Model of MMCHVDC
A threephase MMC topology is shown in Fig. 1 [6–9]. According to KCL, the threephase current can be expressed as
$$ \left\{\begin{array}{c}\hfill {i}_a={i}_{pa}+{i}_{na}\hfill \\ {}\hfill {i}_b={i}_{pb}+{i}_{nb}\hfill \\ {}\hfill {i}_c={i}_{pc}+{i}_{nc}\hfill \end{array}\right. $$
(1)
For three singlephase units, applying KVL to the upper and lower arms yields:
$$ \left\{\begin{array}{c}\hfill {u}_a\left(\frac{U_{dc}}{2}{u}_{pa}\right)=2L\frac{d{i}_{pa}}{dt}+2R{i}_{pa}\hfill \\ {}\hfill {u}_b\left(\frac{U_{dc}}{2}{u}_{pb}\right)=2L\frac{d{i}_{pb}}{dt}+2R{i}_{pb}\hfill \\ {}\hfill {u}_c\left(\frac{U_{dc}}{2}{u}_{pc}\right)=2L\frac{d{i}_{pc}}{dt}+2R{i}_{pc}\hfill \end{array}\right. $$
(2)
$$ \left\{\begin{array}{c}\hfill {u}_a\left({u}_{na}\frac{U_{dc}}{2}\right)=2L\frac{d{i}_{na}}{dt}+2R{i}_{na}\hfill \\ {}\hfill {u}_b\left({u}_{nb}\frac{U_{dc}}{2}\right)=2L\frac{d{i}_{nb}}{dt}+2R{i}_{nb}\hfill \\ {}\hfill {u}_c\left({u}_{nc}\frac{U_{dc}}{2}\right)=2L\frac{d{i}_{nc}}{dt}+2R{i}_{nc}\hfill \end{array}\right. $$
(3)
In the above equations, 2L and 2R denote the equivalent arm inductance and resistance, respectively. Adding Eqs. (2) and (3) leads
$$ \left\{\begin{array}{c}\hfill {u}_a\left({u}_{na}{u}_{pa}\right)/2=L\frac{d{i}_a}{dt}+R{i}_a\hfill \\ {}\hfill {u}_a\left({u}_{nb}{u}_{pb}\right)/2=L\frac{d{i}_b}{dt}+R{i}_b\hfill \\ {}\hfill {u}_c\left({u}_{nc}{u}_{pc}\right)/2=L\frac{d{i}_c}{dt}+R{i}_c\hfill \end{array}\right. $$
(4)
According to Eq. (4), the time domain mathematic model of a MMC in abc coordinate is given by [10, 11].
$$ \left\{\begin{array}{c}\hfill \frac{d{i}_a(t)}{dt}=\frac{R}{L}{i}_a(t)+\frac{1}{L}\left[{u}_a(t)\left({u}_{na}(t){u}_{pa}(t)\right)/2\right]\hfill \\ {}\hfill \frac{d{i}_b(t)}{dt}=\frac{R}{L}{i}_b(t)+\frac{1}{L}\left[{u}_b(t)\left({u}_{nb}(t){u}_{pb}(t)\right)/2\right]\hfill \\ {}\hfill \frac{d{i}_c(t)}{dt}=\frac{R}{L}{i}_c(t)+\frac{1}{L}\left[{u}_c(t)\left({u}_{nc}(t){u}_{pc}(t)\right)/2\right]\hfill \end{array}\right. $$
(5)
Written Eq. (5) in phasor form yields
$$ \begin{array}{l}\frac{d}{dt}\left[\begin{array}{l}{i}_a\\ {}{i}_b\\ {}{i}_c\end{array}\right]=\left[\begin{array}{l}\frac{R}{L}\kern2em 0\kern2.33em 0\\ {}\kern.7em 0\kern1.5em \frac{R}{L}\kern2.12em 0\\ {}\kern.7em 0\kern2.3em 0\kern1.62em \frac{R}{L}\end{array}\right]\left[\begin{array}{l}{i}_a\\ {}{i}_b\\ {}{i}_c\end{array}\right]\\ {}+\left[\begin{array}{l}\frac{1}{2L}\kern1em \frac{1}{2L}\kern1.8em 0\kern2.2em 0\kern2.5em 0\kern2.1em 0\\ {}\kern1em 0\kern1.9em 0\kern1.12em \frac{1}{2L}\kern1.36em \frac{1}{2L}\kern2.12em 0\kern2.1em 0\\ {}\kern1em 0\kern1.9em 0\kern2.2em 0\kern2.2em 0\kern1.5em \frac{1}{2L}\kern1.24em \frac{1}{2L}\end{array}\right]\left[\begin{array}{l}{u}_{na}\\ {}{u}_{pa}\\ {}{u}_{nb}\\ {}{u}_{pb}\\ {}{u}_{nc}\\ {}{u}_{pc}\end{array}\right]\\ {}\kern4em +\left[\begin{array}{l}\frac{1}{L}\kern1em 0\kern1.1em 0\\ {}0\kern1em \frac{1}{L}\kern1em 0\\ {}0\kern1.2em 0\kern1em \frac{1}{L}\end{array}\right]\left[\begin{array}{l}{u}_a\\ {}{u}_b\\ {}{u}_c\end{array}\right]\end{array} $$
