Robust and Scalable Management of Power Networks in Dual-Source ...

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 17, NO. 4, APRIL 2016

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Robust and Scalable Management of Power Networks in Dual-Source Trolleybus Systems: A Consensus Control Framework Di Zhang, Jiuchun Jiang, Senior Member, IEEE, Le Yi Wang, Fellow, IEEE, and Weige Zhang

Abstract—Dual-source trolleybuses powered by onboard battery and grid electricity offer unique advantages in fuel economy, cost reduction, and passenger capacity, which are particularly appealing for public transportation in populated cities. Their mobility and power supply network configurations introduce challenging power management issues on their dedicated supply power grids. To ensure safe, reliable, and efficient operation of trolleybuses, their high-current and dynamic loads must be distributed to the feeders properly and promptly. Based on certain emerging networks of feeders and supply stations for city trolleybus systems, this paper introduces a new framework for current flow balancing based on recently developed weighted-and-constrained consensus control methods to manage the power supply network. Using only neighborhood information exchange among feeder lines in the network, the consensus control can achieve global current balancing with fast convergence to a balanced state, robustness to load perturbations, reconfiguration with feeder addition and deletion, and rebalancing under feeder capacity variation. The methodology is scalable in the sense that system expansion will not substantially increase the control system complexity. The power system configurations of the Beijing dual-source trolleybus system are used for simulation case studies on the new power management methods. Robustness and scalability are demonstrated, together with discussions on the feasibility, flexibility, and implementation issues of the methodology. Index Terms—Dual-source trolleybus, power supply network, consensus control, current flow control, robustness, scalability.

I. I NTRODUCTION

V

EHICLE electrification has emerged as a critical driving force for global green economy. Complementing gradual market penetration of passenger electric vehicles, electrification of public transportation has been pursued with increased Manuscript received July 29, 2015; revised October 9, 2015; accepted October 12, 2015. Date of publication November 2, 2015; date of current version March 25, 2016. This work was supported in part by the Beijing Higher Young Elite Teacher Project under Grant YETP0570 and in part by the United States National Science Foundation under Grant ECCS-1507096. The Associate Editor for this paper was Z. H. Mao. D. Zhang is with Beijing Jiaotong University, Beijing 100044, China, and also with the Department of Electrical and Computer Engineering, College of Engineering, Wayne State University, Detroit, MI 48202 USA (e-mail: dizhang1560@gmail.com). J. Jiang and W. Zhang are with the School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China (e-mail: jcjiang@bjtu. edu.cn; wgzhang@bjtu.edu.cn). L. Y. Wang is with the Department of Electrical and Computer Engineering, College of Engineering, Wayne State University, Detroit, MI 48202 USA (e-mail: lywang@wayne.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITS.2015.2492564

intensity recently [1]. Efficient and reliable electric buses have been investigated as a prominent solution for reducing traffic congestion and air pollution [2], whose urgency has been recognized by severe and frequent haze and PM2.5 exposure in populated cities. For example, the Beijing government has a comprehensive strategic plan for “high-priority development of public transportation” deploying electric buses, as part of sustainable city development. Conventional trolleybuses lack flexibility due to their complete dependence on the electricity lines. In contrast, pure electric buses on battery power enjoy flexibility but their charging and battery swap infrastructures are very expensive and require large space [3]. Consequently, dual-source trolleybuses powered by both battery and grid electricity have been recognized as a promising platform of public transportation. They are allowed to run autonomously without limitations by grid electricity, can operate under different traffic conditions with extended range, have high reliability, and do not require charging or battery swap stations. Similar Online Electric Vehicle (OLEV), developed by South Korea, picks up electricity from power transmitters buried underground, to charge batteries while the vehicle is in motion [4]. While dual-source trolleybuses offer unique advantages in fuel economy, cost reduction, and passenger capacity, their mobility introduces challenging power management issues on their dedicated supply power grids. Desired control and management strategies must offer several features: (1) To ensure safe, reliable, and efficient operation of trolleybuses, their high-current and dynamic loads must be distributed to feeders properly and promptly. Otherwise, the total capacity of the system will be completely limited by the weakest lines or busiest segments. Due to traffic congestions and road conditions, power demands from trolleybuses distribute very unevenly throughout a city. The situation is further compounded by diversified on-board battery state-of-charges on different buses. Proper balancing of loads on feeders ensure safety and maximum utility of the power system capacity. (2) The strategies must be robust against traffic conditions and system status. For example, during peak hours, more buses will join with different densities in different locations; feeder maintenance and protection circuitry conditions imply that feeder capacities can change; under contingency of some parts of the system, the network topology changes. (3) The strategies must have low complexity which will not become overwhelming when the system expands. One implication is that the strategy should be distributed, rather than centralized.

