**4. Steady State Analysis**

In this section, the steady state analysis of the distributed cooperative control is carried out to prove the consensus of the CL average voltage and battery SOC.

#### *4.1. Consensus of the CL Average Voltage*

By di fferentiating (13),

$$\mathrm{d}\overline{V}\_{\mathrm{Ci}} = \mathrm{d}V\_{\mathrm{Ci}} - \sum\_{j=1}^{n} a\_{\overline{ij}} (\overline{V}\_{\mathrm{Ci}} - \overline{V}\_{\mathrm{C}j}) \tag{20}$$

The global observer dynamic can be formulated as

$$\mathbf{d}\overline{\mathbf{V}}\_{\mathbb{C}} = \mathbf{d}\mathbf{V}\_{\mathbb{C}} - \mathbf{L}\overline{\mathbf{V}}\_{\mathbb{C}} \tag{21}$$

where **L** = **D** − **A**, in which **D** = *diag*{ *n j*=1 *aij*}, **A** = [*aij*] ∈ *Rn*<sup>×</sup>*n*, **V**C = [ *VC*1, *VC*2, ··· , *VCn*] T and

**VC** = [ *VC*1, *VC*2, ··· , *VCn*] T is the CL DC-bus voltage vector and CL average voltage vector, respectively. Since **VC**(0) = **VC**(0), (21) can be written in the frequency domain as

$$\overline{\mathbf{V\_{C}^{\mathbf{s}}}} = s(\mathbf{sI} + \mathbf{L})^{-1} \mathbf{V\_{C}^{\mathbf{s}}} = \mathbf{H} \mathbf{V\_{C}^{\mathbf{s}}} \tag{22}$$

where **VsC**and **VsC**are the form in frequency domain of **VC** and **VC**.

The correction term in (14) can be written in the frequency domain as

$$\mathcal{S}\_1 = \mathbf{G} \mathbf{U} (\mathbf{V}\_{\mathbf{Cref}} - \mathbf{V}\_{\mathbf{C}}^{\mathbf{u}}) \tag{23}$$

where **VCref** is the Laplace transform of the reference voltage vector, and lim *s*→0 *s***VCref** = *VCre f* 1**n** in which **1n** is a column vector with all elements equal to 1 [27]. **GU** = diag{*GUi*}, in which

$$\mathbf{G}\_{Lli} = k\_{PLli} + \frac{k\_{Lli}}{s} \tag{24}$$

combining (8) and (15), then write it in the frequency domain by substituting (22) and (23)

$$\mathbf{V\_{Cref}} = \mathbf{V\_{Cref1}} - \mathbf{I\_C}\mathbf{R\_d} = \mathbf{V\_{Cref}} + \mathbf{G\_U(V\_{Cref} - HV\_C^n)} - \mathbf{I\_C}\mathbf{R\_d} \tag{25}$$

where **Rd** = diag{*Rdi*}is the droop coefficient matrix, and **IC** is the Laplace transform of the CL current vector [*IC*1, *IC*2, ··· , *ICn*]<sup>T</sup>

Thus, the dynamic behavior of the CL DC bus voltage with a closed-loop voltage regulator can be expressed as

$$\mathbf{V\_{C}^{s}} = \mathbf{G\_{C1}} \mathbf{V\_{Cref}^{\*}} \tag{26}$$

where **GC1** = diag{*GC*1*i*} is the transfer function matrix, in which *GC*1*i* is the closed-loop transfer function of the *i*th TPC of DCES, and can be simplified as an inertial element

$$\lim\_{s \to 0} \mathbf{G}\_{\mathbf{C}1} = \mathbf{I}\_{\mathbf{n}} \tag{27}$$

where **In** is a n-order unit matrix.

> The CL current matrix **IC** can be obtained by the admittance matrix **Y** of the DCESs

$$\mathbf{I}\_{\mathbb{C}} = \mathbf{Y} \mathbf{V}\_{\mathbb{C}}^{\mathbf{s}} \tag{28}$$

By substituting (26) and (28) into (25), the dynamic behavior of the CL DC bus voltage can be written as

$$\mathbf{V\_{C}^{s}} = (\mathbf{I\_{n}} + \mathbf{G\_{C1}G\_{U}H} + \mathbf{G\_{C1}R\_{d}Y})^{-1} \times \mathbf{G\_{C1}}(\mathbf{I} + \mathbf{G\_{U}})\mathbf{V\_{Cref}}\tag{29}$$

The steady state of the CL DC bus voltage can be obtained as

$$\mathbf{V\_{C}^{\text{ss}}} = \lim\_{s \to 0} \mathbf{V\_{C}^{\text{g}}} = \lim\_{s \to 0} \left[ s(s\mathbf{I\_{n}} + s\mathbf{G\_{C1}}\mathbf{G\_{U}}\mathbf{H} + s\mathbf{G\_{C1}}\mathbf{R\_{d}}\mathbf{Y})^{-1} \times s\mathbf{G\_{C1}}(\mathbf{I\_{n}} + \mathbf{G\_{U}})\mathbf{V\_{Cref}} \right] \tag{30}$$

Based on the definition of **GU**, it can be obtained as

$$\lim\_{s \to 0} \mathbf{G}\_{\mathbf{U}} = \mathbf{K}\_{\mathbf{I}\mathbf{U}} \tag{31}$$

Therefore,

$$\mathbf{QV\_{C}^{\mathbf{ns}}} = \mathop{\rm lims}\_{\mathbf{c} \to 0} \mathbf{H} \mathbf{V\_{C}^{\mathbf{n}}} = \left(\mathbf{K\_{IU}}\right)^{-1} \times \mathbf{K\_{IU}} V\_{\mathbb{C}ref} \mathbf{1\_{n}} = V\_{\mathbb{C}ref} \mathbf{1\_{n}} \tag{32}$$

where **Q** = lim *s*→0 **H** is the *n* × *n* matrix with all the elements equal to 1/*<sup>n</sup>*. So, the steady state of the CL average voltage can be written as

$$\overline{\mathbf{V\_{C}^{\mathbf{as}}}} = \mathbf{Q} \mathbf{V\_{C}^{\mathbf{as}}} = V\_{cref} \mathbf{1\_{n}} \tag{33}$$

Equation (33) implies that the CL average voltage will all reach consensus in the steady state at *VCref*.

*Energies* **2019**, *12*, 3422

#### *4.2. Consensus of the Battery SOC*

The correction term in (18) can be written in the frequency domain as

$$\mathbf{S}\_1 = \mathbf{G}\_\mathbf{S} \mathbf{L} \mathbf{S} \tag{34}$$

where **S** is the Laplace transform vector of the state variable in (16).

By substituting (34), (19) can be written in the frequency domain

$$\mathbf{V\_{Cref2}} = \mathbf{V\_{Cref}} + \mathbf{G\_SLS} \tag{35}$$

Similar to (26),

$$\mathbf{V\_{C}^{s}} = \mathbf{G\_{C2}} \mathbf{V\_{Cref2}} \tag{36}$$

where **GC2** = diag{*GC*2*i*} is the transfer function matrix, in which *GCi* is the closed-loop transfer function of the *i*th BBC of DCES, and can be simplified as an inertial element.

$$\lim\_{s \to 0} \mathbf{G}\_{\mathbf{C}1} = \mathbf{I}\_{\mathbf{n}} \tag{37}$$

Therefore, **S** can be written as

$$\mathbf{S} = \left(\mathbf{G}\mathbf{c}\mathbf{2}\mathbf{G}\mathbf{s}\mathbf{L}\right)^{-1} \times \left(\mathbf{V}\_{\mathbf{C}}^{\mathfrak{s}} - \mathbf{G}\mathbf{c}2\mathbf{V}\_{\mathbf{C}\mathbf{ref}}\right) \tag{38}$$

Based on the definition of **GS**, it can be obtained as

$$\lim\_{s \to 0} \mathbf{G}\_{\mathbf{S}} = \mathbf{K}\_{\mathbf{IS}} \tag{39}$$

The form of the state variable **S** in the frequency domain is **Ss**, and its steady state can be obtained as

$$\mathbf{S}^{\mathsf{ss}} = \lim\_{s \to 0} \mathbf{S}^{\mathsf{s}} = \lim\_{s \to 0} [s(s \mathbf{G}\_{\mathsf{C2}} \mathbf{G}\_{\mathsf{S}} \mathbf{L})^{-1} \times (s \mathbf{V}\_{\mathsf{C}}^{\mathsf{s}} - s \mathbf{G}\_{\mathsf{C2}} \mathbf{V}\_{\mathsf{Cref}})] \tag{40}$$

Therefore, **QSss** can be written as

$$\begin{array}{lcl} \textbf{Q}\textbf{S}^{\textbf{ss}} &= \underset{s \to 0}{\text{lim}} \textbf{HS} = \underset{s \to 0}{\text{lim}} [s\textbf{H}(s\textbf{G}\_{\textbf{C}2}\textbf{G}\_{\textbf{S}}\textbf{L})^{-1} \times (s\textbf{V}\_{\textbf{C}}^{\textbf{s}} - s\textbf{G}\_{\textbf{C}2}\textbf{V}\_{\textbf{Cref}})] \\ &= \underset{s \to 0}{\text{lim}} [s(\textbf{K}\textbf{s}\textbf{L})^{-1} \times (\textbf{I}\_{\textbf{n}} - \textbf{Q}) \boldsymbol{V}\_{\textbf{Cref}}\textbf{1}\_{\textbf{n}}) \\ &= \textbf{0}\_{\textbf{n}} \end{array} \tag{41}$$

It can be seen that **Sss** is the eigenvector of **Q** associated with the eigenvalue zero, which ensures the state variable *Si* consensus. [27] According to the property of the Laplace matrix [28], the final consensus values can be obtained as

$$\mathbf{S}^{\mathfrak{so}} = \mathbf{S}^{\mathfrak{so}} \mathbf{1}\_{\mathfrak{n}} \tag{42}$$

where *Sss* is the positive real value to which all the state variables *Si* will be converged.

Equations (33) and (42) shows that the average bus voltage and battery SOC will all reach consensus in the steady state.
