin{array}{l}U_{a 1} \\ U_{a 2} \\ U_{b 1} \\ U_{b 2}\end{array}\right]}\end{array}\right.$     (34)

Presuming that the state variables of the total system are a juxtaposition of subsystems state variables in order to build the system ⅀ based to [28]. As well, a combination of subsystem inputs constitutes the inputs of the total system. To retrieve the state variables of subsystems within the total system, subsystems are arranged in the order in the total system.

The form of the state space matrices A, B, C and D of the global system which are resulted from the associations of state space matrices of each subsystem are defined by [28]:

$A=\left[\begin{array}{cc}A_{1} & B_{1} T_{12} C_{2} \\ B_{2} T_{21} C_{1} & A_{2}\end{array}\right] B=\left[\begin{array}{cc}{[0]} & B_{1} T_{12} D_{2} \\ B_{2} T_{21} D_{1} & {[0]}\end{array}\right]$

$C=\left[ \begin{matrix}   \mathop{C}_{1} & \left[ 0 \right]  \\   \left[ 0 \right] & \mathop{C}_{2}  \\\end{matrix} \right]$; $D=\left[ \begin{matrix}   \mathop{D}_{1} & \left[ 0 \right]  \\   \left[ 0 \right] & \mathop{D}_{2}  \\\end{matrix} \right]$     (35)

A. State space model of station1

In order to find the state space model of station1, we studied the link between the state space model of the voltage controller and the state space model of the active power controller with the state space model of the current controller as presented in Figure 6.

First, a development of the state space model of the voltage controller with the state space model of the reactive power controller is done. The reached model is deduced directly from the state space model of each subsystem, so, we obtain:

$\left\{\begin{array}{l}\mathrm{\frac{d}{d t}\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta Z_{v} \\ \Delta i_{s q 1-r e f} \\ \Delta Z_{r 1}\end{array}\right]=\left[\begin{array}{cccc}0 & 0 & 0 & 0 \\ p_{5} & p_{6} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & p_{3} & p_{4}\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta Z_{v} \\ \Delta i_{s q 1-r e f} \\ \Delta Z_{r 1}\end{array}\right]+\left[\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & I_{s q 0} & 0 & 0 & 1 & V_{g d 0}\end{array}\right]\left[\begin{array}{l}\Delta U_{s 1-r e f} \\ \Delta V_{g d 1} \\ \Delta U_{s 1} \\ \Delta i_{l 1} \\ \Delta Q_{g 1-r e f} \\ \Delta U_{i s q 1}\end{array}\right]} \\ \mathrm{\frac{d}{d t}\left[\begin{array}{c}i_{s d 1-r e f} \\ i_{s q 1-r e f}\end{array}\right]=\left[\begin{array}{cccc}0 & -\frac{U_{s 0}}{V_{g d}} & 0 & 0 \\ 0 & 0 & 0 & -\frac{1}{V_{g d 0}}\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta Z_{v} \\ \Delta i_{s q 1-r e f} \\ \Delta Z_{r 1}\end{array}\right]+\left[\begin{array}{cccccc}0 & -\frac{I_{s d 0}}{V_{g d 0}} & \frac{I_{m 0}}{V_{g d 0}} & \frac{U_{s 0}}{V_{g d 0}} & 0 & 0 \\ 0 & -\frac{I_{s q 0}}{V_{g d 0}} & 0 & 0 & -\frac{1}{V_{g d 0}} & 0\end{array}\right]\left[\begin{array}{l}\Delta U_{s 1-r e f} \\ \Delta V_{g d 1} \\ \Delta U_{s 1} \\ \Delta i_{l 1} \\ \Delta Q_{g 1-r e f} \\ \Delta U_{i s q 1}\end{array}\right]}\end{array}\right.$     (36)

where, $p_{j} ; j \in\{3,4,5,6\}$ is a constant value, representing the elements of the state matrix of the system (36).