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Based on the emerging configurations of feeders and supply stations for city dual-source trolleybus systems, this paper introduces a new framework for power flow balancing based on recently developed weighted and constrained consensus control methods [5], [6] to manage the power supply network. Here, power balancing on the feeders is formulated as a “consensus” problem; the physical condition that the total power supply equals the total load is an imposed “constraint”; and different capacities of the feeders are modeled by the “weighting.” There are related methodologies for managing power flows in such systems which may be viewed as a special type of micro-grids (MGs). The traditional PQ control methods for large power systems adjust active and reactive power to maintain a desired power factor, achieve load distribution via droop control, and support voltage stability. These methods have been employed in grid-connected MGs, especially during islanding operations [7], [8]. These basic control strategies can be integrated to develop an autonomous management system for MGs [9]. In [10], the concept of frequency-coordinating virtual impedance which enables both time-scale and powerscale coordination is proposed for autonomous control in a distributed micro-grid system. Micro-grid management systems have also been developed by using agent-based technologies [11]. The supervisory controller in [12] employs a formal state machine framework in which maximum utilization of renewable energy sources is pursued. The framework encounters typical complexity issues in state machines and discrete event systems. MG supervisory control in [13] is used as a supervisor layer that interacts with the central control and physical systems via communication interfaces to collect data for control strategy computation and manage mode switching at the physical layer. The framework is of central control types. For networked control systems with applications to MGs, there are general methodologies on both decentralized and centralized control schemes, such as control of distributed generators [14] and global optimization of MGs by centralized control [15]. In [16], a novel wireless load-sharing controller for islanding parallel inverters in an AC-distributed system is proposed, which only uses local measurements of the unit to increase the modularity, reliability, and flexibility of the system. It points out that the control strategies ideally should be implemented in a plug-and-play fashion which requires no prior knowledge on the grid structure and minimal little communications with neighboring power sources [17]. These schemes employ a central processing unit to communicate bidirectionally with resources, which become less desirable when the number of DGs grows high and network interconnections become more complex due to increased data communications, implying more complex information processing and higher cost of equipment and computation devices. For reduced costs and complexity, decentralized control uses local controllers for DGs, which can limit network communications to some common reference signals only [18]. Such schemes must resolve the challenging issues of robustness and global optimization [19], [20]. Our methodology offers some distinct advantages: (1) It uses only neighborhood information exchange among feeder lines in the network; and consensus control can achieve global cur-

rent balancing by using only low-complexity local information exchange. This is especially important for a large network with physically distributed subsystems. (2) It has provable properties of fast convergence to balanced states. (3) Its robustness to load perturbations (trolleybus acceleration, speed variations, charging/discharge of batteries, etc.) allows reconfiguration with feeder addition and deletion (such as maintenance and line failure). It can adapt for re-balancing under feeder capacity variation (if a feeder experiences transient periods after a switch tripping, partial loss of line capacity, etc.). (4) It is scalable in the sense that system expansion will not significantly increase control system complexity. For methodology evaluation, the power system configurations of the Beijing Dual-Source Trolleybus System, which will be in operation in 2017, are used for simulation case studies on the new power management methods. Robustness and scalability are demonstrated, together with discussions on the feasibility, flexibility, and implementation issues of the methodology. The rest of the paper is organized into the following sections. In Section II, the main problems are motivated with a description of the development of dual-source trolleybuses in Beijing. The main network configuration of the supply stations and feeders is defined. Algorithms of the new current flow control framework are presented in Section III. The algorithms are further enhanced by post-iterate averaging for improved convergence under large observation noise. In Section IV, the weighted-and-constrained consensus control method is employed to distribute loads to different feeders whose capacity limits, specified by their overcurrent protection values and other limiting factors, are used to achieve relative (weighted) load balancing. Robustness and scalability of the methodology are discussed in Section V, followed by conclusions in Section VI. II. M OTIVATING S CENARIOS AND S YSTEM C ONFIGURATION In this section, the planned power supply system of the Beijing Dual-Source Trolleybus System is employed to motivate the main problems and to specify a system configuration for methodology development. It should be emphasized that our methodology is not intrinsically limited by the basic scenarios, scopes of case studies, or system configurations. A. Background on the Dual-Source Trolleybus Power Supply Network in Beijing At present, there are 16 dual-source trolleybus lines and more than 588 trolleybuses in Beijing. The total length of the trolleybus routes is about 214 km, which are supplied by 15 power stations. In early 2012, 180 new dual-source trolleybuses have been put into use by the Beijing Public Transport Group, and are demonstrating in #103, #104 and other 3 city bus lines. Extensive operation data and relevant experience have been accumulated during the past two years. To implement the “Beijing Clean Air Action Plan 2013–2017,” the Beijing Public Transport Group has developed a “Bus Group Vehicle Development Action Plan 2013–2017,”

