Distributed Control Strategy Based on a Consensus Algorithm for the Inter-cell and Inter-cluster Voltage Balancing of a Cascaded H-Bridge Based STATCOM

Cascaded H-bridge converters are widely used in the implementation of medium voltage static synchronous compensators (STATCOMs). This is because of the advantages of relatively high-power density and the improved utilisation of low-voltage semiconductor devices. Major concerns with this topology are (i) to ensure a proper balance of the individual DC capacitor voltage in its cells, and (ii) the balancing of the average DC voltage between its clusters. In the research literature, these issues are typically addressed by using centralised control approaches, meaning that both an extensive processing capability and multiple digital outputs and communication channels for the switching signals are required, increasing system complexity. In contrast to this trend, in this paper, a distributed control scheme based on a consensus algorithm is proposed to deal with these issues in a three-phase STATCOM based on a cascaded H-bridge converter. The main advantages of the proposed control scheme are: (i) it does not require a centralised controller, since the cells work autonomously in a cooperative fashion to achieve voltage regulation, distributing the control effort among the cells, and (ii) it increases the fault tolerance of the converter and thus, the reliability of the system by adequately considering its redundancy. Extensive simulation work is provided to validate this proposal, and the characteristics described above.


I. In t r o d u c t io n
The Cascaded H-Bridge (CHB) multilevel converter, illustrated in Fig. 1, also known as Single-Star Bridge-Cells (SSBC) [1], is one of the most utilised topologies for STATCOM applications. As depicted in Fig. 1, this topology considers three clusters composed of cascade-connected Hbridge cells. The fundamental challenge of this topology is the control of every floating capacitor voltage to the desired voltage level. To this end, the control of the floating capacitor voltages is typically addressed into two-stage, the balancing of the average value of the cluster capacitor voltages (inter cluster) and the balancing of the capacitor voltages belonging to the same cluster (inter-cell). Traditionally, these concerns have been addressed in the literature by using centralised control schemes. This means that all the quantities (capacitor voltages, currents, etc.) required for implementing the control systems of the converter are sent to a central controller, in charge of processing all this information and controlling the power converter [1]. However, in recent years, distributed control approaches have been studied intensively (e.g. [2]). In this case, a centralised controller is not necessary owing to the control effort is distributed along with the cells of the converter, as shown in Fig. 1. Under this distributed approach, the cells of the converter can operate cooperatively to obtain global objectives such as the inter-cell and inter-cluster voltage balancing schemes. This approach has the following advantages: better reliability, flexibility, scalability, plug-andplay operation, and tolerance to single-point failures [3].
implementation are discussed. Section IV presents the simulation results, and section V presents the conclusions of this work.

A. Communication Structure and Consensus Algorithm
A distributed communication network is required for the implementation of the proposed distributed cooperative control scheme. The bidirectional network considered in this work is modelled as an undirected graph G = (N , A) among the cells JV = {1, , where is the set of communication links and A is the non-negative N X N weighted adjacency matrix. The elements of are /iy = , with if and only if {i,j} e £ [17]. In this sense, let consider the case where all the N nodes of the graph G have scalar first-order single-integrator dynamics: where and u £ 6 R. Also, let assumes that x £ denotes the value of some quantity of interest at node . It is said that the variables x£ achieve consensus if [*i(t) -xj(ty\ -> o as t -> oo . Therefore, the consensus can be achieved via a feedback loop by applying the protocol u £ given by (2) [18] [19]. This control is distributed in that it only depends on the immediate neighbours of node in the graph topology. This protocol is known as a local voting protocol [18] [19].

Ui = -j r y .
Atf ■ ( * -* ; ) When the consensus is achieved, the quantity of interest xt of each node of the system will be equal to each other and they will converge to the average of the initial states x £(0) as follows [18] [19]: It is worth to highlight that the consensus of variables x £ is achieved if and only if the graph G is a spanning tree [18] [19].
In (2), parameters fry are associated with the distributed communication network, whereas /q is a gain that modifies the transient behaviour of the controller [17] [20]. Fig. 2 shows the consensus-based distributed controller. Finally, based on (6), it can be concluded that the inter-cell capacitor voltage control loop shown in Fig. 3 can be represented by the consensus form given by (2). Thus, the inter-cell voltage balancing can be performed in a distributed manner.