6.png

Figure 6. Global model of station1

Then, to have the state space model of station1, we determined the transfer matrix $T_{a b 1}$ between the inputs of subsystem obtained by the voltage and reactive power controllers’ models connected together and the output of the current controller model. This matrix is defined as:

$T_{a b 1}=\left[\begin{array}{l}\Delta U_{s 1-r e f} \\ \Delta V_{g d 1} \\ \Delta U_{s 1} \\ \Delta i_{l 1} \\ \Delta Q_{g 1-r e f} \\ \Delta i_{s q 1}\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 1} \\ \Delta i_{s q 1} \\ \Delta U_{s 1}\end{array}\right]$     (37)

The transfer matrix $T_{a b 1}$ given in Eq. (38), links the inputs of the voltage controller model with the reactive power controller model and the output of the current controller model.

$T_{b a 1}=\left[\begin{array}{c}\Delta i_{s d 1-r e f} \\ \Delta i_{s q 1-r e f}\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta i_{s q 1-r e f}\end{array}\right]$     (38)

So, we deduced the global state space model of station1, as given in system (39):

$\left\{\begin{array}{l}\mathrm{\frac{d}{d t}\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta Z_{v} \\ \Delta i_{s q 1-r e f} \\ \Delta Z_{r 1} \\ \Delta i_{s d 1} \\ \Delta i_{s q 1} \\ \Delta U_{s 1} \\ \Delta Z_{11} \\ \Delta Z_{21}\end{array}\right]=\left[\begin{array}{ccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p_{5} & p_{6} & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & p_{3} & p_{4} & 0 & V_{g d 0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{-R_{s}}{L_{s}} & 0 & 0 & \frac{1}{L_{s}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{-R_{s}}{L_{s}} & 0 & 0 & \frac{1}{L_{s}} \\ 0 & 0 & 0 & 0 & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ 0 & \frac{-U_{s 0}}{V_{g d 0}} & 0 & 0 & & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{-1}{V_{g d}} & 0 & -1 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 1-r e f} \\ \Delta Z_{v} \\ \Delta i_{s q 1-r e f} \\ \Delta Z_{r 1} \\ \Delta i_{s d 1} \\ \Delta i_{s q 1} \\ \Delta U_{s 1} \\ \Delta Z_{11} \\ \Delta Z_{21}\end{array}\right]+\left[\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s d 0}}{V_{g d 0}} & \frac{-I_{m 0}}{V_{g d 0}} & \frac{-U_{s 0}}{V_{g d 0}} & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s q 0}}{V_{g d 0}} & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta U_{s 1-r e f} \\ \Delta V_{g d 1} \\ \Delta U_{s 1} \\ \Delta i_{l 1} \\ \Delta Q_{g 1-r e f} \\ \Delta i_{s q 1} \\ \Delta i_{s d 1-r e f} \\ \Delta i_{s q 1-r e f}\end{array}\right]} \\ \mathrm{\left[\begin{array}{l}\Delta i_{\text {sd1-ref}} \\ \Delta i_{\text {sq1-ref}} \\ \Delta i_{s d 1} \\ \Delta i_{\text {sq1}}\end{array}\right]=\left[\begin{array}{ccccccccc}0 & \frac{-U_{s 0}}{V_{g d 0}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{\text {sd1-ref}} \\ \Delta Z_{v} \\ \Delta i_{\text {sq1-ref}} \\ \Delta Z_{r 1} \\ \Delta i_{\text {sd} 1} \\ \Delta i_{\text {sq} 1} \\ \Delta U_{s 1} \\ \Delta Z_{11} \\ \Delta Z_{21}\end{array}\right]+\left[\begin{array}{cccccccc}0 & \frac{-I_{s d 0}}{V_{g d 0}} & \frac{I_{m 0}}{V_{g d 0}} & \frac{U_{s 0}}{V_{g d 0}} & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s q 0}}{V_{g d 0}} & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta U_{s 1-r e f} \\ \Delta V_{g d 1} \\ \Delta U_{s 1} \\ \Delta i_{l 1} \\ \Delta Q_{g 1-r e f} \\ \Delta i_{s q 1} \\ \Delta i_{s d 1-r e f} \\ \Delta i_{s q 1-r e f}\end{array}\right]}\end{array}\right.$     (39)