ZHANG et al.: ROBUST AND SCALABLE MANAGEMENT OF POWER NETWORKS IN TROLLEYBUS SYSTEMS

TABLE I T HE E LECTRIC D RIVE B USES D EVELOPMENT P ROGRAMS 2014–2017 IN B EIJING

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TABLE II S ET VALUES OF THE OVERCURRENT P ROTECTION OF S OME S EGMENTS

Fig. 2. The current of different feeder lines in a selected time interval.

node of the network, although the internal structure, circuits, protection system, and load condition can be highly complex. Fig. 1. The structure of trolleybus power supply network.

C. New Power Supply Network in which dozens of bus lines of gasoline or diesel vehicles in the city are expected to be converted to electric drive vehicles. By the end of 2017, 55 new trolleybus lines and more than 2,000 new dual-source trolleybuses will be added. The scope of the dual-source trolleybus development programs 2014–2017 in Beijing is shown in Table I.

B. Present Power Supply Network The old trolleybus power supply network is shown in Fig. 1. The original AC bus voltage of the system is 10 kV; the transformer ratio is 10 kV/485 V; and two non-controlled sixpulse bridges are paralleled and used as a 12-pulse rectifier. Two 12-pulse rectifiers are connected to the DC bus with the rated voltage 650 V and voltage range from 640 V to 680 V in actual operation. The DC bus is linked to different segments by feeder lines. The dual-source trolleybuses in the power supply network can be driven by electricity from either the lines or on-board batteries. There are 130 km supply lines which are divided into 75 segments powered by the feeder lines. The trolleybuses pass them by inertia and each of the segments is segregated. 4–8 power segments are connected to a power station, while one station’s supply radius is about 2 km. The feeder line is about 1.5 km, and the vehicle line of each segment is about 1–2 km. Since each segment supplies several trolleybuses, the network structure is very complex. Viewing from the terminal of a feeder line, a segment can be represented abstractly as one

To ensure the operation security of dual-source trolleybuses and the power supply network, more advanced overcurrent protection systems are added in the new system. When the duration of overcurrent in one feeder line is more than the set value, the switch would trip, and the dual-source trolleybuses of this segment would have to drive by battery until the switch recloses. The set values of the overcurrent protection for six representative segments are listed in Table II. There are different numbers of trolleybuses in different segments, which are in different operation conditions. The currents of different feeder lines in a selected time interval of three hours are shown in Fig. 2. We can see that the current of each feeder line fluctuates rapidly and substantially. There is obvious imbalance of currents among different feeder lines, which may severely limit the total number of dual-source trolleybuses that the power supply network can serve. Clearly, if one can balance the currents among the feeder lines, the feeder peak currents can be reduced substantially to the “Average” curve shown in Fig. 2, with an improvement of more than 40% on the overall capacity of dual-source trolleybuses that the same power supply system can serve without increasing the power ratings. Since the existing power supply network uses relatively backward technology which was built in the 60 s and 70 s, and can’t support balanced power operation, we have proposed to the Beijing authority to build new distributed stations with the voltage source PWM rectifiers, instead of the centralized station. With interconnected segments shown in Fig. 3, the system power losses can be greatly reduced. On the other hand,

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Fig. 4. Information network among segments in one power supply station.

Fig. 3. Diagram of the new trolleybus power supply network.

one can balance the power output of each converter by using the neighboring PWM rectifiers’ information so that the constructed power supply network can serve as many trolleybuses as possible. This new power supply system configuration will be implemented in the new system under the Beijing Public Transport Group. In the subsequent sections, a new control strategy will be introduced to manage such networked supply power systems. As a general framework, we present general methodologies first, followed by case studies on a special station of the Beijing Trolleybus System. III. P OWER F LOW M ANAGEMENT FOR U NIFORM S UPPLY N ETWORKS : C ONSTRAINED C ONSENSUS We start our methodology development for uniform power supply networks in which all feeder lines have identical characteristics. Its extension to non-uniform power supply networks will be presented in Section IV. A. Basic Control Structures A power supply network contains r segments. Under a distributed communication scheme and distributed control strategy, the segments are linked by communication systems, forming a network topology. A simple example of a 6-feeder network and information exchange topology is shown in Fig. 4. In this topology, each segment (including PWM rectifier, feeder line, and vehicle line) is represented by one node. Suppose that there are r segments (nodes) in one power supply station, and each node knows the state information (the current of the feeder line) of its linked neighbors. By certain rules in adjusting its own state using the local information, all nodes are controlled to achieve a global balance, namely, the currents of feeder lines are equally dispatched. In the power supply network, the real-time current of the i-th feeder line is xi (t) at time t, and the initial current of the i-th feeder line is xi0 . We control the output current of

