Droop control can be implemented in two forms: currentbased (I–V characteristic curve), and powerbased (P–V characteristic curve) which is used in this paper [44]. Accurate power sharing in droop control, regardless of DC line resistances and system structure, occurs by common voltage feedback, known as Pilot Voltage Droop (PVD) [45]. Accordingly, the proposed control method uses common voltage feedback in the presence of highbandwidth communication.
Optimal power sharing should be maintained as much as possible with minimum variation to minimize the negative effects of disturbances on neighboring AC grids (e.g., frequency deviation) during postcontingency conditions. The optimal droop coefficients satisfy this purpose, while also minimizing the DC voltage variation and maintaining all converters (with or without droop control mode) within their limits. They are determined based on the initial loading of converters, the stability constraint, and converter limitations by linearizing the system around the steadystate operating point in various contingencies.
The optimal droop coefficients are stored in the lookup table to be used immediately in the first control layer after detection of disturbances by the central controller. The proposed lookup table is an array of data that relates input values (operating conditions) to output values (optimal droop coefficients). As a result, the processing time is minimized by retrieving a set of values from memory instead of online calculations. Figure 2 shows the P–V droop characteristic curve and the modified power control loop considering the proposed method.
P_{n} and V_{n} are the rated power and voltage of the converter. The central controller detects disturbances through online monitoring of the active power (P_{i}) and DC voltage (V_{i}) of the VSC stations. Droop coefficients (R_{droop}) are inversely related to DC voltage deviation. Therefore, they can be increased to minimize DC voltage deviation, though high droop coefficients reduce stability. To overcome this drawback, they are limited to the maximum droop coefficients (R_{i,max}), which are determined based on smallsignal stability analysis of the VSCMTDC system [21, 27, 46]. Accordingly, the differential–algebraic equations (DAEs) of the system are linearized around the operating point and expressed in statespace form by:
$$\Delta \dot{X} = A\Delta X + B\Delta U$$
(3)
where ΔX, ΔU, A, and B are the statevector, inputvector, statematrix, and inputmatrix, respectively. The maximum droop coefficients can be increased until all eigenvalues remain in the left half of the complex plane to ensure system stability.
The derivative of the power flow equations with respect to the voltage vector (V) results in the Jacobian matrix, which is expressed by:
$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {P = V \odot (YV)} \\ {J = {\raise0.7ex\hbox{${\partial P}$} \!\mathord{\left/ {\vphantom {{\partial P} {\partial V}}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V}$}} \, } \\ \end{array} } \right. \to J = diag(V)Y + diag(YV) \\ & \quad \Rightarrow J = \left[ {\begin{array}{*{20}c} {{\raise0.7ex\hbox{${\partial P_{1} }$} \!\mathord{\left/ {\vphantom {{\partial P_{1} } {\partial V_{1} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{1} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{1} }$} \!\mathord{\left/ {\vphantom {{\partial P_{1} } {\partial V_{m} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{m} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{1} }$} \!\mathord{\left/ {\vphantom {{\partial P_{1} } {\partial V_{n} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{n} }$}}} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {{\raise0.7ex\hbox{${\partial P_{m} }$} \!\mathord{\left/ {\vphantom {{\partial P_{m} } {\partial V_{1} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{1} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{m} }$} \!