$\left\{\begin{array}{l}\mathrm{\frac{d}{d t}\left\lfloor\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2}\end{array}\right]=\left[\begin{array}{cccc||c}0 & 0 & 0 & 0 \\ p_{1} & p_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & p_{3} & p_{4}\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2}\end{array}\right]+\left[\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\ 1 & -V_{g d 0} & -I_{s d 0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I_{s q 0} & 1 & V_{g d 0}\end{array}\right]\left[\begin{array}{l}\Delta P_{g 2-r e f} \\ \Delta i_{s d 2} \\ \Delta V_{g d 2} \\ \Delta Q_{g 2-r e f} \\ \Delta i_{s q 2}\end{array}\right]} \\ \mathrm{\left[\begin{array}{c}\Delta i_{s d 1-r e f} \\ \Delta i_{s q 1-r e f}\end{array}\right]=\left[\begin{array}{cccc}0 & \frac{1}{V_{g d 0}} & 0 & 0 \\ 0 & 0 & 0 & \frac{-1}{V_{g d 0}}\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2}\end{array}\right]+\left[\begin{array}{ccccc}\frac{1}{V_{g d 0}} & 0 & \frac{-I_{s d 0}}{V_{g d 0}} & 0 & 0 \\ 0 & 0 & \frac{-I_{s q 0}}{V_{g d 0}} & \frac{-1}{V_{g d 0}} & 0\end{array}\right]\left[\begin{array}{l}\Delta P_{g 2-r e f} \\ \Delta i_{s d 2} \\ \Delta V_{g d 2} \\ \Delta Q_{g 2-r e f} \\ \Delta i_{s q 2}\end{array}\right]}\end{array}\right.$     (40)

B. State Space model of station2

In the same way, we determined the state space model of station2, as presented in Figure 7. This station is controlled by the following controllers: active power, reactive power, and current controller.

7.png

Figure 7. Model of station2

First, a development of the state space model of the active power controller with the state space model of the reactive power controller is done. The reached model is deduced directly from the state space model of each subsystem, so, we obtain:

$p_{j} ; j \in\{1,2,3,4\}$ is a constant value, representing the elements of the state matrix of the system (40).

Then, in order to get the state space model of station2, we should determine the transfer matrix T_ab2 between the inputs of subsystem obtained by the active and reactive power controllers’ models connected together and the output of the current controller model. This matrix is defined as:

$T_{a b 2}=\left[\begin{array}{l}\Delta P_{g 2-r e f} \\ \Delta i_{s d 2} \\ \Delta V_{g d 2} \\ \Delta Q_{g 2-r e f} \\ \Delta i_{s q 2}\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2} \\ \Delta i_{s q 2} \\ \Delta U_{s 2}\end{array}\right]$     (41)

The transfer matrix T_ab2 given in Eq. (42), links the inputs of the active power controller model with the reactive power controller model and the output of the current controller model.

$T_{b a 2}=\left[\begin{array}{l}\Delta i_{s d 2-\text {ref}} \\ \Delta i_{s q 2-r e f}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta i_{s q 2-r e f}\end{array}\right]$     (42)

So we deduced the state space model of station2:

$\left\{\begin{array}{l}\mathrm{\frac{d}{d t}\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2} \\ \Delta i_{s d 2} \\ \Delta i_{s q 2} \\ \Delta U_{s 2} \\ \Delta Z_{12} \\ \Delta Z_{22}\end{array}\right]=\left[\begin{array}{ccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p_{1} & p_{2} & 0 & 0 & -V_{g d 0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & p_{3} & p_{4} & 0 & V_{g d 0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{-R_{s}}{L_{s}} & 0 & 0 & \frac{1}{L_{s}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{-R_{s}}{L_{s}} & 0 & 0 & \frac{1}{L_{s}} \\ 0 & 0 & 0 & 0 & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & -1 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2} \\ \Delta i_{s d 2} \\ \Delta i_{s q 2} \\ \Delta U_{s 2} \\ \Delta Z_{12} \\ \Delta Z_{22}\end{array}\right]+\left[\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s d 0}}{V_{g d 0}} & \frac{I_{m 0}}{V_{g d 0}} & \frac{I_{s d 0}}{V_{g d 0}} & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s q 0}}{V_{g d 0}} & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta P_{g 2-r e f} \\ \Delta i_{s d 2} \\ \Delta V_{g d 2} \\ \Delta Q_{g 2-r e f} \\ \Delta i_{s q 2} \\ \Delta i_{s d 2-r e f} \\ \Delta i_{s q 2-r e f}\end{array}\right]} \\ \mathrm{\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta i_{s q 2-r e f} \\ \Delta i_{s d 2} \\ \Delta i_{s q 2} \\ \Delta U_{s 2}\end{array}\right]=\left[\begin{array}{ccccccccc}0 & \frac{1}{V_{g d 0}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta i_{s d 2-r e f} \\ \Delta Z_{a} \\ \Delta i_{s q 2-r e f} \\ \Delta Z_{r 2} \\ \Delta i_{s d 2} \\ \Delta i_{s q 2} \\ \Delta U_{s 2} \\ \Delta Z_{12} \\ \Delta Z_{22}\end{array}\right]+\left[\begin{array}{cccccccc}0 & \frac{-I_{s d 0}}{V_{g d 0}} & \frac{I_{m 0}}{V_{g d 0}} & \frac{U_{s 0}}{V_{g d 0}} & 0 & 0 & 0 & 0 \\ 0 & \frac{-I_{s q 0}}{V_{g d 0}} & 0 & 0 & \frac{-1}{V_{g d 0}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{l}\Delta P_{g 2-r e f} \\ \Delta i_{s d 2} \\ \Delta V_{g d 2} \\ \Delta Q_{g 2-r e f} \\ \Delta i_{s q 2} \\ \Delta i_{s d 2-r e f} \\ \Delta i_{s q 2-r e f}\end{array}\right]}\end{array}\right.$     (43)

