Voltage source converter based DC grid study

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DOI: 10.1049/gtd2.12262

O R I G I NA L R E S E A RC H PA P E R

Optimal sparse control of interval networks: Voltage source converter based DC grid study

Behrouz Nabi Saeed Seyedtabaii

Electrical Engineering Deptartment, Shahed University, Tehran 3319118651, Iran

Correspondence

S. Seyedtabaii, Elec. Eng. Dept., Shahed Univ., Tehran 3319118651, Iran.

Email:[email protected]

Abstract

Decentralized optimal control of interval state space systems involves significant compli- cations due to the required sparsity, robustness, and convergence rate. By adding an l1 term to the standard linear quadratic optimization, the obtained composite index, accommodating the sparsity, contains both differentiable and non-differentiable terms. Knowing that the choice of optimization techniques are problem-dependent, in this paper, several smart provisions such as “Active set”, “Coordinate descent”, and separately managing the differentiable part of the cost function are put together to form a relatively fast algorithm.

In addition, a rule is introduced to adjust automatically the degree of sparsity and a constraint to confine the interval closed-loop system eigenvalues to the desired s plane region.

As a result, a robust sparse decentralized control (RSDC) approach for interval networks is established. The RSDC method is utilized for the control of a multi-terminal VSC (Voltage Source Converter ) -based DC (Direct Current) grid with parametric uncertainty under various external random perturbations. The test results show that the proposed method with just 20 feedback links (out of available 100) performs superiorly to the alternative ones in terms of sparsity, convergence rate, and time-domain performances, verified through extensive simulations.

1 INTRODUCTION

One of the significant advantages of Voltage Source Converter for High Voltage Direct Current (VSC-HVDC) networks is the potential in implementing radial, circular, or meshed topology [1]. Besides, it facilitates the combination of distributed gener- ation units (DG’s) with large energy consumption centres [2].

It can also provide several ancillary services to the AC grids owing to its rapid and flexible power control [3]. Inversely, such interconnections among tightly controlled active devices (VSCs) will give rise to harmful interactions and possible instability [2].

Another prime topic in this respect is uncertainty that has to be effectively addressed; otherwise, secure operation of the network may be lost.

DC grid voltage may be controlled using local measurements to restore power balance [4]. However, as the network topology becomes too complex, more effective strategies are required. The master-slave, DC voltage droop, amplitude-phase,

This is an open access article under the terms of theCreative Commons AttributionLicense, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

and multi-point voltage controls are developed to administer the challenges [2].

A centralized Quasi-Luenberger observer-based control scheme for non-linear DC linked hybrid microgrids (MG) has been explored in [5]. In [6], distributed control of the network based on a consensus algorithm has been detailed. A distributed cooperative control design for a multi-terminal DC (MTDC) power system employing the consensus protocol is applied in [3]. Decentralized voltage and power control of Hybrid HVDC has been explored in [7].

Regarding robust design, a comparison of fractional order versus H_∞ andμ synthesis has been discussed in [8, 9]. The design of fractional controllers for uncertain systems has been studied in [10].Η2 decentralized controller for DC-segmented network has been investigated in [11]. A heuristic method based on regularized H₂ optimal control to promote sparsity for the decentralized controller has been studied in [12]. In [1], a robust H_∞ decentralized supplementary control for VSC-MTDC

IET Gener. Transm. Distrib.2021;1–11. wileyonlinelibrary.com/iet-gtd 1

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networks based on impedance minimization technique has been elaborated. A robust sparse design has been discussed in [13], where the H2technique for an interconnected system with poly- topic uncertainties is simulated. The application of sliding mode in control of uncertain systems has been analysed in [14, 15].

The controllers provided by the robust feedback synthesis are usually not the optimal ones. Balancing between robustness and optimality is always a complicated mathematical task. Linear Matrix Inequality (LMI) is one solution to the problem. In [16], a robust state feedback centralized controller using LMI has been developed. Implementation of the non-linear reset control in an island MG including multiple renewable energy sources and DGs with parametric uncertainty has been discussed in [17].

Moreover, during the last decades and after the publica- tion of the celebrated Kharitonov’s theorem [18], the robust control design for parametric uncertain systems has gained tremendous attention. Robust transient droop function based on Kharitonov’s stability theorem is proposed in [19] to attenu- ate the system low-frequency oscillations originated from con- ventional droop control. Robust control of DC MGs based on LMI optimization using linear programming and applying Kharitonov’s theorem to mitigate the destabilizing effect of power loads has been addressed in [18, 20]. However, the technique is not well adapted and easily implemented in multivari- able systems.