the i-th converter with the information of neighboring converters to achieve global balance, xi (t) → β, i = 1, . . . r r namely i for the average β, β = i=1 x0 /r. Denote the state vector x(t) = [x1 (t), . . . , xr (t)] , where the prime means transpose of the vector. The nodes are linked by an information topology, represented by a directed graph ζ whose element (i, j) (called a directed edge from node i to node j) indicates an information communication line between node i and node j., (i, j) ∈ ζ indicates estimation of the state xj by node i via a communication link. For node i, (i, j) ∈ ζ is a departing edge and (j, i) ∈ ζ is an entering edge. The total number of communication links in ζ is ls . From its physical meaning, node i can always observe its own state, which will not be considered as a link in ζ.

B. Control Algorithms and Convergence Properties In a power supply network with r nodes, for a selected time interval τ for current balancing updates, the consensus control is performed at the discrete-time steps nτ , n = 1, 2, . . .. At the n-th control step, the state value will be denoted by xn = [x1n , . . . , xrn ] . Power flow control updates xn to xn+1 by the amount un (1)

xn+1 = xn + un

where un = [u1n , . . . , urn ] is the node control. In this control problem, a current (load) transfer λij n (called link control) from node i to node j at the n-th step is the decision variable. The node control amount uin is determined by the link control amount λij n via the following relationship uin = −

(i,j)∈ζ

λij n +

λji n.

(2)

(j,i)∈ζ

The most relevant constraint in this control scheme is that for all n, ⎧ r ⎨ xi = L n (3) i=1 ⎩ i xn ≤ Iset , i = 1, 2, . . . , r


where L is the total load (the summation of all currents from the feeders), and Iset is the set value of the overcurrent protection of the feeder line. j A link (i, j) ∈ ζ entails an estimate x îj n of xn by node i with ij observation noise dn j ij x îj n = xn + dn .

continuous-time Markov chain is irreducible if the system of equations

(5)

where H1 is an ls × r matrix whose rows are elementary îj then the l-th vectors such that if the l-th element of ηn is x row in H1 is the row vector of all zeros except for a “1” at the j-th position. Each link in ζ provides the local balancing error îj information δnij = xin − x n , an estimated difference between i j xn and xn . This information may be represented by a vector δn of size ls containing all δnij in the same order as ηn . δn can be written as δn = H2 xn − ηn = H2 xn − H1 xn − dn = Hxn − dn (6) where H2 is an ls × r matrix whose rows are elementary îj then the l-th vectors such that if the l-th element of ηn is x row in H2 is the row vector of all zeros except for a “1” at the i-th position, and H = H2 − H1 . Due to network constraints, the information δnij can only be used by nodes i and j. When the power control is linear, time ij invariant, and memoryless, we have λij n = μn gij δn where gij is the (local) link control gain and μn is a global time-varying scaling factor which will be used in state updating algorithms as the recursive step size. Let G be the ls × ls diagonal matrix that has gij as its diagonal element. In this case, the node current control becomes un = −μn H Gδn . For convergence analysis, we note that μn is a global control variable and we may represent un equivalently as un = −μn H G(Hxn − dn ) = −μn (H GHxn − H Gdn ) (7)

with M = −H GH and W = H G. This, together with (2), leads to xn+1 = xn + μn (M xn + W dn ).

(9)

has a unique solution, where the row vector ν = [ν 1 , . . . , ν r ] ∈ R1×r with ν i > 0 for each i = 1, . . . , r is the associated stationary distribution and 1 is the column vector of all 1’s. Under Assumption (A0), we can show that (1) M has rank r − 1 and is negative semi-definite, and (2) M is a generator of a continuous-time Markov chain, and is irreducible; see [21] for a proof. Assumption (A1): (1) The step size satisfies the following conditions: μn ≥ 0, μn → 0 as n → ∞, and n μn → ∞. (2) The noise {dn } is a stationary φ-mixing sequence such that Edn = 0, E|dn |2+Δ < ∞ for some Δ > 0, and that the mixing measure φñ satisfies ∞

Δ

φñ(1+Δ) < ∞

(8)

Assumption (A0): (1) All link gains are positive, gij > 0; (2) ζ is strongly connected. ˜ = (˜ Recall that a square matrix Q qij ) is a generator of a continuous-time Markov chain if q ˜ ≥ 0 for all i = j and ij q ˜ = 0 for each i. Also, a generator or the associated ij j

(10)

k=0

⎧ ⎨φñ =

(1+Δ)

sup A∈F n+m

= μn (M xn + W dn )

˜=0 νQ ν1 = 1

(4)

Let ηn and dn be the ls dimensional vectors that contain all ij x îj n and dn in a selected order, respectively. Then, (4) can be written compactly as ηn = H1 xn + dn

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(1+Δ)

E (2+Δ) |P (A|Fm )−P (A)| (2+Δ)

⎩F = σ{dk; k < n}, n

F n = σ{dk; k ≥ n}.

(11)

It is noted that the mixing condition on the noise allows the observation noises to be correlated, extending the case of independent and identically distributed (i.i.d.) noises. Theorem 1: Under Assumption (A1), the iterates generated by the stochastic approximation algorithm (8) satisfies xn → β1 with probability 1 (w.p.1), as n → ∞. C. Simulation Case Study To illustrate and evaluate the above methodologies, we will use the scenario of a station with 6 feeders, as shown in Table II. Its communication network topology is assumed to be the one in Fig. 4. Although the current limits on this system are different (see the line for “Junbaose” segment), in this simulation study, we assume that all lines are to be operated within the limit of 1100 A. As Fig. 4 shows, there are communication lines between 1 and 2, 2 and 3, 2 and 4, 4 and 5, 5 and 6 in this power supply network. The initial currents (not balanced) of feeder lines are x10 = 960 A, x20 = 1050 A, x30 = 680 A, x40 = 800 A, x50 = 780 A, x60 = 920 A. As a result, r = 6, ζ = { (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (4, 5), (5, 4), (5, 6), (6, 5) }, x = [x1 , x2 , x3 , x4 , x5 , x6 ] . The new advanced technology of converters can respond to commands as fast as 10 ms. As a result, τ = 25 ms is selected so that all control updates will be completed before the next step. From the initial current distribution (not balanced), the average current is. Hence, the goal of consensus control is to achieve xi (t) → 865 A. By choosing the order for the links as (1,2),

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Fig. 5. Current flow control simulation trajectories. (a) Current distribution trajectories. (b) Consensus error trajectories.

Fig. 6. Current flow control simulation under large observation noise. (a) Current distribution trajectories. (b) Consensus error trajectories.

(2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (4, 5), (5, 4), (5, 6), (6, 5), we have x ˜ = [ˆ x12 , x ˆ21 , x ˆ23 , x ˆ32 , x ˆ24 , x ˆ42 , x ˆ45 , x ˆ54 , x ˆ56 , xˆ65 ] , ⎡ ⎤ 0 1 0 0 0 0 ⎢1 0 0 0 0 0⎥ ⎢ ⎥ ⎢0 0 1 0 0 0⎥ ⎢ ⎥ ⎢0 1 0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 1 0 0⎥ ⎢ ⎥ H1 = ⎢ 1 0 0 0 0⎥ ⎢0 ⎥ ⎢0 0 0 0 1 0⎥ ⎢ ⎥ ⎢0 0 0 1 0 0⎥ ⎢ ⎥ ⎣0 0 0 0 0 1⎦ 0 0 0 0 1 0 ⎤ ⎡ 1 0 0 0 0 0 ⎢0 1 0 0 0 0⎥ ⎥ ⎢ ⎢0 1 0 0 0 0⎥ ⎥ ⎢ ⎢0 0 1 0 0 0⎥ ⎥ ⎢ ⎢0 1 0 0 0 0⎥ ⎥ ⎢ H2 = ⎢ 0 0 1 0 0⎥ ⎥ ⎢0 ⎢0 0 0 1 0 0⎥ ⎥ ⎢ ⎢0 0 0 0 1 0⎥ ⎥ ⎢ ⎣0 0 0 0 1 0⎦ 0 0 0 0 0 1

D. Post-Iterate Averaging for Improved Convergence Under Large Observation Noise

and H = H2 − H1 . Suppose that the control gains on the links are selected as g12 = g21 = 10, g23 = g32 = 12, g24 = g42 = 15, g45 = g54 = 13, g56 = g65 = 16. Then G = diag[10, 10, 12, 12, 15, 15, 13, 13, 16, 16]. As a result, M = −H GH (equation shown at the bottom of the next page.) Based the conditions above, simulation is performed in which the link observation noises are i.i.d. sequences of Gaussian noises with mean zero and variance 6. The simulation results are shown in Fig. 5, the subplot (a) shows how currents of different feeder lines whose initial currents are different from each other are gradually distributed to the desired consensus. The subplot (b) shows the consensus error trajectories which reduce to zero, implying that the currents of different feeder lines become balanced.

The basic stochastic approximation algorithm (8) demonstrates desirable convergence properties under relatively small observation noises. However, its convergence rate is not optimal and it is sensitive to large noises. When noises are large, its convergence may not be sufficiently fast and its states show fluctuations. For example, for the same system as in the previous case study, if the noise variance is increased from 6 to 100, its state trajectories demonstrate large variations, as shown in Fig. 6. To improve accuracy and convergence rate, a post-iterate averaging step is added, resulting in a two-stage stochastic approximation algorithm. For definiteness and simplicity, take μn = 1/nα for some 0.5 < α < 1. The algorithm is modified to xn+1 = xn + n1α M xn + n1α W dn (12) 1 1 x¯n+1 = x ¯n − n+1 x ¯n + n+1 xn+1 . The above algorithm belongs to the class of iterate averaging procedures. The idea is that in the first stage one obtains a rough estimate using large step sizes so that the iterations will reach the vicinity of the true parameter fast. Then a second stage of averaging is added, which will smooth out the iterations, resulting in smaller variances. Such an algorithm is known to achieve asymptotic optimality; see [22] for the work on consensus-type algorithms and [23, Chapter 11] for general framework, references, and further details. Strong convergence of the averaged x ¯n follows from that of xn . This is stated in the following theorem with its proof omitted. Theorem 2: Suppose the conditions of Theorem 1 are satisfied. For iterates generated by algorithm (12), xn → β1 w.p.1 as n → ∞. We now use the same case-study system to illustrate the effectiveness of post-iterate averaging. Suppose that the link observation noises are i.i.d. sequences of Gaussian noises of mean zero and variance 100. Now, the consensus control is


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Define the weighting (capacity) coefficients as γ = [γ 1 , . . . , γ r ] , and the state scaling matrix ψ = diag[1/γ 1 , . . . , 1/γ r ]. Then the goal of current flow control is to achieve consensus on weighted currents xi /γ i , namely, xi (t)/γ i → β, for some constant. It can be easily derived that r r i x0 / γi. (13) β= i=1

i=1

Accordingly, some of the other matrices also need to be modified as follows ˜ n = Hxn − ψd ˜ n δn = H2 ψxn − ψη

Fig. 7. Current flow control simulation with post-iterate averaging. (a) Current distribution trajectories. (b) Consensus error trajectories.

expanded with post-iterate averaging. Fig. 7 shows the current distribution trajectories. The current distributions converge to the consensus faster with much less fluctuation. IV. W EIGHTED AND C ONSTRAINED C ONSENSUS C ONTROL A. Algorithms of Weighted and Constrained Consensus Control The currents of feeder lines in Section III are balanced for uniform supply networks. However, the set values of feeders are often different from each other in power supply networks. This may stem from uneven segments, different projected loads, various estimated traffic conditions. In addition, during realtime operations, operating conditions of different feeders may experience variations, such as temperature, protection circuit tripping, overload protection condition modifications, etc. The set values of the feeders will be termed as the “capacities” of the nodes in the network topology. Under non-uniform node capacities, the control objective is modified so that the relative current flows on the feeders to their capacities are balanced. We now use the same power supply network to derive the weighted-and-constrained consensus control algorithms.

(14)

where the link scaling matrix ψ˜ is the ls ×ls diagonal matrix whose k-th diagonal element is 1 if the k-th element ˜ îj of ηn is x n ; and H = H2 ψ− ψH1 . In this case, define J = H2 −H1 . Then the node current flow control becomes un = −μn J Gδn , and ˜ n) un = −μn J G(Hxn − ψd ˜ n) = −μn (J GHxn − J Gψd = μn (M xn + W dn )

(15)

˜ This, together with (1), with M = −J GH and W = J Gψ. leads to xn+1 = xn + μn M xn + μn W dn .

(16)

B. Simulation Case Study For comparison and evaluation, the same scenario of a station with 6 feeders is used, and its communication networks and network topology are assumed to be the one in Fig. 4. To illustrate the full capability of the methodologies, more diversified feeder capacities than Table II are used to capture potential larger variations in capacities due to construction and operating conditions. We now use the same dual-source trolleybuses power supply network with feeder capacity values γ = [1100, 1200, 1300, 1200, 1400, 1100] . Then ψ = diag[1/1100, 1/1200, 1/1300, 1/1200, 1/1400, 1/1100]. Accordingly, ψ˜ = diag[1/ 1200, 1/1100, 1/1300, 1/1200, 1/1200, 1/1200, 1/1400, 1/ 1200, 1/1100, 1/1400]. We can calculate the desired weighted

⎤ −20 20 0 0 0 0 ⎢ 20 −74 25 30 0 0 ⎥ ⎥ ⎢ ⎢ 0 24 −24 0 0 0 ⎥ ⎥ M = −H GH = ⎢ ⎢ 0 30 0 −56 26 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 26 −58 32 ⎦ 0 0 0 0 32 −32 ⎡ 10 −10 0 0 0 0 0 ⎢−10 10 12 −12 15 −15 0 ⎢ ⎢ 0 0 −12 12 0 0 0 W = H G = ⎢ ⎢ 0 0 0 0 −15 15 13 ⎢ ⎣ 0 0 0 0 0 0 −13 0 0 0 0 0 0 0 ⎡

0 0 0 −13 13 0

0 0 0 0 16 −16

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ −16⎦ 16

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Fig. 8. Current flow control with weighted consensus.

Fig. 10. Robustness of current distribution against sudden current changes.

Fig. 9. Weight current and error norms with weighted consensus control. (a) Weighted current distribution trajectories. (b) Consensus error trajectories.

Fig. 11. Robustness of current distribution against sudden current changes. (a) Weighted current distribution trajectories. (b) Consensus error trajectories.

consensus is x1 = 782.1 A, x2 = 853.2 A, x3 = 924.2 A, x4 = 853.2 A, x5 = 995.3 A, x6 = 782.1 A. Then with the same control G, we can calculate and get the H, J, M , and W . Fig. 8 shows the current distribution trajectories of feeder lines which are gradually distributed according to the capacities of the feeder lines. Fig. 9(a) illustrates that the weighted currents converge to a constant; and the weighted consensus error trajectories reduce to zero, as shown in the subplot (b). V. ROBUSTNESS AND S CALABILITY A. Robustness to Disruption of Currents of Feeder Lines When several trolleybuses enter or exit one or more of the segments in the power supply network, or when some trolleybuses speed up or slow down, there would be a sudden current change of the feeder lines which would perturb the network from its consensus. Then current flow control will re-distribute through the network to reach a new equilibrium of consensus. We now use the same system to illustrate the robustness against current disruptions. Suppose that at t = 10 s, a sudden increase of 100 A occurs on feeder line 1. Consensus control

then distributes it fairly from line 1 to the other feeder lines in the power supply network. Figs. 10 and 11 show the current and weighted current distribution trajectories. They confirm that the current distributions can converge to the new consensus in a short time period after current disruption. B. Scalability There will be rapid expansion of dual-source trolleybuses in the subsequent several years. Consequently, there will be more segments. When new segments are added to the network, the network topology will change. In the neighborhood-based network control method, an addition or deletion of segments will only affect some neighboring nodes. In fact, all other nodes will never be aware of changes in other parts of the network. However, by iterative control, the total load will still be properly distributed throughout the entire network. Similarly, a new segment (a new feeder line) will get its fair share of the load through the consensus control, even though all other nodes not in its neighborhood implement the exactly same control strategies as before.


1037

Fig. 12. An expanded power supply network from 6 nodes to 7 nodes.

To be more concrete, suppose that a new segment, labeled r + 1, is added to the network with its own initial current xr+1 . 0 Assume that the new node is linked to node r only. Then, by (2), the control of node r will be modified from the original λrj λjr (17) xrn+1 = xrn − n + n (r,j)∈A

Fig. 13. Current dispatch control under network topology changes.

(j,r)∈A

to a slightly modified scheme r,r+1 xrn+1 = xrn − λrj λjr + λr+1,r n + n − λn n (r,j)∈A