\mathord{\left/ {\vphantom {{\partial P_{m} } {\partial V_{m} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{m} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{m} }$} \!\mathord{\left/ {\vphantom {{\partial P_{m} } {\partial V_{n} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{n} }$}}} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {{\raise0.7ex\hbox{${\partial P_{n} }$} \!\mathord{\left/ {\vphantom {{\partial P_{n} } {\partial V_{1} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{1} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{n} }$} \!\mathord{\left/ {\vphantom {{\partial P_{n} } {\partial V_{m} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{m} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{n} }$} \!\mathord{\left/ {\vphantom {{\partial P_{n} } {\partial V_{n} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial V_{n} }$}}} \\ \end{array} } \right] \\ \end{aligned}$$
(4)
where P and Y refer to the active power vector and admittance matrix of the DC grid. The symbol ʘ is the Hadamard product operator.
The formation procedure of the lookup table according to N1 contingency is presented in the following sections. It is to be noted that converters ‘1: m’ and ‘m + 1: n’ operate in droop and constant power modes, respectively.
3.1 Wind power changes
The system equations during wind power changes (ΔP^{*}) are expressed by:
$$\begin{aligned} & \left. {\left\{ {\begin{array}{*{20}c} {\Delta P_{i} =  \alpha R_{i} \Delta V_{1} } \\ {\Delta P_{i} = \sum\limits_{j = 1}^{n} {J_{i,j} \Delta V_{j} } } \\ \end{array} } \right.} \right_{i = 1:n} \mathop{\longrightarrow}\limits_{{R_{i} = 0\mathop {}\nolimits_{{}} i \ge m + 1}}^{{\Delta P_{m + 1} = \Delta P^{*} }} \\ & \quad \left[ {\begin{array}{*{20}c} {\Delta V_{1} } \\ {\Delta V_{2} } \\ \vdots \\ {\Delta V_{m} } \\ {\Delta V_{m + 1} } \\ \vdots \\ {\Delta V_{n} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {J_{1,1} + \alpha R_{1} } & {J_{1,2} } & \cdots & {J_{1,n} } \\ {J_{2,1} + \alpha R_{2} } & {J_{2,2} } & \cdots & {J_{2,n} } \\ \vdots & \vdots & \ddots & \vdots \\ {J_{m,1} + \alpha R_{m} } & {J_{m,2} } & \cdots & {J_{m,n} } \\ {J_{m + 1,1} } & {J_{m + 1,2} } & \cdots & {J_{m + 1,n} } \\ \vdots & \vdots & \ddots & \vdots \\ {J_{n,1} } & {J_{n,2} } & \cdots & {J_{n,n} } \\ \end{array} } \right]}}_{X}^{  1} \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \vdots \\ 0 \\ {\Delta P^{*} } \\ \vdots \\ 0 \\ \end{array} } \right] \\ \end{aligned}$$
(5)
The active power and DC voltage deviations of VSC stations according to (5) are determined by:
$$\left\{ {\begin{array}{*{20}c} \begin{gathered} \Delta V_{i} = \frac{1}{\left X \right}X_{i,m + 1}^{A} \Delta P = \frac{1}{\left X \right}X_{m + 1,i}^{C} \Delta P \hfill \\ \quad = \frac{{(  1)^{m + 1 + i} X_{m + 1,i}^{M} }}{{\sum\limits_{i^{\prime} = 1}^{n} {(  1)^{i^{\prime} + 1} X_{i^{\prime},1} X_{i^{\prime},1}^{M} } }}\Delta P\quad {(}i = 1:n) \hfill \\ \end{gathered} \\ {\Delta P_{i} =  \alpha R_{i} \Delta V\quad \quad \quad \quad \quad {(}i = 1:m)} \\ \end{array} } \right.$$
(6)
where X^{A}, X^{C}, and X^{M} matrices refer to the adjugate matrix, cofactor matrix, and minor of X, respectively. The constrained optimization problem is expressed by:
$$\begin{gathered} \, \hfill \\ \hfill \\ \begin{array}{*{20}l} {max} \hfill & {\Delta P^{*} } \hfill \\ {subject\,to} \hfill & {} \hfill \\ \quad {\Delta V_{i,min} \le \Delta V_{i} \le \Delta V_{i,max} } \hfill &{(i = 1:n)} \hfill \\ \quad {\Delta P_{i} \le \Delta P_{i,max} } \hfill & {(i = 1:m)} \hfill \\ \quad {R_{i} = R_{i,0} } \hfill & {(i = 1:m;i \ne k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ \quad {0 \le \alpha R_{i} \le R_{i,max} } \hfill & {(i = k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ \end{array} \hfill \\ \end{gathered}$$
(7)
The following process is proposed to find the optimal droop coefficients related to the output power of OWFs:

1.
The maximum allowable wind power considering the initial droop coefficients (k_{i} = 0) is computed by solving the problem (7).

2.
If any power constraint is activated by the problem (7) solving process, the droop coefficient of its related converter is permitted to change.

3.
If any voltage constraint is activated by the problem (7) solving process, the droop coefficient that minimizes (8) is permitted to change.
$$min\quad \alpha (R_{i,max}  R_{i,0} )\left( {{\raise0.7ex\hbox{${R_{max,0} }$} \!\mathord{\left/ {\vphantom {{R_{max,0} } {R_{i,0} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${R_{i,0} }$}}} \right)$$
(8)

4.
Problem (7) is modified based on updated droop constraints obtained from the 2nd and 3rd steps. Then, it is solved to find the new maximum allowable wind power and its corresponding optimal droop coefficients.

5.
For a more accurate estimate, the constrained optimization problem (9) can be expressed based on updated droop constraints obtained from the 2nd step. It determines the optimal droop coefficients related to the specific wind power (ΔP_{spec}) that is less than the maximum allowable power computed in the 3rd step. To increase accuracy of the lookup table, problem (9) should be solved with several specific wind powers (the higher they are, the more accurate it is).
$$\begin{gathered} \hfill \\ \hfill \\ \begin{array}{*{20}l} {min\quad \sum\limits_{i = 1}^{m} {\left( {\frac{{R_{i}  R_{i,0} }}{{R_{i,0} }}} \right)^{2} } } \hfill & {} \hfill \\ {subject\,to} \hfill & {} \hfill \\ \quad {\Delta V_{i,min} \le \Delta V_{i} \le \Delta V_{i,max} } \hfill & {(i = 1:n)} \hfill \\ \quad {\Delta P_{i} \le \Delta P_{i.max} } \hfill & {(i = 1:m)} \hfill \\ \quad {\Delta P_{m + 1} = \Delta P_{spec} } \hfill & {} \hfill \\ \quad {R_{i} = R_{i,0} } \hfill & {(i = 1:m;i \ne k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ \quad {0 \le \alpha R_{i} \le R_{i,max} } \hfill & {(i = k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ \end{array} \hfill \\ \end{gathered}$$
(9)
Steps 2–5 are repeated until all converters reach their ratings. In each iteration, the optimal droop coefficients related to the specific output power of OWFs are obtained. Finally, the optimal droop coefficients are approximated by interpolation as a function of wind power changes.
3.2 Converter outage
With the outage of the jth converter, the system equations are expressed by:
$$\begin{aligned} & \left. {\left\{ {\begin{array}{*{20}c} {\Delta P_{i} =  \alpha R_{i} \Delta V_{1} } \\ {\Delta P_{i} = \sum\limits_{j = 1}^{n} {J_{i,j} \Delta V_{j} } } \\ \end{array} } \right.} \right_{i = 1:n} \mathop{\longrightarrow}\limits_{{R_{i} = 0\mathop {}\nolimits_{{}} i \ge m + 1}}^{{\Delta P_{j} =  P_{j}^{{}} }} \\ & \quad \left[ {\begin{array}{*{20}c} {\Delta V_{1} } \\ \vdots \\ {\begin{array}{*{20}c} {\Delta V_{j} } \\ \vdots \\ \end{array} } \\ {\Delta V_{m} } \\ {\Delta V_{m + 1} } \\ \vdots \\ {\Delta V_{n} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {J_{1,1} + \alpha R_{1} } & {J_{1,2} } & \cdots & {J_{1,n} } \\ {\begin{array}{*{20}c} \vdots \\ {J_{j,1} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {J_{j,2} } \\ \end{array} } & {\begin{array}{*{20}c} \ddots \\ \cdots \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {J_{j,n} } \\ \end{array} } \\ \vdots & \vdots & \ddots & \vdots \\ {J_{m,1} + \alpha R_{m} } & {J_{m,2} } & \cdots & {J_{m,n} } \\ {J_{m + 1,1} } & {J_{m + 1,2} } & \cdots & {J_{m + 1,n} } \\ \vdots & \vdots & \ddots & \vdots \\ {J_{n,1} } & {J_{n,2} } & \cdots & {J_{n,n} } \\ \end{array} } \right]}}_{X}^{  1} \left[ {\begin{array}{*{20}c} 0 \\ {\begin{array}{*{20}c} \vdots \\ {  P_{j}^{{}} } \\ \end{array} } \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0 \\ \end{array} } \right] \\ \end{aligned}$$
(10)
where P_{j} refers to the active power of the jth converter before its outage. The DC voltage and active power deviations of VSC stations according to (10) and the constrained optimization problem are respectively determined by:
$$\left\{ {\begin{array}{*{20}c} \begin{gathered} \Delta V_{i} = \frac{1}{\left X \right}X_{i,j}^{A} (  P_{j} ) = \frac{1}{\left X \right}X_{j,i}^{C} (  P_{j} ) \hfill \\ \, = \frac{{(  1)^{j + i} X_{j,i}^{M} }}{{\sum\limits_{i^{\prime} = 1}^{n} {(  1)^{i^{\prime} + 1} X_{i^{\prime},1} X_{i^{\prime},1}^{M} } }}(  P_{j} ){ (}i = 1:n;i \ne j) \hfill \\ \end{gathered} \\ {\Delta P_{i} =  \alpha R_{i} \Delta V_{1} \, (i = 1:m;i \ne j)} \\ \end{array} } \right.$$
(11)
and
$$\begin{array}{*{20}l} {min\quad \sum\limits_{\begin{subarray}{l} i = 1 \\ i \ne j \end{subarray} }^{m} {\left( {\frac{{R_{i}  R_{i,0} }}{{R_{i,0} }}} \right)^{2} } } \hfill & {} \hfill \\ {subject\,to} \hfill & {} \hfill \\ {\quad \Delta V_{i,min} \le \Delta V_{i} \le \Delta V_{i,max} } \hfill & {(i = 1:n;i \ne j)} \hfill \\ {\quad \Delta P_{i} \le \Delta P_{i.max} } \hfill & {(i = 1:m;i \ne j)} \hfill \\ {\quad R_{i} = R_{i,0} } \hfill & {(i = k_{1} ,k_{2} , \ldots ,k_{j  1} ;i \ne j)} \hfill \\ {\quad 0 \le \alpha R_{i} \le R_{i,max} } \hfill & {(i = 1:m;i \ne j,k_{1} ,k_{2} , \ldots ,k_{j  1} )} \hfill \\ \end{array}$$
(12)
The following process is proposed to find the optimal droop coefficients during the jth converter outage:

1.
Problem (12) is solved assuming that all droop coefficients are permitted to change (k_{i} = 0).

2.
If any power constraints are activated by the problem (12) solving process, the droop coefficients of their related converters are still permitted to change.

3.
If any voltage constraint is activated by the problem (12) solving process, the droop coefficient that minimizes (8) is still allowed to change.

4.
The rest of the droop coefficients are considered equal to their initial value (R_{i,0}).

5.
Problem (12) is modified based on updated droop constraints obtained from the 2nd, 3rd and 4th steps.

6.
If the problem (12) solving process cannot find an optimal solution, the (λ+1)th droop coefficient that minimizes (8) is also allowed to change (λ=iteration number).
Steps 5, 6 are repeated until problem (12) can find an optimal solution. The 1st converter outage deteriorates the reliable operation of the MTDC system because of missing common voltage feedback (\({\Delta \mathrm{V}}_{1}\)). Accordingly, the system equations in (10) should also be defined based on another voltage feedback (e.g., \({\Delta \mathrm{V}}_{2}\)) to maintain stability during the 1st converter outage.
3.3 DC line disconnection
By disconnecting any DC line (e.g., T_{j,m}), the system equations are expressed by:
$$\begin{aligned} & \left. {\left\{ {\begin{array}{*{20}c} {\Delta P_{i} =  \alpha R_{i} \Delta V_{1} } \\ {\Delta P_{i} = \sum\limits_{j = 1}^{n} {J_{i,j}^{*} \Delta V_{j} } + \sum\limits_{j = 1}^{n} {J^{\prime}_{i,j} \Delta Y_{j,i}^{T} } } \\ \end{array} } \right.} \right_{i = 1:n} \mathop{\longrightarrow}\limits^{{R_{i} = 0\mathop {}\nolimits_{{}} i \ge m + 1}} \\ & \quad \left[ {\begin{array}{*{20}c} {\Delta V_{1} } \\ \vdots \\ {\begin{array}{*{20}c} {\Delta V_{j} } \\ \vdots \\ \end{array} } \\ {\Delta V_{m} } \\ {\Delta V_{m + 1} } \\ \vdots \\ {\Delta V_{n} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {J_{1,1}^{*} + \alpha R_{1} } & \cdots & {J_{1,n}^{*} } \\ \vdots & \ddots & \vdots \\ {J_{j,1}^{*} + \alpha R_{j} } & \cdots & {J_{j,n}^{*} } \\ \vdots & \ddots & \vdots \\ {J_{m,1}^{*} + \alpha R_{m} } & \cdots & {J_{m,n}^{*} } \\ {J_{m + 1,1}^{*} } & \cdots & {J_{m + 1,n}^{*} } \\ \vdots & \ddots & \vdots \\ {J_{n,1}^{*} } & \cdots & {J_{n,n}^{*} } \\ \end{array} } \right]}}_{X}^{  1} \left[ {\begin{array}{*{20}c} 0 \\ {\begin{array}{*{20}c} \vdots \\ {  (J^{\prime}_{j,j}  J^{\prime}_{j,m} )Y_{j,m} } \\ \end{array} } \\ \vdots \\ {  (J^{\prime}_{m,m}  J^{\prime}_{m,j} )Y_{m,j} } \\ 0 \\ \vdots \\ 0 \\ \end{array} } \right] \\ \end{aligned}$$
(13)
where J^{*} and J \(\mathrm{^{\prime}}\) are the modified Jacobin matrix and the derivative of the power flow equations with respect to DC line admittances. They are obtained by:
$$\left\{ \begin{gathered} J^{\prime} = \left[ {\begin{array}{*{20}c} {{\raise0.7ex\hbox{${\partial P_{1} }$} \!\mathord{\left/ {\vphantom {{\partial P_{1} } {\partial Y_{1,1} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial Y_{1,1} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{1} }$} \!\mathord{\left/ {\vphantom {{\partial P_{1} } {\partial Y_{1,n} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial Y_{1,n} }$}}} \\ \vdots & \ddots & \vdots \\ {{\raise0.7ex\hbox{${\partial P_{n} }$} \!\mathord{\left/ {\vphantom {{\partial P_{n} } {\partial Y_{n,1} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial Y_{n,1} }$}}} & \ldots & {{\raise0.7ex\hbox{${\partial P_{n} }$} \!\mathord{\left/ {\vphantom {{\partial P_{n} } {\partial Y_{n,n} }}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial Y_{n,n} }$}}} \\ \end{array} } \right] = V \cdot V^{T} \hfill \\ J^{*} = diag(V)Y^{*} + diag(Y^{*} V) \hfill \\ \end{gathered} \right.$$
(14)
where Y^{*} refers to the modified admittance matrix that is affected by DC line disconnection. The active power and DC voltage deviations of VSC stations and the constrained optimization problem are determined respectively by:
$$\left\{ {\begin{array}{*{20}l} \begin{gathered} \Delta V_{i} = \frac{  1}{{\left X \right}}(X_{i,j}^{A} (J^{\prime}_{j,j}  J^{\prime}_{j,m} )Y_{j,m} + X_{i,m}^{A} (J^{\prime}_{m,m}  J^{\prime}_{j,m} )Y_{j,m} ) \hfill \\ \quad = \frac{  1}{{\left X \right}}(X_{j,i}^{C} (J^{\prime}_{j,j}  J^{\prime}_{j,m} )Y_{j,m} + X_{m,i}^{C} (J^{\prime}_{m,m}  J^{\prime}_{j,m} )Y_{j,m} ) \hfill \\ \quad = \frac{{(  1)^{j + i + 1} X_{j,i}^{M} (J^{\prime}_{j,j}  J^{\prime}_{j,m} )}}{{\sum\limits_{i^{\prime} = 1}^{n} {(  1)^{i^{\prime} + 1} X_{i^{\prime},1} X_{i^{\prime},1}^{M} } }}(Y_{j,m} ) \hfill \\ \quad + \frac{{(  1)^{m + i + 1} X_{m,i}^{M} (J^{\prime}_{m,m}  J^{\prime}_{j,m} )}}{{\sum\limits_{i^{\prime} = 1}^{n} {(  1)^{i^{\prime} + 1} X_{i^{\prime},1} X_{i^{\prime},1}^{M} } }}(Y_{j,m} ) \, \quad (i = 1:n;i \ne j) \hfill \\ \end{gathered} \hfill \\ {\Delta P_{i} =  \alpha R_{i} \Delta V\quad \quad (i = 1:m;i \ne j)} \hfill \\ \end{array} } \right.$$
(15)
and
$$\begin{array}{*{20}l} {min\quad \sum\limits_{i = 1}^{m} {\left( {\frac{{R_{i}  R_{i,0} }}{{R_{i,0} }}} \right)^{2} } } \hfill & {} \hfill \\ {subject\,to} \hfill & {} \hfill \\ {\quad \Delta V_{i,min} \le \Delta V_{i} \le \Delta V_{i,max} } \hfill & {(i = 1:n)} \hfill \\ {\quad \Delta P_{i} \le \Delta P_{i.max} } \hfill & {(i = 1:m)} \hfill \\ {\quad R_{i} = R_{i,0} } \hfill & {(i = k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ {\quad 0 \le \alpha R_{i} \le R_{i,max} } \hfill & {(i = 1:m;i \ne k_{1} ,k_{2} , \ldots ,k_{j} )} \hfill \\ \end{array}$$
(16)
If any converter is separated from the MTDC system because of DC line disconnection, its related power and voltage constraints are ignored. The constrained optimization problem (16) can be solved according to the proposed process in Sect. 3.2 to obtain the optimal droop coefficients during DC line disconnection. It should be noted that during sequential disturbances, system equations are modified by combining of (5), (10), and (13) based on disturbances that occur.