The parametric uncertainties are easily captured by interval models. Thus, some approaches for reformulating the interval Lyapunov and Riccati equations as a convex optimization problem with LMI constraints have been suggested. Some aim at restricting the closed-loop characteristic equation coefficients to a predefined stable interval polynomial and others to enclose their eigenvalues to a specific region of the s plane and enforce the relative stability. Modelling uncertainty by interval state- space allows using the traditional optimal synthesis such as the Linear Quadratic (LQ) control design. In this respect, a robust state-feedback controller synthesis for an interval state-space has been investigated in [21].

This paper provides a solution for the design of a decentralized controller for interval networks. Sparsity is enforced by introducing an l₁ term to the objective function and robustness is ensured by a constraint that limits the closed-loop eigenvalues to the desired sub-region in the s plane. By incorporat- ing the “active set” idea and separately managing the differentiable part of the composite cost function a relatively fast iterative non-linear optimization scheme is developed. The proposed robust sparse decentralized controller (RSDC) is tested over a 5-bus DC network. The network is modelled in an interval state-space form, and the required conditions for the desired performance (locating poles in any sub-region of the s plane) are derived. The RSDC performance is compared with the outcome of some contestant algorithms. The simulations reveal that the algorithm effectively secures the transient system performance despite uncertainty far ahead of the others.

The rest of this paper is organized as follows. Section 2 discusses the RSDC design based on the eigenvalue clustering considering system uncertainties. Simulation results of the

application of RSDC to a five terminal test system are presented in Section3, and lastly, the conclusion comes in Section 4.

2 THE OPTIMAL DECENTRALIZED CONTROL OF INTERVAL SYSTEMS

Consider the interval system

̇

x =A_iox+B_iou+w, y=Cx

A_io∶=[A

o, ̄A_o],B_io∶=[B

o, ̄B_o]

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whereuis the control,A_io∈ℝ^n×nandB_io∈ℝ^n×mare the system interval parameters,wis zero-mean white process noise, andC defining the measured states subsequently reflects the sparsity pattern.

2.1 Optimality

The system in Equation (1) is controlled by the following state feedback stabilizer,

u(t)=K Cx=Kx (2) whereKis the control gain. Design of optimalKis conducted by minimizing the following infinite horizon LQR cost function,

J(K)= lim

t→∞

1 T [∫

T

0

(

x(t)^TQx(t)+u(t)^TRu(t) )

]dt (3) whereQ∈ℝⁿ^×ⁿandR∈ℝ^m^×^mare the LQR parameters balancing the disturbance rejection and the settling speed. The initial centralized full state control gainK0stabilizing the nominal system is calculated using the continuous-time algebraic Riccati equation (CARE) [22],

A_o^TP+PA_o+Q−PBR⁻¹B^TP =0

K₀= −R⁻¹B^TP (4)

2.2 Robustness constraints

The controller is designed for the nominal system, but constraints have to be applied forKto preserve robustness in terms of settling time and overshoot. In this respect, the desiredΩd

region of the s plane is defined as shown in Figure1.

The right vertical border side isζωnline whereζis the damping ratio, andωnis the natural frequency. The settling time and overshoot are the functions of these two parameters [23]:

𝜏s = 4 𝜍wn

, P.O=100 e^−𝜍𝜋

√1− 𝜍² (5)

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F I G U R E 1 The desiredΩdregion

Consider the following closed loop interval state-space system,

̇

x=A_i(K)x, {

A_i∈Rⁿ^×ⁿ;A(K)≤A_i≤A(K̄ )} (6) whereA(K)≤(K) element-wise. From here, argumentKis omit- ted for brevity. ForAi, the midpointAand the radiusA_△are introduced as below,

A=1∕2(Ā +A),A_Δ =1∕2(Ā −A) (7) The eigenvalue setΛ(Ai)are all eigenvalues overA∈Ai.

The desired poles region, shown in Figure1, is expressed by, Ωd ={

(x,y)||𝛾00+ 𝛾10x+ 𝛾01y<0}

(8) where the vertical, horizontal, or inclined borderlines can be adjusted. The sufficient conditions for the eigenvalues of the nominal systemAto remain inΩd(γ00;γ10;γ01) sub-region are that for any positive definite Hermitian q∈H₊ⁿ there exists a unique positive definite Hermitianp∈H₊ⁿsatisfying the following equation [24],

Δ =c₀₀p+c₁₀A^Tp+c₀₁pA+q=0 c00= 𝛾00, c10=1

2(𝛾10+i𝛾01), c01= 1

2(𝛾10−i𝛾01) (9) To extend the condition to the interval system, A matrix in Equation (9) is replaced byA_ias follows,

Δi= Δ +c10(Ai−A)^Tp+c01p(Ai−A)

< Δ +c10A^T_Δp+c01pA_Δ<0

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Now, the interval system poles are in the desired regionΩdas long asΔiremains at least negative semidefinite.

Since eigenvalues are invariant to similarity transformation, an alternative approach for a special caseΩd(γ00; 1; 0) (poles on the left of a vertical line) is expressed by [25],

Re(𝜆i)+∑n j=1

r_j ri

f_{i j}< −𝛾00

F =Jo+|T⁻¹R⁻¹|A_Δ|RT|=[fi j]

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whereTis the Jordan andRis an optional diagonal similarity transformation matrix,

T⁻¹R⁻¹ART =J =J_d+J_o

R=diag(r_i), i =1, … ,n, r_i>0 (12) In Equation (12),J_d is the diagonal andJ_o is the off-diagonal parts ofJ.

2.3 Sparsity constraints

To accommodate the sparsity in the design, the cost function in Equation (3) is augmented by aKrelatedg(K) as below,

MinK

(J(K)+g(K))

(13) g manages the trade-off between the sparsity and the system performance. The sparsity penalty function applied here is the weightedl₁-norm expressed by [26],

g(K)=‖Λ◦K‖₁=∑

i j

𝜆i j|||K_{i j}||| (14) whereΛ(λ’s) could be set independently.

To automatically (instead of manually) assign the optimal degree of sparsity; at each iteration,λijare set inversely propor- tional to the absolute value of the previousKijas below,

𝜆m,i j =1∕(|Ki j,m−1|+ 𝜀) (15) where 0<ϵ<1 guards againstK_ij=0 [12]. LowK_ijvalues cause highλij,leading to lowerK_ijwhich eventually may cross out its counterpart link.

2.4 Iterative design

Now, the problem of robust sparse design is to find a robust gainKfor the closed-loop system to satisfy,

eig(Aⁱ(K))⊆ Ωd (16)

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Since Equation (13) is no longer completely differentiable, Equation (4) is not applicable and its iterative equivalent has to be employed. For a guess ofK that makes the system stable, Equation (17) has a uniqueP(K)>0solution,

Aⁱ(K)^TP+PAⁱ(K)+Q+K^TRK =0 (17) and theJ(K) value is given by [27],

J(K)=

{tr PW =tr L(Q+K^TRK), Aⁱ(K) stable

∞ Otherwise

Aⁱo+BⁱoK ==Aⁱ(K)

Aⁱ(K)L+LAⁱ(K)^T +W =0

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whereWis the covariance matrix of the process noise,w(1).

The solution to Equations (17) and (18) for an interval system is not feasible thus alternative faster low-cost search approaches are required to be developed.

In the formed composite objective function in Equation (13), terms are convex, but g is not differentiable. To simplify the solution, the differentiableJis approximated by its Taylor equivalent where its Hessian matrix is replaced by 1/t.I as below,

∼

J

K

(D)=J(K+D)≈J(K)+ ∇J(K)^TD+ 1

2t ‖D‖²₂ (19) In Equation (19),t>0 is the step size andDis the renewedK step. The gradient ofJ(K) is given by [27],

{∇J(K)=2(

B^TP+RK) L

E =PB+K^TR ⇒ ∇J(K)=2LE (20) As a result, the objective function Equation (13) is reformulated to,

D̂ =arg min

D∈D_A×(

J(K)+2(B^TP+RK)LD+ 1 2t ‖D‖²₂ +‖‖(K +D)◦Λ‖‖)

(21) Or in a short form expressed by,

[D]̂ =arg min

D∈D_A

(_∼

J_K(D)+‖(K+D)◦Λ‖₁ )

(22) Now, to reduce the computation burden and speeding up the convergence rate, the “active set” idea is introduced as below,

S ={ (i,j)||(

∇J(K))

i j|> Λi j or Ki j≠o }

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whereSis a binary matrix, one for active directions and zero for the others. Directions with low gradient values are considered inactive. By this measure instead of search inm×ndirection, a reduced subset of it satisfying Equation (23) is executed.

Subsequently, theKstep size,Dis also limited to those directions that existed in S,

DS ≡{

D∶Di j =0 ∀(i,j)∉S}

(24) The search follows the “coordinate descent” strategy where at each step only an update in one of the directions is performed.

The method has been shown that improves the algorithm convergence rate.

