System description
Figure 1 shows the understudy DCMG including two DC distributed generation (DCDG) units, a multi-bus DC network, and cluster of local and common loads. Each DCDG unit is modelled by a DC voltage source and a DC–DC Buck–Boost converter (BBC). The point of common coupling (PCC) bus can be connected to the main DC grid through a circuit breaker. Depending on the status of the circuit breaker, the DCMG either can operates in grid-connected mode, or can autonomously operates in islanded mode. The parameters of the DCMG and DCDGs are given in “Appendix” [24].
Modelling of a DCDG unit
Figure 2 shows a BBC-based DCDG unit [25]. Taking into account the parameters in the nominal conditions and neglecting the discontinuous conduction mode, the averaged dynamics model of the BBC can be expressed as:
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{{i}}= & {} {\theta _1}{\delta }-{\theta _2}{(1-\delta )v}-{\theta _3}{i} \end{aligned}$$
(1)
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{{v}}= & {} {\theta _4}{(1-\delta )i}-{\theta _4}{i_{\mathrm{o}}} \end{aligned}$$
(2)
where the state variables i and v are the average current and voltage of the BBC inductor and capacitor, respectively, \(i_{\mathrm{o}}\) is the output current of the BBC that is considered as a measurable external disturbance, \(\delta\) is the duty cycle of BBC switch, and
$$\begin{aligned} \theta _1=\frac{v_d}{L},\quad \theta _2=\frac{1}{L},\quad \theta _3=\frac{R}{L},\quad \theta _4=\frac{1}{C} \end{aligned}$$
(3)
are the BBC parameters in the nominal conditions. If the parameters \(\theta _1-\theta _4\) deviate from their nominal values, the BBC state equations can be modified as
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{{i}}= & {} \,{\theta _1}{\delta }-{\theta _2}{(1-\delta )v}-{\theta _3}{i}+\xi _{\mathrm{i}} \end{aligned}$$
(4)
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{{v}}= & {}\, {\theta _4}{(1-\delta )i}-{\theta _4}{i_{\mathrm{o}}}+\xi _{\mathrm{v}} \end{aligned}$$
(5)
where \(\xi _{\mathrm{i}}\) and \(\xi _{\mathrm{v}}\) are the lumped uncertainties imposed on system dynamics containing parametric uncertainties and system dynamic disturbances.
Control strategy
In the grid-connected mode, all of the DCDG units operate in the current control mode. Therefore, both the DCDG1 and DCDG2 units are equipped with a designed adaptive current controller which regulates their current at pre-specified values. Adopting MS control strategy in the islanded mode, while DCDG2 operates in current controlled mode, as a slave unit, fulfilling the voltage control of the DCMG is deputed to DCDG1 as a master unit. Therefore, an adaptive voltage controller is designed in this paper that mounts on the inner current controller in a cascade structure, to force the voltage of DCMG to track pre-defined trajectory. In order to enhance the dynamic response, reference tracking performances, and robustness of the proposed control scheme, all of the controllers are adaptive designed in such a way that the lumped uncertainties imposed the system dynamics are estimated based on Lyapunov control theory. In the following subsections, the process of designing controllers is presented in details.
Current controller
The basis of both the master and slave controllers in the proposed control strategy is an adaptive current controller. Letting
$$\begin{aligned} e_{\mathrm{i}}=i^{{\mathrm{ref}}}-i \end{aligned}$$
(6)
be the reference current tracking error, and
$$\begin{aligned} \tilde{\xi }_{\mathrm{i}}=\hat{\xi }_{\mathrm{i}}-\xi _{\mathrm{i}} \end{aligned}$$
(7)
be the error between \(\xi _{\mathrm{i}}\) and its estimated value \(\hat{\xi }_{\mathrm{i}}\), then, to generate an adaptive control law, the following positive-definite Lyapunov function is selected
$$\begin{aligned} {{V_{\mathrm{i}}}}=\tfrac{1}{2}{e}_{\mathrm{i}}^{2}+\tfrac{1}{2}{{\gamma }_{\mathrm{i}}}\tilde{\xi }_{\mathrm{i}}^{2} \end{aligned}$$
(8)
where \({\gamma }_{\mathrm{i}}>{0}\) is an estimation gain. The time derivate of \(V_{\mathrm{i}}\) can be expressed as
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{i}}}={e}_{\mathrm{i}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{e}_{\mathrm{i}}+{{\gamma }_{\mathrm{i}}}\tilde{\xi }_{\mathrm{i}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{\xi }_{\mathrm{i}} \end{aligned}$$
(9)
Using (6), (7), and (4), it can be obtain that
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{i}}}={e}_{\mathrm{i}}\left( \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{i}^{\mathrm{ref}}-{\theta _1}{\delta }+{\theta _2}{(1-\delta )v}+{\theta _3}{i}-\xi _{\mathrm{i}}\right) +{{\gamma }_{\mathrm{i}}}\tilde{\xi }_{\mathrm{i}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\hat{\xi }_{\mathrm{i}} \end{aligned}$$
(10)
By cancelling the second term in right side of (10), the estimation law can be determined as [4]:
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\hat{\xi }_{\mathrm{i}}=-{{\gamma }_{\mathrm{i}}^{-1}}e_{\mathrm{i}} \end{aligned}$$
(11)
Considering (11), \(\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{i}}}\) can be given by
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{i}}}={e}_{\mathrm{i}}\left( \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{i}^{\mathrm{ref}}-{\theta _1}{\delta }+{\theta _2}{(1-\delta )v}+{\theta _3}{i}-\hat{\xi _{\mathrm{i}}}\right) \end{aligned}$$
(12)
If the control input \(\delta\) is selected as
$$\begin{aligned} \delta =\left( \theta _1+\theta _2{v}\right) ^{-1}\left( \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{i}^{\mathrm{ref}}+{\theta _3}{i}-\hat{\xi _{\mathrm{i}}}+\theta _2+{k_{\mathrm{i}}}{e_{\mathrm{i}}}\right) \end{aligned}$$
(13)
where \({k_{\mathrm{i}}}>{0}\) is a control gain, then, it can be obtain that
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{i}}}=-{k_{\mathrm{i}}}{e}^2_{\mathrm{i}}<{0} \end{aligned}$$
(14)
that is a negative-definite function and based on Barballat’s Lemma [26], it can be conclude that the current controller system is asymptotically stable.