(6)
Considering the uncertainty, external disturbances and system faults, the Eq. (6) can be written as follows
$$ \left\{\begin{array}{l}\overset{.}{\mathbf{x}}(t)=\mathbf{Ax}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{h}(t)+\boldsymbol{\Delta} (t)+{\mathbf{B}}_f\mathbf{f}(t)\\ {}\mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{M}(t)\end{array}\right. $$
(7)
where \( \mathbf{x}(t)=\left[\begin{array}{c}\hfill {i}_a\hfill \\ {}\hfill {i}_b\hfill \\ {}\hfill {i}_c\hfill \end{array}\right] \), \( \mathbf{u}(t)=\left[\begin{array}{c}\hfill {u}_{na}\hfill \\ {}\hfill {u}_{pa}\hfill \\ {}\hfill {u}_{nb}\hfill \\ {}\hfill {u}_{pb}\hfill \\ {}\hfill {u}_{nc}\hfill \\ {}\hfill {u}_{pc}\hfill \end{array}\right] \), \( \mathbf{A}=\left[\begin{array}{ccc}\hfill \frac{R}{L}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{R}{L}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{R}{L}\hfill \end{array}\right] \),\( \mathbf{B}=\left[\begin{array}{cccccc}\hfill \frac{1}{2L}\hfill & \hfill \frac{1}{2L}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2L}\hfill & \hfill \frac{1}{2L}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2L}\hfill & \hfill \frac{1}{2L}\hfill \end{array}\right] \), \( \mathbf{h}(t)=\left[\begin{array}{ccc}\hfill \frac{1}{L}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{1}{L}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{L}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {u}_a\hfill \\ {}\hfill {u}_b\hfill \\ {}\hfill {u}_c\hfill \end{array}\right] \)
_{.}
In Eq. (7), Δ(t) represents uncertainty and external disturbances generated during the actual modeling process. f(t) represents the system faults which are required to detect and identify. M(t) is measurement noise introduced by the measurement system. B
_{
f
}, C and D are known matrices with appropriate dimensions. y(t) is the system output. If Ω(t) is used to represent Δ(t) + h(t), Eq. (7) can be written as follows.
$$ \left\{\begin{array}{l}\overset{.}{\mathbf{x}}(t)=\mathbf{Ax}(t)+\mathbf{B}\mathbf{u}(t)+\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\\ {}\mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{M}(t)\end{array}\right. $$
(8)
For further analysis, the following assumptions are made:

1)
Uncertainties in the system are normbounded, that is ‖Δ(t)‖ ≤ V
_{Δ}.

2)
System matrix pair (A, C) is observable.

3)
Uncertainties in the system meet the condition ‖Ω(t)‖ ≤ γ‖x(t)‖ ≤ V
_{Ω}.

4)
The M(t) in the system is normbounded, that is ‖M(t)‖ ≤ V
_{
M
}.

5)
The initial state of the system is zero.
Design of full dimension state observer
Fault diagnosis is carried out by full dimension state observer based on the state equation of flexible HVDC system [12–14]. It can be seen from Eq. (8) that Ω(t) contains the system uncertainties and external disturbance, and the grid voltage u_{a}, u_{b}, u_{c}, which can be regarded as unknown nonlinear perturbation term.