B. Relation Between Consensus Theory and the Typical Control System for Inter-Cell Voltage Balancing
In the operation of the CHB-based STATCOM, two control systems are used for capacitor voltage balancing purposes. The first one is for achieving the inter-cells voltage balancing (see Fig. 3), whereas the second one aims to balance the average DC voltage between the clusters of the converter (see Fig. 4). In both cases, proportional controllers are used (see Fig. 3 and Fig. 4, respectively). Also, it must be noted that the plants associated with each control loop have the form given by (1), i.e., they are integrative plants. [21] C. Relation Between Consensus Theory and the Typical Control System for Inter-Cluster Voltage Balancing Fig. 4 shows the inter-cluster voltage balancing control loop of the converter. Note that this control system uses v c as reference, which is calculated as the average of the DC voltage of the three clusters that composed the STATCOM. A proportional controller is used to control the cluster voltage v Cx to vc [21]. The output of the controller is multiplied by the line current i*, producing the control action v£x to regulate the inter-cluster DC voltages. In Fig. 3, the typical control scheme for inter-cell voltage balancing for the CHB-based STATCOM is depicted. As seen, the cell average voltage v* in cluster "je" is calculated and used as the reference value of every cell associated with that cluster. A proportional controller is used to drive the voltage in each cell of the cluster (u*.) to the reference value vx. Finally, the output of this controller is multiplied by the line current in that cluster ¿*, producing a voltage variation vxf l for each cell of the cluster "x ", which is added to the output of the overall balance loop. From Fig. 3, the error signal is determined as: The previous equation can be rewritten as: In Fig. 4, the average cluster voltage vc is calculated as (8). In this equation, vC 7-(withy = 1,2,3) corresponds to the average DC voltage in each cluster, calculated as the average of the voltages of the capacitors of the cells in that cluster. (8) Vc~3 L°c i 7=1 In this case, considering that vCl = v Ca , v C2 = v Cb and Vc3=vCc the error signal can be expressed as Analogously, the error signal can be rewritten as By using (5) and taking into account the control scheme shown in Fig. 3, the final voltage variation vxf l for each cell of cluster "x " is given by: Finally, the power flow generated to achieve the inter cluster voltage balancing is given by: In t e r -c e l l a n d In t e r -Clu s t e r Vo l t a g e Ba l a n c in g In the previous section, it was demonstrated that both the inter-cell and inter-cluster capacitor voltage balancing control loops depicted in Fig. 3 and Fig. 4 can be written in terms of a consensus algorithm (2). In particular, the inter cell controller can be implemented by using the consensus algorithm (6). Moreover, the inter-cluster controller can be implemented via the consensus algorithm (11). Note that so far, (6) and (11) do not consider the communication topology between the elements of the system (adjacency matrix in (2)). To do that, let consider a graph G as that defined in section II-A. For the inter-cell consensus algorithm (6), the graph G is defined as follows: a graph where the cells of a particular cluster are the vertices and the communications links among them are the edges. Examples of possible graphs are depicted on the top of Fig. 5 (graphs (a)-(c)). In this figure, if the controller uses the graph (a) means that the average of all cells of the cluster is considered. On the other hand, if graph (b) is used, means that the controller taking into account the average of a portion of the voltages of the cells that composed that cluster. Finally, if the graph (c) is considered, it means that the controller is taking the average of just the neighbouring cells voltage.
In a similar way, graph (d) shown on the top of Fig. 5 shows a communication topology that can be used to implement the inter-cluster consensus algorithm (11).
Finally, the consensus algorithms (6) and (11), considering a distributed communication network, look like (13) and (14) respectively. It worth remembering that in (13) represents the elements of the adjacency matrix (AceU) used to model the inter-cell communication topology. Similarly, ^ are the elements of the inter-cluster adjacency matrix (Acluster).
It should be highlighted that in this paper unless otherwise stated, the inter-cell communication topology is that represented by the graph (a) (see on the top of Fig. 5), thus Acea is given by (15). Moreover, the communication topology used to implement the inter-cluster distributed controller is that given by graph (d) (see the top of Fig. 5), being (16) the mathematical representation of the adjacency matrix Acluster. It should be noted that gain of (13) and (14) are time-variant because the line current is a periodic quantity. However, as is discussed in [16], these gains only affect the convergence time of the controller [20], and they will not impact the consensus.