(j,r)∈A

(18) with all other nodes control functions unchanged. The additional communication resources will be limited to the communication channel between node r and node r + 1. This distributed and scalable control strategy is essential for network operations in reducing communication requirements and control complexity. Consider the same case-study system as before. Suppose that at t = 10 s, a new segment is added to the network. The new feeder line has its initial current 700 A and is on node 7 which is linked to node 6 only. This addition results in a network topology change, leading to the new matrices derived below. The expanded network is shown in Fig. 12. As a result, ξ = {(1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (4, 5), (5, 4), (5, 6), (6, 5), (6, 7), (7, 6)}, and the capacity of this new segmentation is supposed to be 1300 A. The new state vector is x = [x1 , x2 , x3 , x4 , x5 , x6 , x7 ] . By choosing the order for the links as (1,2), (2,1), (2,3), (3,2), (2,4), (4,2), (4,5), (5,4), (5,6), (6,5), (6,7), (7,6), we have x ˜= ˆ21 , xˆ23 , x ˆ32 , x ˆ24 , xˆ42 , x ˆ45 , x ˆ54 , x ˆ56 , x ˆ65 , x ˆ67 , xˆ76 ] and [ˆ x12 , x ˜ H, J can be generthe corresponding matrices H1 , H2 , ψ, ψ, ated accordingly. The new consensus current is x1 = 753.4 A, x2 = 821.9 A, x3 = 890.3 A, x4 = 821.9 A, x5 = 958.8 A, x6 = 753.4 A, x7 = 890.3 A. Suppose that the control gains on the new links are g67 = g76 = 15. Then G = diag[10, 10, 12, 12, 15, 15, 13, 13, 16, 16, 15, 15], which defines M and W . After the new segment is added into the system, consensus control distributes currents fairly to all feeder lines in the new power supply network. Figs. 13 and 14 show the current and weighted current distribution trajectories. Initially, the feeder

Fig. 14. Current dispatch control under network topology changes. (a) Weighted current distribution trajectories. (b) Consensus error trajectories.

line 6 changes most, due to its direct link to the new line 7. But, afterward the control adjusts loads to other feeder lines throughout the entire network. The current distributions converge to the new consensus. VI. C ONCLUSION This paper introduces a new control methodology for current balancing in new dual-source trolleybuses power supply networks. This methodology is based on the weighted-andconstrained consensus control that can achieve global current balancing in the entire power supply network using only neighborhood communications. Post-iterate averaging under large observation noise can improve the convergence. The robustness of this control framework can effectively support systems with feeder current fluctuations. The scalability of our control framework can enhance the flexibility of power supply networks without increasing communication, control, and computation complexities, which is exactly what our intelligent public transportation system needs. This control methodology can also be used in other related power management problems, especially micro-grids with distributed generators.

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Di Zhang received the B.S. degree in electrical engineering from Beijing Jiaotong University, Beijing, China, in 2011. He is currently working toward the Ph.D. degree in electrical engineering with Beijing Jiaotong University. He is also currently a Visiting Scholar with the Department of Electrical and Computer Engineering, College of Engineering, Wayne State University, Detroit, MI, USA. His research interests include the charging management of electric vehicles, and systems analysis, modeling, and simulation, with applications in energy systems, transportation systems, and microgrids.

Jiuchun Jiang (M’10–SM’14) received the B.S. degree in electrical engineering and the Ph.D. degree in power system automation from Beijing Jiaotong University, Beijing, China, in 1993 and 1999, respectively. He is currently a Professor with the School of Electrical Engineering, Beijing Jiaotong University. His research interests include battery application technology, electric vehicle charging stations, and microgrid technology. Dr. Jiang was the recipient of the National Science and Technology Progress Second Award for his work on EV bus systems and the Beijing Science and Technology Progress Second Award for his work on EV charging systems.

Le Yi Wang (S’85–M’89–SM’01–F’12) received the Ph.D. degree in electrical engineering from McGill University, Montreal, QC, Canada, in 1990. Since 1990, he has been with Wayne State University, Detroit, MI, USA, where he is currently a Professor with the Department of Electrical and Computer Engineering, College of Engineering. His research interests include complexity and information, system identification, robust control, H∞ optimization, time-varying systems, adaptive systems, hybrid and nonlinear systems, information processing and learning, and medical, automotive, communications, power systems, and computer applications of control methodologies. Dr. Wang was a Keynote Speaker in several international conferences. He serves on the IFAC Technical Committee on Modeling, Identification and Signal Processing. He was an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL and several other journals, and he is currently an Associate Editor of the Journal of System Sciences and Complexity and the Journal of Control Theory and Applications.

Weige Zhang received the M.S. and Ph.D. degrees in electrical engineering from Beijing Jiaotong University, Beijing, China, in 1997 and 2013, respectively. From 1993 to 1994, he was an Engineer with Hohhot Railway Bureau. From 2013 to 2014, he was a Research Assistant with the Department of Electrical and Computer Engineering, University of Michigan–Dearborn, Dearborn, MI, USA. He is currently an Associate Professor with the School of Electrical Engineering, Beijing Jiaotong University. His research interests include battery pack application technology, power electronics, and intelligent distribution systems.