A state observer is established according to Eq. (8):
$$ \left\{\begin{array}{l}\overset{.}{\widehat{\mathbf{x}}}(t)=\mathbf{A}\widehat{\mathbf{x}}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{H}\left(\mathbf{y}(t)\widehat{\mathbf{y}}(t)\right)\\ {}\widehat{\mathbf{y}}(t)=\mathbf{C}\widehat{\mathbf{x}}(t)\end{array}\right. $$
(9)
In Eq. (9), H is a gain matrix to be designed. According to Eqs. (8) and (9), system residual state equation is given by:
$$ \left\{\begin{array}{l}\overset{.}{\mathbf{x}}(t)\overset{.}{\widehat{\mathbf{x}}}(t)=\left(\mathbf{A}\mathbf{H} \mathbf{C}\right)\left(\mathbf{x}(t)\widehat{\mathbf{x}}(t)\right)+\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\mathbf{H}\mathbf{D}\mathbf{M}(t)\\ {}\mathbf{y}(t)\widehat{\mathbf{y}}(t)=\mathbf{C}\left(\mathbf{x}(t)\widehat{\mathbf{x}}(t)\right)+\mathbf{D}\mathbf{M}(t)\end{array}\right. $$
(10)
If \( \mathbf{e}(t)=\mathbf{x}(t)\widehat{\mathbf{x}}(t) \) and r(t) = y(t) − ŷ(t), Eq. (10) can be written as follows:
$$ \left\{\begin{array}{l}\overset{.}{\mathbf{e}}(t)=\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)\mathbf{e}(t)+\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\mathbf{H}\mathbf{D}\mathbf{M}(t)\\ {}\mathbf{r}(t)=\mathbf{C}\mathbf{e}(t)+\mathbf{D}\mathbf{M}(t)\end{array}\right. $$
(11)
In the design of the full dimension state observer, the impact of the fault and external disturbance or uncertainties on the residual should be considered. The transfer function from system uncertainties and external disturbance to residual r(t) is presented by T
_{
rd
} whereas T
_{
rf
} presents the transfer function from system fault to residual r(t). If the value of ‖T
_{
rd
}‖ is small enough and the value of ‖T
_{
rf
}‖ is big enough in the design, the designed observer will be robust. Therefore, the following performance index is used to design the fault diagnosis observer:
$$ J=\frac{\left\Vert {\mathbf{T}}_{rf}\right\Vert }{\left\Vert {\mathbf{T}}_{rd}\right\Vert } $$
(12)
In practical applications, the above performance index can be equivalent to the following equation:
$$ {\left\Vert r(t)\right\Vert}_{\propto}\le \beta {\left\Vert d(t)\right\Vert}_{\propto },{\left\Vert r(t)\right\Vert}_{}\ge \eta {\left\Vert f(t)\right\Vert}_{} $$
(13)
Equation (13) can be solved by using multiobjective optimization methods and thus the robust control theory will be used to solve the performance optimization problem proposed by Eq. (13).
Theorem 1
The system represented by Eq. (8) satisfies the assumptions (1)(5). Optimization index β is given and satisfies the condition D
^{T}
D − β
^{2}
I < 0. In case of no fault in the system, if there are a positive definite symmetric matrix R and a constant factor ε
_{1} > 0 meeting the following linear matrix inequality:
$$ \left[\begin{array}{ccc}\hfill \mathbf{R}\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)+{\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)}^T\mathbf{R}+{\varepsilon}_1^{1}{\mathbf{R}}^T\mathbf{R}+{\mathbf{C}}^T\mathbf{C}\hfill & \hfill 0\hfill & \hfill {\mathbf{C}}^T\mathbf{D}\mathbf{R}\mathbf{H}\mathbf{D}\hfill \\ {}\hfill 0\hfill & \hfill {\varepsilon}_1{\gamma}^2\mathbf{I}\hfill & \hfill 0\hfill \\ {}\hfill {\mathbf{D}}^T\mathbf{C}{\left(\mathbf{R}\mathbf{H}\mathbf{D}\right)}^T\hfill & \hfill 0\hfill & \hfill {\mathbf{D}}^T\mathbf{D}{\beta}^2\mathbf{I}\hfill \end{array}\right] $$
(14)
the residual system (11) is asymptotically stable and meets ‖r(t)‖_{∝} ≤ β‖d(t)‖_{∝}.