Acell
The implementation of the proposed consensus-based distributed controller is depicted in Fig. 5. In this scheme, three control systems can be identified. The controller labelled as overall balance loop is in charge of controlling the reactive power (Q*) injected by the STATCOM to the system and to perform an overall voltage control [22]. The inter cluster control loop aims to regulate the mean-DC voltage among the clusters of the STATCOM. The inter-cell controller regulates the individual DC-voltage in each capacitor of each cluster. Finally, the output signals cluster ( v auv b i, v ci for clusters a , b and c respectively) of this control system are sent to the modulation stage, where the phase-shift PWM technique is used [23].

IV. Sim u l a t io n Resu l t s
In this section, the performance of the proposed control scheme shown in Fig. 5 is evaluated. To do this, the system depicted in Fig. 1 is simulated by using PLECS software, with the parameters displayed in TABLE I. The performance of the proposed controller is verified for the following scenarios: (i) performance of the control system front of a cell failure, (ii) the effects of communication time delays in the communication network, and (iii) performance of the controller in front of failures on the communication links. It should be highlighted that for the simulation tests shown in this section unless otherwise stated; the inter-cell communication topology is that represented by the graph (a) (see the top of Fig. 5). Equation (15) shows the adjacency matrix ( Aceu ) associated with this graph. Similarly, the communication topology used to implement the inter-cluster distributed controller is that given by the graph (d) (see the top of Fig. 5). Equation (16) shows the mathematical representation of this matrix. In this test, the performance of the proposed distributed control scheme is evaluated. In particular, a failure (a short circuit) in the cell a 4 (see Fig. 1) is emulated. In this case, cell a 4 is bypassed. Thus, the a-phase cluster continues working with just three cells after the failure. Four steps are considered in this test: step 1 (0s^t<5s), where only the overall balance control loop is working to regulate the reactive power at Q*=10MVar and at 3750V; step 2 (5s^t<10s), where the proposed inter-cell control loop is enabled (see Fig. 5). In step 3 (10s^t<15s), the inter-cluster control loop is enabled to achieve an equal mean DC voltage among the cluster of the STATCOM. Finally, at the beginning of step 4 (15s^t<20s) a failure in cell a 4 is emulated, thus, that cell is bypassed. Fig. 6(a) shows the voltage in each one of the twelve cells that compose the STATCOM of Fig. 1 during the four steps studied in this work. In step 1, it can be noted that these voltages are not equal due to the different capacitances used and their unequal initial voltages: if these voltages are not controlled, they will diverge until the system becomes unstable. At the beginning of step 2 (and onwards), this trend is changed, and now, the cell voltages are regulated in each cluster. Note in step 2 that, the cell voltages in a particular cluster reach a different convergence voltage compared with the other clusters because the STATCOM is connected to the system through unequal line impedances, which produce unbalanced voltages at its output. This behaviour is shown in steps 1-2 depicted in Fig. 6 (b), where the mean DC voltage of each cluster is depicted (during the whole test). Indeed, in steps 1 and 2 (when the consensus algorithm (14) is disabled), there is an unequal average DC voltage among the clusters. This is overcome, at the beginning of step 3 (and onwards), where the proposed inter-cluster control loop is working. In step 3 of Fig. 6 (a)-(b), both consensus algorithms are working, and it can be concluded that they are effective for balancing both cell and cluster voltages of the STATCOM. In this situation, at the beginning of step 4 (t=15s), a failure of cell a 4 is emulated. In this test, it can be concluded that despite this critical scenario, the inter-cell and inter-cluster voltage balancing is continuously achieved by the proposed control scheme (see Fig. 5). In step 4 of Fig. 6(a), the three cells in the a-phase cluster reach a higher convergence voltage than that achieved in step 3. This is because these cells have to supply the voltage that was provided by cell a 4 before its failure. