Stability Analysis for the DC Microgrid of Chained Communication Network with Cluster Treatment of Characteristic Roots Paradigm

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Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, Applied Energy Symposium and Forum, Energy Integration REM 2017, 18–20 Renewable October 2017, Tianjin, China with Mini/Microgrids, REM 2017, 18–20 October 2017, Tianjin, China

Stability Analysis for the DC Microgrid of Chained Communication The 15th International Symposium on District Heating andCommunication Cooling Stability Analysis for the DC Microgrid of Chained Network with Cluster Treatment of Characteristic Roots Paradigm Network with Cluster Treatment of Characteristic Roots Paradigm Assessing the feasibility of using the heat demand-outdoor Chaoyu Dongaa, Qingbin Gaobb*, Hongjie Jiaaa†, Guohong Wucc, Xiaomeng Liaa, Zhenyu Chaoyu Dong , Qingbin Gaofor *, Hongjie Jia †,d Guohong , Xiaomeng Li ,forecast Zhenyu temperature function a long-term districtWuheat demand Zhang d Zhang *, A. Pina , P. Ferrão , J. Fournier ., B. Lacarrière , O. Le Corre

Electrical and Tianjin University, Tianjin, 300072 c a Information Engineering, a b c b MechanicalaElectrical and Aerospace Engineering, California State University, Long Beach, California, 9084 and Information Engineering, Tianjin University, Tianjin, 300072 c b Electrical Engineering & Information Technology, Tohoku Gakuin University, Miyagi Prefecture, 985-8537 Mechanical d and Aerospace Engineering, California State University, Long Beach, California, 9084 a c Servo Department, Western Digital Corporation, California, 92614 IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Electrical Engineering & Information Technology, Tohoku GakuinIrvince, University, Miyagi Prefecture, 985-8537Lisbon, Portugal d b Servo Department, Western Digital Corporation, Irvince, California, 92614France Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, c Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France

a,b,c

I. Andrić

a

Abstract Abstract The chained communication network is a common topology for the application of control strategies, in which distributed generators Abstract The chained communication network is a common topology the application of control strategies, in which distributed generators (DGs) interact successively in DC microgrids. Through the for chained communication network of a distributed control strategy, the (DGs) successively in DCmaintenance microgrids. Through the chained network of a distributed strategy, the power interact allocation and the voltage of DC microgrids cancommunication be achieved without central controllers.control However, the time District heating networks are addressed in the literature oneproblem of the most effective solutionsHowever, for decreasing the power allocation and the voltage maintenance ofwhich DC microgrids beasachieved without controllers. delay issues are unavoidable andcommonly critical herein, might leadcan to stability and central lack clear investigations. The the facttime that greenhouse gasunavoidable emissions from theoverall building sector. These systems investments are returned through thethat heat delay issues and are and critical herein, which might lead to require stability problem lackwhich clear investigations. The fact the stability performance of the system depend on the time delay high renders itsand analysis paramount. For this purpose, the sales. Due and to the climate conditions and building heat demand future could decrease, the stability performance of theroots overall systemis depend on the time delay rendersofits analysis For this purpose, the cluster treatment of changed characteristic (CTCR) employed torenovation reveal the policies, effect time delayparamount. oninthethechained communication prolonging investment return scenario period. cluster treatment of characteristic roots (CTCR) employedwith to reveal the effect of time delaynetwork on the is chained communication network. As the a result, an unstable of DC ismicrogrids the chained communication detected. Besides, the The mainAsscope of this paper isoutput toscenario assess the of using heat demand – outdoor temperature heat demand network. a result, an power unstable of feasibility DC microgrids withthe the chained communication network isfunction detected. Besides, the relationship between the and time delay is investigated, which is helpful to the stability maintenance offor DC microgrid forecast. The district Alvalade, located Lisbon (Portugal), which was used as a case The district is of consisted of 665 relationship between the power output and timeindelay is investigated, is helpful to thestudy. stability maintenance DC microgrid with the consideration ofof physics and information simultaneously. buildings that vary inofboth construction period simultaneously. and typology. Three weather scenarios (low, medium, high) and three district with the consideration physics and information renovation© scenarios wereLtd. developed (shallow, Copyright 2018 Elsevier All rights reserved.intermediate, deep). To estimate the error, obtained heat demand values were Copyright ©with 2018 The Authors. Published bydemand Elseviermodel, Ltd. previously developed and validated by the authors. comparedand results fromLtd. a dynamic heat Copyright © 2018 Elsevier All rights reserved. Selection peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, The results showed that when only weather change is considered, the marginof of the errorApplied could beEnergy acceptable for someand applications Selection and peer-review under responsibility of the scientific committee Symposium Forum, Renewable Energy Integration with Mini/Microgrids, REM 2017. Renewable Energy Integration with Mini/Microgrids, REM 2017 (the error Energy in annual demand was than 20% for all weather Renewable Integration withlower Mini/Microgrids, REM 2017. scenarios considered). However, after introducing renovation scenarios,DCthemicrogrids; error value up to 59.5%of(depending onroots the (CTCR); weather chained and renovation scenarios combination considered). Keywords: timeincreased delay; cluster treatment characteristic communication network. Keywords: DCofmicrogrids; time delay;increased cluster treatment of characteristic roots (CTCR); chained network. The value slope coefficient on average within the range of 3.8% upcommunication to 8% per decade, that corresponds to the decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and improve the accuracy of heat demand estimations. * Corresponding author. Tel.: +1 (562) 985-1503; fax: +1 (562) 985-1503.

© 2017 The Authors. Published by Elsevier Ltd. E-mail address:author. [email protected] * Corresponding Tel.: +1 (562) 985-1503; fax: +1 (562) 985-1503. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and * Corresponding Tel.: +022-27892811; fax: +022-27892811. E-mail address:author. [email protected] Cooling. E-mail address:author. [email protected] * Corresponding Tel.: +022-27892811; fax: +022-27892811.

E-mail address: [email protected] Keywords: Heat demand; Forecast; Climate change 1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility the scientific 1876-6102 Copyright © 2018 Elsevier Ltd. All of rights reserved. committee of the Applied Energy Symposium and Forum, Renewable Energy Integrationand with Mini/Microgrids, REM 2017. of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Selection peer-review under responsibility Integration with Mini/Microgrids, REM 2017. 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. 1876-6102 Copyright © 2018 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2017 10.1016/j.egypro.2018.04.045

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1. Introduction With the increasing installation and utilization of DC distributed generators (DGs), DC microgrids are enjoying a promising development because of the natural interface for sources and loads [1]. For the growing DC loads [2], it is convenient for them to be directly supplied by the DC power instead of the multiple conversions between AC and DC. In order to allocate the power and to restore the bus voltage for DC microgrids, intensive investigations have been paid in the distributed control strategy to avoid the unreliability and cost issue of central controllers [3,4]. The distributed control strategy usually has two layers: the primary control layer and the secondary control one. According to the feedback of each DG output, the power can be allocated locally in the primary control layer. Then the secondary controller is employed to restore the DC bus voltage through the communication network among DGs. The chained communication network is a common topology for the application of distributed control strategies, in which DGs interact successively in a chain [5]. With one or only a few receivers of the real-time DC bus voltage, the DC microgrid can restore its voltage to the setting value without affecting the power allocation. Despite central controllers are avoided, it is noted that the new problems might arise because of the communication network. One of the most critical problems is caused by the time delay in the communication network. However, there are few reports about this regard in the literature. To tackle this, we introduce a systematic stability analyzing tool called Cluster Treatment of Characteristic Roots (CTCR) to analyze the DC microgrids of chained communication network. The CTCR paradigm consists of three procedures: 1) The potential stability switching loci in the delay domain is firstly classified and revealed; 2) As a byproduct of the previous step, the complete imaginary spectra of the delayed system are obtained; 3) The exact and exhaustive stability region in the domain of the delay is declared for the communication network. The application of CTCR can not only reveal the inherent time-delay stability issues but also provide systemic analysis for DC microgrids. 2. Time delay in the DC Microgrid of Chained Communication Network A parallel DC microgrid with two-layer distributed control strategy is shown in Fig.1 (a). The primary control layer can work in a decentralized way with the conventional droop controllers to allocate the power among DGs, while the secondary control layer utilizes the chained communication network in Fig.1 (b) to restore the bus voltage deviation by the control signal uk, k=1,2,…,n.

a

b Fig. 1. (a) The parallel DC microgrid with two-layer distributed control strategy; (b) The chained communication network.

The real-time DC bus voltage deviation e1k is e1k = V ∗ − vDC = V ∗ − [V ∗ + uk − ( M k + Rk )ik ] = ( M k + Rk )ik − uk

where Mk is the droop coefficient and Rk is the line resistance. They cause the DC bus voltage deviation.

(1)

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3

According to the Kirchoff’s current law, the DC bus voltage vDC can be expressed by the load resistance R and the branch current of DGk, k=1,2,…,n. vDC = R  ik

(2)

k

According to (1)-(2), the real-time voltage error vector e1=[e11…e1k…e1n]T is obtained

e1 = diag {Rk + M k } Z −1 (V ∗ ⋅ I n×1 + u) − u

(3)

where Z=diag{Rk+Mk}+R·In×n, u1=[u11…u1k…u1n]T and all elements in I are 1. Besides the voltage error, the consensus error e2=[e21…e2k…e2n]T is employed to keep the original power sharing ratio by comparing the control signals, which will be affected by the time delay τ in the communication network e2 (t ) = Au(t − τ ) − Bu(t )

(4)

where B=diag(b1,…,bk,…,bn), bk is the number of neighboring secondary controllers for DGk, A=-L/μ+B, L is the Laplacian matrix of the chained communication network, μ is the unified network weight. As the existence of timedelay coefficient A, the system dynamic might be more complicated compared to the original e2=-Lu(t). Remark 1: A unified time delay value τ is assumed in the chained communication network. Although A=-L/μ+B avoids the effect of weight value μ in A, the effect of μ will be reflected by the combination weight vector a1 as follows. Denoted a1=diag(a11,…,ak1,…,an1) and a2=diag(a12,…,ak2,…,an2) as the combination weight vectors for e1 and e2, the summation of error e is

e = a1e1 + a2e2 = a1diag { Rk + M k } Z −1V ∗ ⋅ I n×1 − a1RI n×n Z −1u(t ) − a2 Bu(t ) + a2 Au(t − τ )

(5)

Remark 2: The weighting effect of the communication network will be reflected by a2, while a1 reflects the information receivers of DC bus voltage and the corresponding weight μ. The output of the secondary controller is then obtained by the proportional and integral (PI) controller t

u(t ) = KPe(t ) + KI  edt 0

(6)

where KP=diag(KP1,…,KPk,…, KPn), KI=diag(KI1,…,KIk,…, KIn). When the output in (6) is transferred in the communication network, the neighboring DG will receive the delayed signal u(t − τ ) . To analyse the stability of secondary controller, the derivative of (6) is taken. The small deviation of u(t) is obtained

d Δu(t ) d Δu(t − τ ) − C0 = A0 Δu(t ) + A1Δu(t − τ ) dt dt

(7)

where A0=-[E+KP(a1RIn×nZ-1+a2B)]-1KI(a1RIn×nZ-1+a2B), A1=[E+KP(a1RIn×nZ-1+a2B)]-1KIa2A, C0=[E+KP(a1RIn×nZ1 +a2B)]-1KPa2A, E is the unit matrix. If we consider the system dynamic from the respect of the sum error, the derivative of e(t) is

de (t )  de (t − τ )  − C0 = A0 e (t ) + A1e (t − τ ) dt dt

(8)

Remark 3: Comparing (8) and (7), the difference in coefficients is the sequence of KP and KI. As KP and KI are diagonal matrices, A0 = A 0 , A1 = A1 and C0 = C 0 hold. Therefore, the dynamics of the summing error e and the output

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deviation u(t) are equivalent. In this work, (8) is taken for analyses. Remark 4: As the summing error e can be eliminated by the integral controller, the proportional coefficients are assumed to be 0. Finally, a retarded time-delay system is obtained from (8) de ( t ) / dt = A 0 K I e ( t ) + A1e ( t − τ )

(9)

3. Cluster Treatment of Characteristic Roots (CTCR) Paradigm The stability analysis of the system in (8) is not trivial as infinitely many characteristic roots are present due to the effect of the time delay. To tackle this, a systematic and effective stability analysis tool called the Cluster Treatment of Characteristic Roots (CTCR) paradigm is employed [6-10]. For (9), the corresponding characteristic equation is CE ( s ,τ ) = det( sI − A 0 − A1e −τ s )

(10)

To analyze the stability impact of e-τs in (10), the definition for the complete set of the imaginary spectra of the time-delay system (10) for all possible delay values is Ω = {ωc | CE ( s = ωc i ,τ ) = 0,τ ∈ ℜ + , ωc ∈ ℜ + }= {ωc | < τ , ωc > }

(11)

From the set Ω, two important definitions arise. We present the following results with details in the references. Definition 1: Kernel delay values: The delay values τ k ∈ ℜ + that correspond to < τ k , ωc > occur and satisfy the constraint 0 < τ k ωc < 2π . Definition 2: Offspring delay values: The delay values obtained from kernel delay values through the following point-wise nonlinear conversion τ ± 2 jπ / ωc , where j is an integer number. Definition 3: Root tendency (RT): The direction of transition of the imaginary root as the delay value increases by ε (0 < ε << 1) , as obtained by RT

τj

s =ω i

(

= sgn  Re ∂s / ∂τ j 

s =ω i

)

(12)

This definition implies that any kernel delay value will generate infinitely many offspring, due to the periodicity of the exponential term. With these two definitions, the analysis is able to be achieved by the kernel delay values instead of countless roots. The corresponding property can then be revealed by the following definition and theorem. Theorem: (Invariant root tendency property) Take one imaginary characteristic root ωci from the set (13), the RT of ωci remains invariant from kernel delay value to its offspring. To implement the CTCR paradigm, the Rekasius substitution [11] is firstly employed e −τ s = (1 − Ts ) / (1 + Ts ), τ ∈ ℜ + , T ∈ ℜ

(13)

which is an exact substitution for purely imaginary root, i.e., s = ω i, ω ∈ ℜ . The mapping condition is

τ = 2 / ω ⋅ [tan −1 (ωT ) ± lπ ], l = 0,1,2,...

(14)

From the above equation, one T is mapped into countless τ for a given ω, with one being the kernel delay value and the others offspring. The replacement of (13) into (10) leads to a rational polynomial

 ak ( s )(1 − Ts ) k / (1 + Ts ) k

k =0

=0

(15)

Chaoyu Dong et al. / Energy Procedia 145 (2018) 394–399 Author name / Energy Procedia 00 (2018) 000–000

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5

Multiplying both sides with (1+Ts)n, the above equation can be rewritten as 2n

 bk s k

(16)

=0

k

Note that (16) shares identically the same imaginary roots of (9). Then the corresponding Routh’s array is formulated. If a pair of imaginary roots exists, the only term R1 (T ) corresponding to s1 must be zero. The above equation is a polynomial in T, the roots of which can be calculated effectively and efficiently. We are interested in real roots only due to (13) and these roots correspond to the kernel delay values. Another important property of the Routh’s array is that if there is a row with all zero elements, then we can form the auxiliary equation based on the terms from the immediate above row. Therefore, for a real root, say T of (13), the auxiliary equation is R21 (T ) s 2 + R22 (T ) = 0

(17)

When the condition R21(T)R22(T)>0 meets, the imaginary root could be determined as

ωc = R22 (T ) / R21 (T )

(18)

With ωc, the corresponding kernel delay value τ k can be obtained as well as the RT for those kernel delay values. With these, the number of unstable roots could be calculated with ωc ∈ [0, +∞ ) . 4. Case Studies The chained communication network for a DC microgrid with three DGs is depicted in Fig. 2. τ is the time delay between neighboring DGs in the communication network.

Fig. 2. The communication network of chain pattern.

The DC bus voltage V ∗ and the line resistance Rk are set as 400V and 0.05Ω respectively. The load P=10kW, i.e. the equivalent load resistance R = 400 2 / P = 16 Ω . a2 = 10 ∗ E , M = [3 6 1]T and the integral matrice K I = 1 ∗ E . The unified network μ is 1 as well as the same weights for neighbouring DGs. Three scenarios are defined for comparisons: Scenario 1 (S1): DG1 is the receiver of DC bus information, i.e. only a1 (1,1) = c1 ≠ 0 for a1 ; Scenario 2 (S2): DG2 is the receiver of DC bus information, i.e. only a1 (2,2) = c2 ≠ 0 for a1 ; Scenario 3 (S3): DG3 is the receiver of DC bus information, i.e. only a1 (3,3) = c3 ≠ 0 for a1 . Assume c1= c2= c3=c, the system eigenvalues when τ=0 is listed in Table 1. Table 1. System eigenvalues when τ=0. c=0.1

S1

S2

S3

-300.12

-299.79

-300.11

S1

S2

S3

-301.05

-297.88

-301.07

-99.79

-100

-0.32

-0.32

-100.21

-97.87

-100

-0.32

-3.25

-3.22

c=1

S1

S2

S3

-310.54

-276.26

-313.11

-102.13

-57.73

-100

-125.69

-3.12

-53.47

-34.70

-24.36

c=10

When the time delay in the chained communication network is neglected in Table 1, the DC microgrid is stable for S1-S3 if c ∈ [0.1,10] . With all the eigenvalues in the left, e will converge to 0 as time elapses. Considering non-zero time delay τ ≠ 0 , the ωc and the corresponding kernel delay value τ k of the DC microgrid with three DGs is determined by the CTCR paradigm as shown in Table 2.

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Chaoyu Dong et al. / Energy Procedia 145 (2018) 394–399 Author name / Energy Procedia 00 (2018) 000–000

Table 2. Calculation results with CTCR paradigm when c S1

0.1

1

5

10

ωc

-

-

-

-

τk

-

-

-

-

RT

-

-

-

-

S2

399

τ ≠0.

c

0.1

1

5

10

ωc

7.68

24.13

52.59

72.53

τk

401.72

122.63

52.10

35.57

RT

1

1

1

1

0.1 S3

1

5

10

ωc

-

-

-

-

τk

-

-

-

-

RT

-

-

-

-

As shown in Table 2, S1 and S3 are delay independent stable. However, the DC microgrid suffers from instability because of the time delay in S2. As the corresponding RT>0, the ωc will cross the imaginary axis eventually as τ increases. Besides, the decrease of τk in S2 reflects that the increasing weight in a1 will alleviate the time-delay stability for the DC microgrid. Thus, the S1 and S3 are preferred instead of sending the information of DC bus voltage to the central DG2. To further analyze the time-delay effect on S2, we vary the droop coefficient of DG2, i.e. M2 within [1, 10]. The results of τ c by CTCR is calculated with c=1. According to the calculation results by CTCR, the kernel delay value τ k decreases from 289.93ms to 116.1ms with the droop coefficients. As the larger M2 represents larger power output, the impact of the time delay will be more serious with the increasing output of DG2. Therefore, the DG2 output should not be too large to keep the system stable. 5. Conclusion This paper analyses the time-delay stability for DC microgrid with chained communication network. Through the derivation of the system model and the application of Cluster Treatment of Characteristic Roots (CTCR), the timedelay issue in the chained communication network of DC microgrid is examined and resolved. It is found out that the time delay would cause the system instability if the selection of DG for DC bus voltage feedback and the corresponding power output allocation are inappropriate. The feedback information is better assigned to the outer DGs and the DGs with less power output, which is helpful for the stability of the overall system. Acknowledgments This work was financially supported by the National High-tech R&D Program of China (2015AA050403) and the National Natural Science Foundation of China (U1766210, 51377117). References [1] Wu Pan, Huang WT, Tai NL, Liang S. A novel design of architecture and control for multiple microgrids with hybrid AC/DC connection. Appl Energy. 2018; 210: 1002-1016. [2] Lotfi H, Khodaei A. Hybrid AC/DC microgrid planning. Energy. 2017; 118: 37-46. [3] Lai JG, Zhou H, Lu XQ, Yu XH, Hu WS. Droop-based distributed cooperative control for microgrids with time-varying delays. IEEE Trans Smart Grid. 2016; 62: 1775–89. [4] Dong CY, Jia HJ, Xu QW, Xiao JF, Xu Y, etc. Time-delay stability analysis for hybrid energy storage system with hierarchical control in DC microgrids. IEEE Trans Smart Grid. 2017; in press. [5] Guo FH, Wen CY, Miao JF, Song YD. Distributed secondary voltage and frequency restoration control of droop-controlled inverter-based microgrids. IEEE Trans Ind Electron. 2015; 62: 4355–4364. [6] Gao QB, Olgac N. Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays. Automatica. 2016; 72: 235– 241. [7] Gao QB, Zalluhoglu U, Olgac N. Investigation of local stability transitions in the spectral delay space and delay space. ASME J of DSME. 2014; 136: 051011. [8] Gao QB, Kammer AS, Zalluhoglu U, Olgac N. Combination of sign inverting and delay scheduling control concepts for multiple-delay dynamics. Systems & Control Letters 2015; 77: 55-62. [9] Gao QB, Olgac N. Optimal sign inverting control for time-delayed systems, a concept study with experiments, International J of Control. 2015; 88: 113-122 [10] Gao QB, Olgac N. Determination of the bounds of imaginary spectra of LTI systems with multiple time delays. Automatica. 2016; 72: 235241. [11] Rekasius, ZV. A stability test for systems with delays. Proc. Joint Automatic Control Conf. 1980; Paper No. TP9-A.