C. State Space model of system reactor and cable

The connection between the improved cable model and the smoothing reactor system model is illustrated in Figure 8.

8.png

Figure 8. Connection of the cable to the smoothing reactor

To determine the state space model of this association, we determined the transfer matrix T_ab3 between the inputs of subsystem built from cable model and the output of the smoothing reactor model. This matrix is defined as:

$T_{a b 3}\left[\begin{array}{c}\Delta U_{s 1} \\ \Delta U_{c c 1} \\ \Delta U_{s 2} \\ \Delta U_{c c 2}\end{array}\right]=\left[\begin{array}{cc}0 & 0 \\ \frac{1}{2} & 0 \\ 0 & 0 \\ 0 & \frac{1}{2}\end{array}\right]\left[\begin{array}{c}\Delta U_{1} \\ \Delta U_{2}\end{array}\right]$     (44)

where, $U_{c c i}=\frac{U_{i}}{2}$

The transfer matrix T_ab3 given in Eq. (45), links the inputs of the cable model and the output of the smoothing reactor model is defined by:

$T_{b a 3}\left[\begin{array}{c}\Delta i_{l 1} \\ \Delta i_{l 2}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}\Delta i_{l 1} \\ \Delta i_{l 2}\end{array}\right]$    (45)

Thus, we obtained the state space model of the reactor, cable system association:

$\left\{\begin{array}{l}\mathrm{\frac{d}{d t}\left[\begin{array}{l}\Delta i_{l 1} \\ \Delta i_{l 2} \\ \Delta U_{c c 1} \\ \Delta U_{c c 2} \\ \Delta i_{L c 1} \\ \Delta i_{L c 2}\end{array}\right]=\left[\begin{array}{cccccc}p_{7} & p_{8} & \frac{1}{L_{s r}} & 0 & 0 & 0 \\ p_{9} & p_{10} & 0 & \frac{1}{L_{s r}} & 0 & 0 \\ \frac{-1}{C_{s}} & 0 & \frac{-G_{c}}{C_{c}} & 0 & \frac{-1}{C_{c}} & 0 \\ 0 & \frac{-1}{C_{c}} & 0 & \frac{-G_{c}}{C_{c}} & \frac{1}{C_{c}} & 0 \\ 0 & 0 & \frac{L_{c 2}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{-L_{c 2}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{-R_{c 1} L_{c 2}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{R_{c 1} M_{c 12}}{L_{c 1} L_{c 2}-M_{c 12}} \\ 0 & 0 & \frac{-M_{c 12}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{M_{c 12}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{R_{c 1} M_{c 12}}{L_{c 1} L_{c 2}-M_{c 12}^{2}} & \frac{-R_{c 2} L_{c 1}}{L_{c 1} L_{c 2}-M_{c 12}}\end{array}\right]\left[\begin{array}{l}\Delta i_{l 1} \\ \Delta i_{l 2} \\ \Delta U_{c c 1} \\ \Delta U_{c c 2} \\ \Delta i_{L c 1} \\ \Delta i_{L c 2}\end{array}\right]} \\ \mathrm{\left[\begin{array}{c}\Delta i_{l 1} \\ \Delta i_{l 2} \\ \Delta U_{1} \\ \Delta U_{2}\end{array}\right]=\left[\begin{array}{llllll}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0\end{array}\right]\left[\begin{array}{c}\Delta i_{l 1} \\ \Delta i_{l 2} \\ \Delta U_{c c 1} \\ \Delta U_{c c 2} \\ \Delta i_{L c 1} \\ \Delta i_{L c 2}\end{array}\right]}\end{array}\right.$     (46)