Voltage controller
The main control objective for master unit is to force the voltage of DCMG to remain at its nominal value. Letting
$$\begin{aligned} e_{\mathrm{v}}=v^{\mathrm{ref}}-v \end{aligned}$$
(15)
be the reference voltage tracking error, and
$$\begin{aligned} \tilde{\xi }_{\mathrm{v}}=\hat{\xi }_{\mathrm{v}}-\xi _{\mathrm{v}} \end{aligned}$$
(16)
be the estimation error, where \(\xi _{\mathrm{v}}\) is estimated value of \(\hat{\xi }_{\mathrm{v}}\), a positive-definite Lyapunov function can be chosen as:
$$\begin{aligned} {{V_{\mathrm{v}}}}=\tfrac{1}{2}{e}_{\mathrm{v}}^{2}+\tfrac{1}{2}{{\gamma }_{\mathrm{v}}}\tilde{\xi }_{\mathrm{v}}^{2} \end{aligned}$$
(17)
where \({\gamma }_{\mathrm{v}}>{0}\) is an estimation gain. Differentiating \(V_{\mathrm{v}}\) with respect to time gives
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{v}}}={e}_{\mathrm{v}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{e}_{\mathrm{v}}+{{\gamma }_{\mathrm{v}}}\tilde{\xi }_{\mathrm{v}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{\xi }_{\mathrm{v}} \end{aligned}$$
(18)
Considering (15), (16) and (5), it can be obtained that
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{v}}}={e}_{\mathrm{v}}\left( \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{v}^{\mathrm{ref}}-{\theta _4}{(1-\delta )i}+{\theta _4}{i_{\mathrm{o}}}-\xi _{\mathrm{v}}\right) +{{\gamma }_{\mathrm{v}}}\tilde{\xi }_{\mathrm{v}}\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\hat{\xi }_{\mathrm{v}} \end{aligned}$$
(19)
If the adaption law is chosen as [4]:
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}\hat{\xi }_{\mathrm{v}}=-{{\gamma }_{\mathrm{v}}^{-1}}e_{\mathrm{v}} \end{aligned}$$
(20)
\(\tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{v}}}\) can be reduced to
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{v}}}={e}_{\mathrm{v}}\left( \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{v}^{\mathrm{ref}}-{\theta _4}{(1-\delta )i}+{\theta _4}{i_{\mathrm{o}}}-\hat{\xi _{\mathrm{v}}}\right) \end{aligned}$$
(21)
With the following control law
$$\begin{aligned} i^{\mathrm{ref}}=(1-\delta )^{-1}\left( i_{\mathrm{o}}+\theta _4^{-1}\left\{ \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{v}^{\mathrm{ref}}-\hat{\xi _{\mathrm{v}}}+{k_{\mathrm{v}}}{e_{\mathrm{v}}}\right\} \right) \end{aligned}$$
(22)
where \({k_{\mathrm{v}}}>{0}\) is a control gain, it can be obtain that
$$\begin{aligned} \tfrac{{\mathrm{d}}}{{\mathrm{d}}t}{V_{\mathrm{v}}}=-{k_{\mathrm{v}}}{e}^2_{\mathrm{v}}<{0} \end{aligned}$$
(23)
Based on Barballat’s Lemma [26], it can be conclude that \(e_{\mathrm{v}}{\rightarrow }0\) asymptotically. While the master unit current controller follows a pre-specified reference current in the grid-connected mode, the current reference in the islanded mode of operation is determined by (22) to force the DCMG voltage to track a pre-defined trajectory. The block diagram of the proposed current and voltage controllers is shown in Fig. 3.