2.5 Design steps

The algorithm pseudo-code for the optimization steps are given below,

Input: Interval system Aio, Bio, w; Algorithm parameters Q, R, Λ;

Calculate initial K0 Equation (4) and locate the corresponding eigenvalues Define Ω_d space around the obtained eigenvalues

Current iterate K;

Define active set S Equation (23)

Compute L and P using Equations (17) and (18) for the nominal system

While (not converged) do

For each coordinate (i; j) in active set S do

calculate Dˆij Equation (22) Update K (Kij+Dˆ_ij→K_ij)

If Equation (10) for robustness is OK

Update solution: Dˆij →D_ij end if

end for

Update solution Dˆ →D Update Active set S Equation (23)

Update Λ Equation (15) end while

Output: Regularized Newton step D

3 SIMULATIONS

In this section, the effectiveness of the proposed RSDC is evaluated against other well-known control algorithms, including centralized, distributed, isolated, and voltage control (VC) approaches.

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TA B L E 1 The 5 bus test system parameters

Parameters N1 N2 N3 N4 N5

R_load(±50%) 0.150 0.450 0.375 0.600 0.225

L_load(±50%) 0.045 0.20 0.1 0.150 0.1

L₁₂ L₂₃ L₃₄ L₄₅ L₅₁

R_line 0.01 0.015 0.075 0.005 0.02

Lline 0.150 0.225 0.1125 0.075 0.300

Cline 0.30 0.20 0.40 0.60 0.15

F I G U R E 2 (a) VSC-based DC line and (b) five bus multi-terminal VSC-based DC network

The test bench is a 5-bus DC grid shown in Figure2(b). The simplified VSC-based back-to-back DC Line is portrayed in Fig- ure2(a). VSCs have been represented by controllable current sources which is a suitable and sufficient model for VSC-DCs to study AC and DC transients and high-level control system designs [28, 29]. C_i and C_m model the VSCs output capaci- tance, and the DC line is represented byL_imandR_im. Base on the model, a multi-terminal VSC-based DC network has the structure portrayed in Figure 2(b). A detailed description of the impedance-based model for VSC’s and its extension to the MTDC grid model has been discussed in [1, 30].

Each of the current sources is controlled by a PI controller based on the voltage error e and its integral σ. Thus, the

state-space mathematical model of the network is obtained as below,

⎡⎢

⎢⎢

⎣

∙e_i i∙Li

𝜎∙i

∙iLi j

⎤⎥

⎥⎥

⎦

=

⎡⎢

⎢⎢

⎢⎣

0 −1

C_iI 0 Mi j

1

L_i.I −R_i

L_iI 0 0

k_p.I 0 0 0

N_{i j} 0 0 −Ri j

L_{i j} I

⎤⎥

⎥⎥

⎥⎦

⎡⎢

⎢⎢

⎣ e_i iLi

𝜎i

iLi j

⎤⎥

⎥⎥

⎦ +

⎡⎢

⎢⎢

⎢⎣ 1 C_i.I

0 0 0

⎤⎥

⎥⎥

⎥⎦ [u_i],

k,i,j =1, … ,5

Mi j =

⎧⎪

⎪⎨

⎪⎪

⎩

−1 Ci

jis theineighbour,j >i 1

C_i jis theineighbour,j <i 0, otherwise

N =

⎧⎪

⎪⎪

⎨⎪

⎪⎪

⎩ N_ii = 1

Li j ,N_{i j} = −1 Li j

jis theineighbour, j >i

N_ii = −1

L_{i j} ,N_{i j}= 1

L_{i j} jis theineighbour,j<i

0 otherwise

(25) wheree,σ,i_L, andI_Lijare the node voltage error, the integral of error, nodeiload current, and lineijcurrent. 0 elements define zero matrixes of appropriate dimensions.

The system consists of 20 states and 5 inputs. So, the size of matrixesA_iandB_iare 20×20, 20×5. The control gainKhas the maximum dimension of 5×20=100. The simulation parameters of the grid are listed in Table1.

Although in the simulation process, a 5-bus DC grid was modelled and controlled, the rules presented in the paper are general and can be extended easily to a large-scale system.

The model in Equation (24) in compact form is expressed by,

̇

x=A_iox+B_iou

y=Cx (26)

TheCpattern is related to the choice of control structure: centralized, decentralized, or distributed.

Based on the assigned parameters, the damping ratio of the loads at nodes 1 to 5 are 0.35, 0.5, 0.6, 0.8, and 0.35, respectively; thus the bus voltages will experience different transient responses. Moreover, the line resistances are kept sufficiently low regarding the load resistances till a network with strong interconnections among nodes is formed.

The primary source of uncertainty in the grid is the loads. Therefore, to the extent of 50% tolerance is applied to Rload and Lload to represent the unavoidable system

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uncertainty. Hence, the network is no longer nominal, and an interval system is formed. Uncertainties in the loads do not perturb all of the parameters of the network; just parts of them are affected. To reduce the computation loads, the system states Equation (25) has been rearranged in a way to confine the uncertainty to a sub-matrix A11 as shown below,

A_io=

[A11 A12

A₂₁ A₂₂ ]

=

⎡⎢

⎢⎢

⎣

0 a₁₂ 0 a₁₄ a21 a22 0 0 a₃₁ 0 0 0 a₄₁ 0 0 a₄₄

⎤⎥

⎥⎥

⎦

, B_io=

⎡⎢

⎢⎢

⎣ b₁₁

0 0 0

⎤⎥

⎥⎥

⎦ (27) where

a₁₂=[

3.33 5 2.5 1.67 6.67] .I5

a₂₁=[

[20,24.44] [4.5,5.5] [9,11] [6,7.33] [9,11]] .I5

a₂₂= −

[[2.73,4.07] [1.84,2.75] [3.07,4.58]

[3.27,4.89] [1.84,2.75]

] .I5

a₃₁=I₅, a₄₄= −6.67.I5

a14=

⎡⎢

⎢⎢

⎣

−3.33 0 0 0 3.33

5 −5 0 0 0

0 2.5 −2.5 0 0

0 0 1.67 −1.67 0

0 0 0 6.67 −6.67

⎤⎥

⎥⎥

⎦

a₄₁=

⎡⎢

⎢⎢

⎣

0 6.67 0 0 −3.33

−6.67 0 4.44 0 0

0 −4.44 0 8.89 0

0 0 −8.89 0 13.33

3.33 0 0 −13.33 0

⎤⎥

⎥⎥

⎦ b₁₁=[

3.33 5 2.5 1.67 6.67] .I5

whereI₅is a 5 by 5 unity matrix.

RSDC is designed in an iterative fashion utilizing the proposed coordinate descent method. In each step, robustness constraint Equation (11) is checked for negative semi definiteness.

Although in the simulation process, a 5-bus DC grid is modelled and controlled, the rules presented in the paper are general and easily can be extended to a large-scale system.

For better illustration of the achievements, four other designs are also conducted, which are:

Centralized LQR control: The Linear Quadratic Regulator (LQR) is a well-known method that provides full state optimal feedback to make the closed-loop system stable with the desired performance. The method is applied to the nominal system as a reference for RSDC performance evaluation [31].

F I G U R E 3 The required links (Ksparsity pattern) for five designs

Distributed LQR control: Distributed LQR approximates the centralized LQR where a distribution index is added to the optimal cost function [32]. Distributed control architecture limits the interconnections to the neighbouring units avoiding com- plete network interactions [26].

Isolated LQR control: In an isolated control, each unit is controlled by its local controller using just local measurements with no signalling from the other network units. This implies that multiple subsystems operate independently without concerning the overall network instability [33]. In this case,A_ois not diagonal and the design ignores the coupling among nodes.

Voltage control (VC): The voltage control method uses the voltage error through a PI-controller to regulate the multi- terminal DC system as it has been detailed in [34, 35]

Designs are applied to Equation (27) and the sparsity pattern of the controller feedback links of the five methods including RSDC is obtained as shown in Figure3.

Based on the figure, a fully decentralized controller using RSDC has been reached which stabilizes the uncertain system.

The numbers of the required feedback links are 20 out of 100, which all are local. The objective function value is 0.185, which is the best among the other algorithms listed in Table2.

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TA B L E 2 The performance, convergence time, and sparsity index of RSDC versus the others

Non-local links

Assigned links

Performance (obj. function)

Convergence.

time Methods

0 20 0.0185599 0.223994 RSDC

35 100 0.0166379 0.276554 Centralized

40 60 0.0309003 0.225665 Distributed

0 25 0.0308631 0.180290 Isolated

0 15 0.0399127 0.208725 VC

F I G U R E 4 Poles of the nominal closed-loop system

The total number of links between the nodes in the centralized, distributed, isolated, and voltage control methods are 100, 60, 25, and 15, respectively. The non-local links for the centralized are 35 and for the distributed are 40.

The centralized design method ends up in a stable network with full 100 links; most of them non-local, requiring costly infrastructure with worse objective function than RSDC. The distributed method with 40 non-local links does not provide performance comparable to RSDC. Isolated and VC designs do not need any non-local links similar to RSDC, but they cannot get close to the performance index of RSDC.

According to the results, despite the sparsity, the RSDC method has better performance indices regarding the others; while, the optimization speed is also pretty acceptable, as depicted in Table2.

Dominant poles of the closed-loop system using RDSC and other control methods have been shown in Figure4. As the figure indicates, the modes of all designs are on the left side of the s plane, but RSDC poles are on the left side ofζωn=-0.2, meaning better relative stability has been achieved.

F I G U R E 5 The nodes one to five nominal system transient responses

3.1 Disturbance rejection performance

To assess the disturbance rejection capacity of the designs, a ran- domly generated perturbation is applied to the system. Time- domain responses of the nodes have been exhibited in Figure5.

As the figure indicates, the RSDC design, despite higher sparsity, can easily withstand the disturbance and secure the system stability with minimum steady-state error. The max disturbance deviation using the proposed RSDC design is improved by at least 11% compared to the other approaches. The bus voltages are restored to the nominal values within 4 s, and the steady- state error touches zero, which is better than the results obtained using the contestant algorithms.

3.2 Robustness against parameter uncertainty

In this test, 50 random systems by applying 30% tolerances to the network loads are generated, and the system response in containing the uncertainty is assessed. The disturbance response of all buses of the system utilizing RSDC design has been exhibited in Figure6. The disturbance response of Node 1of all five methods has been shown in Figure7. From Figure6and7, the numerical index of the system behaviour is calculated, which has been listed in Table3. Based on the indexes of Table3, the RSDC disturbance rejection achievement has the lowest max

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F I G U R E 6 The RSDC disturbance responses to a group of 30%

tolerance random systems

F I G U R E 7 Node 1 disturbance responses for a group of 30% tolerance random systems using four designs

TA B L E 3 The average transient behaviour indexes for a group of 30%

Settling time(s) Max deviation Jindex Methods

NAN 1.99169e⁴² 3.71e⁸⁵ V control

1.190878 0.147766195 0.005746 Distributed

1.284884 0.148026481 0.005842 Isolated

0.55769 0.128291926 0.00286 RSDC

1.200368 0.152046278 0.006152 Centralized

F I G U R E 8 Disturbance responses for a group of 50% tolerance random systems using RSDC

deviation, cost functionJ, and settling time; while it has lower feedback links than the others. The VC control that has a lower number of links than RSDC completely fails in securing the system performance, Figure7. Thus, it is concluded that the RSDC design better handles the 30% tolerances in the system dynam- ics than the other algorithms.

To dig up more the RSDC design strength, this time, 50%

tolerances are applied to the network loads, and the design performances are examined. The results similar to the previous test are repeated. Figures8and9show the disturbance rejection characteristics of the designs. Based on the figures, numerical indexes are computed, which have been listed in Table4.

The RSDC cost functionJ, the max deviation, and the settling time are pretty lower than the contestant algorithms. As Fig- ure9presents, the VC control cannot even secure the system’s stability.

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F I G U R E 9 Node 1 disturbance responses for a group of 50% tolerance random systems using four designs

TA B L E 4 The average transient behaviour indexes for a group of 50%

Settling time (s) Max deviation Jvariance Method

NAN 1.19043e⁵² 1.1e¹⁰⁶ V control

1.187497 0.106845923 0.005627 Distributed

1.273379 0.107134994 0.005736 Isolated

0.467287 0.089193895 0.001962 RSDC

1.186483 0.111067282 0.006033 Centralized

3.3 Robustness against random disturbances

To examine the capacity of the design in accommodating various perturbations, a group of 50 random step-wise disturbances is generated and applied to the system. The transient responses of the network buses have been shown in Figure10.

The behaviour of Node 1 in response to the set of perturbations utilizing various designs has also been portrayed in Figure11. From the figures, numerical merits of performances are produced, which have been illustrated in Table5. They are the average performance index J, the max deviation, and the settling time. The indexes indicate that the minimum average cost function belongs to the execution of RSDC as well as the lowest max deviation and settling time.

Another test is conducted using 50 random pulse-wise perturbations. The disturbance responses of network nodes utiliz-

F I G U R E 1 0 The RSDC disturbance response to the group of step-type random perturbations

F I G U R E 1 1 Node 1 disturbance responses against the collection of step-type random perturbations using four designs

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TA B L E 5 The average transient behaviour indexes for the step-wise perturbations

Settling time (s) Max deviation Variance Method

NAN 4.46254e⁴² 1.01e⁸⁷ V control

1.698409 3.56679e³⁹ 1.44e⁸¹ Distributed

1.636871 1.71465e³⁹ 3.33e⁸⁰ Isolated

0.534408 0.064783449 0.007899 RSDC

1.557451 2.11819e³⁸ 5.08e⁷⁸ Centralized

F I G U R E 1 2 The RSDC disturbance response to the collection of pulse-type random perturbations

ing the RSDC method have been portrayed in Figure12. The perturbation responses of Node 1, under various designs, have also been presented in Figure13. Again, the numerical indexes are derived from the figures, which have been shown in Table6.

The previous results are certified from the table that the RSDC is a superior design regarding the others in terms of the min-

TA B L E 6 The average transient behaviour indexes for the pulse-type perturbations

Settling time (s) Max deviation Variance Method

NAN 7.3e⁸⁴ 1.2269e⁴² V control

2.520466 0.03111 0.08029402 Distributed

2.32852 0.031064 0.081942011 Isolated

1.862376 0.013224 0.068818377 RSDC

2.231395 0.028363 0.080276019 Centralized

F I G U R E 1 3 Node 1 disturbance responses against the collection of step-type random perturbations using four designs

imum cost function, lower max deviation, and faster settling time.

4 CONCLUSION

In this paper, a robust sparse decentralized controller for interval networks is suggested. In this regard, a new design method is developed which automatically adjusts the sparsity degree, enforces performance robustness, and improves convergence rate. The controller is applied to a five-bus VSC-based DC grid subject to 50% load tolerances. The design indexes are compared with the centralized, distributed, isolated, and voltage control methods. The results show that the RSDC method renders a controller with 20 feedback links out of available 100 and the objective function value as low as 0.018, which is better than the other algorithms. The convergence time is 0.22 s, which is a little bit higher than the algorithms with costly non- local links. Therefore, it is concluded that the proposed method is promising, which enjoys overall superiority over the others.

A C K N OW L E D G M E N T

This work has been partially supported by the research depart- ment of Shahed University, Tehran, IRAN.

O RC I D

Behrouz Nabi https://orcid.org/0000-0002-4779-4755 Saeed Seyedtabaii https://orcid.org/0000-0001-7617-5542

(11)

R E F E R E N C E S

1. Agbemuko, A.J., et al.: Advanced impedance-based control design for decoupling multi-vendor converter HVDC grids. IEEE Trans. Power Delivery 35(5), 2459–2470 (2020)

2. Li, Z., et al.: Universal power flow algorithm for bipolar multi-terminal VSC-HVDC. Energies 13(5), 1053 (2020)

3. Wang, W., et al.: A distributed cooperative control based on consensus protocol for VSC-MTDC systems. IEEE Trans. Power Syst. 36(4), 2877–2890 (2021)

4. Simiyu, P., et al.: Review of the DC voltage coordinated control strategies for multi-terminal VSC-MVDC distribution network. J. Eng. 2019(16), 1462–1468 (2018)

5. Zolfaghari, M., Gharehpetian, G.B., Anvari-Moghaddam, A.: Quasi- Luenberger observer-based robust DC link control of UIPC for flexible power exchange control in hybrid microgrids. IEEE Syst. J. 15(2), 2845–

2854 (2021)

6. Chen, L., et al.: Resilient active power sharing in autonomous microgrids using pinning-consensus-based distributed control. IEEE Trans. Smart Grid 10(6), 6802–6811 (2019)

7. Yang, S., et al.: Decentralized and autonomous voltage balancing control approach for hybrid multi-terminal Ultra-HVDC system. Int. J. Electr.

Power Energy Syst. 115, 105472 (2020)

8. Seyedtabaii, S.: A modified FOPID versus H∞andμsynthesis controllers:

Robustness study. Int. J. Control Autom. Syst. 17(3), 639–646 (2019) 9. Seyedtabaii, S., Zaker, S.: Lateral control of an Aerosonde facing tolerance

in parameters and operation speed: Performance robustness study. Trans.

Inst. Meas. Control 41(8), 2319–2327 (2019)

10. Seyedtabaii, S.: New flat phase margin fractional order PID design: Per- turbed UAV roll control study. Rob. Auton. Syst. 96, 58–64 (2017) 11. Vellaboyana, B., Taylor, J.A.: Optimal decentralized control of DC-

segmented power systems. IEEE Trans. Autom. Control 63(10), 3616–

3622 (2018)

12. Wu, X., Jovanovi´c, M.R.: Sparsity-promoting optimal control of systems with symmetries, consensus and synchronization networks. Syst. Control Lett. 103, 1–8 (2017)

13. Argha, A., Su, S.W., Ye, L., Celler, B.G.: Optimal sparse output feedback for networked systems with parametric uncertainties. Numer. Algebra Control Optim. 9(3), 283 (2019)

14. Seyedtabaii, S.: Modified adaptive second order sliding mode control:

Perturbed system response robustness. Comput. Electr. Eng. 81, 106536 (2020)

15. Rezaee, M., Seyedtabaii, S.: On an optimized fuzzy supervized multiphase guidance law. Asian J. Control 18(6), 2010–2017 (2016)

16. Mehdi, M., et al.: Robust control of a DC microgrid under parametric uncertainty and disturbances. Electr. Power Syst. Res. 179, 106074 (2020)

17. Banki, T., Faghihi, F., Soleymani, S.: Frequency control of an island microgrid using reset control method in the presence of renewable sources and parametric uncertainty. Syst. Sci. Control Eng. 8(1), 500–507 (2020) 18. Marcillo, K.E.L., et al.: Interval robust controller to minimize oscillations

effects caused by constant power load in a DC multi-converter buck-buck system. IEEE Access 7, 26324–26342 (2019)

19. Dehkordi, N.M., Sadati, N., Hamzeh, M.: Robust tuning of transient droop gains based on Kharitonov’s stability theorem in droop-controlled microgrids. IET Gener., Transm. Distrib. 12(14), 3495–3501 (2018)

20. Lucas-Marcillo, K.E., et al.: Novel robust methodology for controller design aiming to ensure DC microgrid stability under CPL power varia- tion. IEEE Access 7, 64206–64222 (2019)

21. Patre, B.M., Deore, P.J.: Robust state feedback for interval systems: An interval analysis approach. Reliab. Comput. 14, 46–60 (2010)

22. Rautert, T., Sachs, E.W.: Computational design of optimal output feedback controllers. SIAM J. Optim. 7(3), 837–852 (1997)

23. Dorf, R.C., Bishop, R.H.: Modern Control Systems. Pearson, London (2011)

24. Horng, I.-R., Horng, H.-Y., Chou, J.-H.: Eigenvalue clustering in subre- gions of the complex plane for interval dynamic systems. Int. J. Syst. Sci.

24(5), 901–914 (1993)

25. Chen, J.: Sufficient conditions on stability of interval matrices: Connections and new results. IEEE Trans. Autom. Control 37(4), 541–544 (1992) 26. Nabi, B., Seyedtabaii, S.: Fast distributed control design for DC linked

microgrids. Int. J. Electr. Power Energy Syst. 122, 106221 (2020) 27. Wytock, M., Kolter, J.Z.: A fast algorithm for sparse controller design.

arXiv preprint arXiv:1312.4892(2013)

28. Barnes, M., et al.: HVDC systems in smart grids. Proc. IEEE 105(11), 2082–2098 (2017)

29. Wachal, R. et al.: Guide for the development of models for HVDC convert- ers in a HVDC grid. Cigré TB604 (WG B4. 57), Paris, Tech. Rep., CIGRE (2014)

30. Agbemuko, A.J., et al.: Dynamic modelling and interaction analysis of multi-terminal VSC-HVDC grids through an impedance-based approach.

Int. J. Electr. Power Energy Syst. 113, 874–887 (2019)

31. Yadav, O., et al.: Controller design for MTDC grid to enhance power sharing and stability. IET Gener. Transm. Distrib. 14(12), 2323–2332 (2020) 32. Vlahakis, E.E., Dritsas, L.D., Halikias, G.D.: Distributed LQR-based sub-

optimal control for coupled linear systems. IFAC-PapersOnLine 52(20), 109–114 (2019)

33. Ishizaki, T., et al.: Retrofit control: Localization of controller design and implementation. Automatica 95, 336–346 (2018)

34. Tang, X., Lu, D.D.-C.: Enhancement of voltage quality in a passive network supplied by a VSC-HVDC transmission under disturbances. Int. J. Electr.

Power Energy Syst. 54, 45–54 (2014)

35. Campos-Gaona, D., Peña-Alzola, R., Ordonez, M.: Nonminimum phase compensation in VSC-HVDC systems for fast direct voltage control.

IEEE Trans. Power Delivery 30(6), 2535–2543 (2015)

How to cite this article: Nabi, B., Seyedtabaii, S.:

Optimal sparse control of interval networks: Voltage source converter based DC grid study. IET Gener.

Transm. Distrib.1–11 2021.

https://doi.org/10.1049/gtd2.12262.