where, $p_{j} ; j \in\{7,8,9,10\}$ is a constant value, representing some elements of the state matrix of the system (46).

D. State space model of the studied VSC-HVDC system

In the preceding parts, each linearized subsystem of the VSC-HVDC liaison is modeled as a linear state space model. With the aim of getting a total closed-loop model of the VSC-HVDC system, inputs and outputs of subsystems are linked together as shown in Figure 9.

After developing the link between the systems composed of station 1, station 2 and the system smoothing reactor with cable, we obtain the final state matrix A of the VSC-HVDC

System. As well, the final state vector ∆X', the final input vector ∆U' and the final output vector ∆Y' of the VSC-HVDC system are given as:

$\begin{align}  & {{X}_{VSC-HVDC}}=\left[ \begin{matrix}   \Delta {{i}_{sd1-ref}} & \Delta {{Z}_{v}} & \Delta {{i}_{sq1-ref}} & \Delta {{Z}_{r1}} & \Delta {{i}_{sd1}} & \Delta {{i}_{sq1}} & \Delta {{U}_{s1}} & \Delta {{Z}_{11}} & \Delta {{Z}_{21}} & \Delta {{i}_{sd2-ref}} & \Delta {{Z}_{a}} & \Delta {{i}_{sq2-ref}}  \\\end{matrix} \right. \\ & {{\left. \begin{matrix}   \Delta {{Z}_{r2}} & \Delta {{i}_{sd2}} & \Delta {{i}_{sq2}} & \Delta {{U}_{s2}} & \Delta {{Z}_{12}} & \Delta {{Z}_{22}} & \Delta {{i}_{l1}} & \Delta {{i}_{l2}} & \Delta {{U}_{cc1}} & \Delta {{U}_{cc2}} & \Delta {{i}_{Lc1}} & \Delta {{i}_{Lc2}}  \\\end{matrix} \right]}^{T}} \\\end{align}$     (47)

$\begin{align}  & \mathop{Y}_{VSC-HVDC}=\left[ \begin{matrix}   \mathop{\Delta i}_{sd1-ref} & \mathop{\Delta i}_{sq1-ref} & \mathop{\Delta i}_{sd1} & \mathop{\Delta i}_{sq1} & \mathop{\Delta U}_{s1} & \mathop{\Delta i}_{sd2-ref} & \mathop{\Delta i}_{sq2-ref} & \mathop{\Delta i}_{sd2} & \mathop{\Delta i}_{sq2} & \mathop{\Delta U}_{s2} & \mathop{\Delta i}_{l1} & \mathop{\Delta i}_{l2}  \\\end{matrix} \right. \\ & {{\begin{matrix}   \mathop{\Delta U}_{1} & \left. \mathop{\Delta U}_{2} \right]  \\\end{matrix}}^{T}} \\\end{align}$     (48)

$\begin{align}  & {{U}_{VSC-HVDC}}=\left[ \begin{matrix}   \Delta {{U}_{s1-ref}} & \Delta {{V}_{gd1}} & \Delta {{U}_{s1}} & \Delta {{i}_{l1}} & \Delta {{Q}_{g1-ref}} & \Delta {{i}_{sq1}} & \Delta {{i}_{sd1-ref}} & \Delta {{i}_{sq1-ref}} & \Delta {{P}_{g2-ref}} & \Delta {{i}_{sd2}} & \Delta {{V}_{gd2}} & \Delta {{Q}_{g2-ref}}  \\\end{matrix} \right. \\ & {{\left. \begin{matrix}   \Delta {{i}_{sq2}} & \Delta {{i}_{sd2-ref}} & \Delta {{i}_{sq2-ref}} & \Delta {{i}_{l1}} & \Delta {{i}_{l2}} & \Delta {{U}_{cc1}} & \Delta {{U}_{cc2}}  \\\end{matrix} \right]}^{T}} \\\end{align}$     (49)

$A{}_{VSC-HVDC}=\left[ \begin{matrix}   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   {{p}_{5}} & {{p}_{6}} & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & {{p}_{3}} & {{p}_{4}} & 0 & {{V}_{gd0}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & \frac{-{{R}_{s}}}{{{L}_{s}}} & 0 & 0 & \frac{1}{{{L}_{s}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & \frac{-{{R}_{s}}}{{{L}_{s}}} & 0 & 0 & \frac{1}{{{L}_{s}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & {{a}_{1}} & {{a}_{2}} & {{a}_{3}} & {{a}_{4}} & {{a}_{5}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{U}_{s0}}}{{{V}_{gd0}}} & 0 & 0 & 0 & 0 & 0  \\   0 & \frac{{{U}_{s0}}}{{{V}_{gd0}}} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & \frac{-1}{{{V}_{gd0}}} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{p}_{1}} & {{p}_{2}} & 0 & 0 & -{{V}_{gd0}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{p}_{3}} & {{p}_{4}} & 0 & {{V}_{gd0}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{-{{R}_{s}}}{{{L}_{s}}} & 0 & 0 & \frac{1}{{{L}_{s}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{-{{R}_{s}}}{{{L}_{s}}} & 0 & 0 & \frac{1}{{{L}_{s}}} & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{a}_{1}} & {{a}_{2}} & {{a}_{3}} & {{a}_{4}} & {{a}_{5}} & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{V}_{gd0}}} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{-1}{{{V}_{gd0}}} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{p}_{7}} & {{p}_{8}} & \frac{1}{{{L}_{sr}}} & 0 & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{p}_{9}} & {{p}_{10}} & 0 & \frac{1}{{{L}_{sr}}} & 0 & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{{{C}_{c}}} & 0 & -\frac{{{G}_{c}}}{{{C}_{c}}} & 0 & -\frac{1}{{{C}_{c}}} & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{{{C}_{c}}} & 0 & -\frac{{{G}_{c}}}{{{C}_{c}}} & \frac{1}{{{C}_{c}}} & 0  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{L}_{c2}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & -\frac{{{L}_{c2}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & -\frac{{{R}_{c1}}{{L}_{c2}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & \frac{{{R}_{c2}}{{M}_{c12}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}}  \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{{{M}_{c12}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & \frac{{{M}_{c12}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & \frac{{{R}_{c1}}{{M}_{c12}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}} & -\frac{{{R}_{c2}}{{L}_{c1}}}{{{L}_{c1}}{{L}_{c2}}-M_{c12}^{2}}  \\\end{matrix} \right]$

9.png

Figure 9. Closed-loop linearized model of the controlled point-to-point VSC-HVDC link

where, $a_{1}=\frac{-V_{m d 0}-i_{S q_{0}} L_{S} w}{C_{S} U_{S 0}}+\frac{z_{1}^{\prime}}{C_{S}}$; $a_{2}=\frac{-V_{m q 0}-i_{S q 0} L_{S} W}{C_{S} U_{S 0}}+\frac{z_{2}^{\prime}}{C_{S}}$; $a_{3}=\frac{i_{m 0}}{C_{s} U_{S 0}}+\frac{z_{3}^{\prime}}{C_{s}}$; $a_{4}=\frac{-i_{s d 0}}{C_{s} U_{s 0}}+\frac{z_{4}^{\prime}}{C_{s}}$; $a_{5}=\frac{-i_{s q 0}}{C_{s} U_{s 0}}+\frac{z_{5}^{\prime}}{C_{s}}$.

Small-signal stability analyses (SSSA) becomes essential in the design of HVDC controllers in order to enhance their flexibility to faults, and to boost their aptitude to contribute to power grid operation.

The stability of two terminal VSC-HVDC system using the small signal stability technique for a linearized state space model is studied in this paper. The state space model of each subsystem was developed then a connection between each block was made to create a global linear state space model. The study of eigenvalues associated to the global linearized state space system is automatically used to improve the stability of the VSC-HVDC link. This work will help us to develop a generic method, with a view of providing flexible models to study the stability of a multi-terminal MTDC grid.