Theorem 2
The system represented by Eq. (8) satisfies the assumptions (1)(5). Optimization index η is given and satisfies the condition D
^{T}
D − η
^{2}
I < 0. In case of no fault in the system, if there are a positive definite symmetric matrix T and a constant factor η
_{1} > 0 meeting the following linear matrix inequality:
$$ \left[\begin{array}{ccc}\hfill \mathbf{R}\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)+{\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)}^T\mathbf{R}+{\eta}_1^{1}{\mathbf{R}}^T\mathbf{R}+{\mathbf{C}}^T\mathbf{C}\hfill & \hfill 0\hfill & \hfill {\mathbf{C}}^T\mathbf{D}+\mathbf{R}\mathbf{H}\mathbf{D}\hfill \\ {}\hfill 0\hfill & \hfill {\eta}_1{\gamma}^2\mathbf{I}\hfill & \hfill 0\hfill \\ {}\hfill {\mathbf{D}}^T\mathbf{C}+{\left(\mathbf{R}\mathbf{H}\mathbf{D}\right)}^T\hfill & \hfill 0\hfill & \hfill {\eta}^2\mathbf{I}{\mathbf{D}}^T\mathbf{D}\hfill \end{array}\right] $$
(15)
The residual system (11) is asymptotically stable and meets ‖r(t)‖_{−} ≥ η‖f(t)‖_{−}.
Theorem 3
The system represented by Eq. (8) satisfies the assumptions (1)(5). Optimization index β > 0, η > 0 are given. If there are positive definite symmetric matrix R, T,H, constant factor ε
_{1} < 0 and η
_{1} > 0 meeting the following linear matrix inequality:
$$ \left[\begin{array}{ccc}\hfill \mathbf{R}\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)+{\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)}^T\mathbf{R}+{\varepsilon}_1^{1}{\mathbf{R}}^T\mathbf{R}+{\mathbf{C}}^T\mathbf{C}\hfill & \hfill 0\hfill & \hfill {\mathbf{C}}^T\mathbf{D}\mathbf{R}\mathbf{H}\mathbf{D}\hfill \\ {}\hfill 0\hfill & \hfill {\varepsilon}_1{\gamma}^2\mathbf{I}\hfill & \hfill 0\hfill \\ {}\hfill {\mathbf{D}}^T\mathbf{C}{\left(\mathbf{R}\mathbf{H}\mathbf{D}\right)}^T\hfill & \hfill 0\hfill & \hfill {\mathbf{D}}^T\mathbf{D}{\beta}^2\mathbf{I}\hfill \end{array}\right] $$
(16)
$$ \left[\begin{array}{ccc}\hfill \mathbf{R}\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)+{\left(\mathbf{A}\mathbf{H}\mathbf{C}\right)}^T\mathbf{R}+{\eta}_1^{1}{\mathbf{R}}^T\mathbf{R}+{\mathbf{C}}^T\mathbf{C}\hfill & \hfill 0\hfill & \hfill {\mathbf{C}}^T\mathbf{D}+\mathbf{R}\mathbf{H}\mathbf{D}\hfill \\ {}\hfill 0\hfill & \hfill {\eta}_1{\gamma}^2\mathbf{I}\hfill & \hfill 0\hfill \\ {}\hfill {\mathbf{D}}^T\mathbf{C}+{\left(\mathbf{R}\mathbf{H}\mathbf{D}\right)}^T\hfill & \hfill 0\hfill & \hfill {\eta}^2\mathbf{I}{\mathbf{D}}^T\mathbf{D}\hfill \end{array}\right] $$
(17)
The residual system (11) is asymptotically stable and meets ‖r(t)‖_{∝} ≤ β‖d(t)‖_{∝} and ‖r(t)‖_{−} ≥ η‖f(t)‖_{−}.
The gain matrix H of the full dimension state observer can be obtained from Theorem 3. The design process of gain matrix H takes into account the impact of system disturbances and uncertainties on the residual, and the sensitivity of the residual for faults.
Determination of fault detection threshold
When disturbances or uncertainties exist in the system, a fault detection threshold method is usually used to determine whether the system has a fault [15–17]. This section provides fault detection thresholds calculated according to the definition of norm.
Theorem 4
The system represented by Eq. (8) satisfies the assumptions (1)(5). If the residual system (11) meets ‖r(t)‖ > (a
V
_{Ω} + b
V
_{
M
})(t − t
_{0}) + c
V
_{
M
}, then system fault is detected,where \( a=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\right\Vert \), \( b=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\mathbf{H}\mathbf{D}\right\Vert \), \( c=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{D}\right\Vert \).
Proof
According to the residual state Eq. (11) the following two equations can be derived:
$$ \mathbf{e}(t)=\boldsymbol{\Psi} \left(t,{t}_0\right)\mathbf{e}\left({t}_0\right)+{\displaystyle {\int}_{t_0}^t\boldsymbol{\Psi} \left(t,s\right)\left[\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\mathbf{H}\mathbf{D}\mathbf{M}(t)\right]ds} $$
(18)
$$ \mathbf{r}(t)=\mathbf{C}\left\{\boldsymbol{\Psi} \left(t,{t}_0\right)\mathbf{e}\left({t}_0\right)+{\displaystyle {\int}_{t_0}^t\boldsymbol{\Psi} \left(t,s\right)\left[\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\mathbf{H}\mathbf{D}\mathbf{M}(t)\right]ds}\right\}+\mathbf{D}\mathbf{M}(t) $$
(19)
According to assumption (5), e(t
_{0}) = 0. Thus
$$ \mathbf{r}(t)={\displaystyle {\int}_{t_0}^t\mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\left[\boldsymbol{\Omega} (t)+{\mathbf{B}}_f\mathbf{f}(t)\mathbf{H}\mathbf{D}\mathbf{M}(t)\right]ds}+\mathbf{D}\mathbf{M}(t) $$
(20)
Equation (21) can be obtained by seeking norm in both sides as
$$ \begin{array}{l}\left\mathbf{r}(t)\right\le {\displaystyle {\int}_{t_0}^t\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\boldsymbol{\Omega} (t)\right\Vert ds}+{\displaystyle {\int}_{t_0}^t\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right){\mathbf{B}}_f\mathbf{f}(t)\right\Vert ds}\\ {}\kern2em +{\displaystyle {\int}_{t_0}^t\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\mathbf{H}\mathbf{D}\mathbf{M}(t)\right\Vert ds}+\left\Vert \mathbf{D}\mathbf{M}(t)\right\Vert \end{array} $$
(21)
where \( a=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\right\Vert \), \( b=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{C}\boldsymbol{\Psi } \left(t,s\right)\mathbf{H}\mathbf{D}\right\Vert \), \( c=\underset{t\in \left[{t}_0,t\right]}{ \sup}\left\Vert \mathbf{D}\right\Vert \)
According to assumptions (1)(4), Eq. (22) is obtained as
$$ \left\Vert \mathbf{r}(t)\right\Vert \le {\displaystyle {\int}_{t_0}^ta\left\Vert \boldsymbol{\Omega} (t)\right\Vert }ds+{\displaystyle {\int}_{t_0}^tb\left\Vert \mathbf{M}(t)\right\Vert }ds+\left\Vert \mathbf{D}\mathbf{M}(t)\right\Vert $$
(22)
Equation (22) can be simplified as
$$ \begin{array}{l}\left\Vert \mathbf{r}(t)\right\Vert \le {\displaystyle {\int}_{t_0}^ta{\mathbf{V}}_{\Omega}}ds+{\displaystyle {\int}_{t_0}^tb{\mathbf{V}}_M}ds+c{\mathbf{V}}_M\\ {}\kern2em =a{\mathbf{V}}_{\Omega}\left(t{t}_0\right)+b{\mathbf{V}}_M\left(t{t}_0\right)+c{\mathbf{V}}_M\\ {}\kern2em =\left(a{\mathbf{V}}_{\Omega}+b{\mathbf{V}}_M\right)\left(t{t}_0\right)+c{\mathbf{V}}_M\end{array} $$
(23)
Proof is done.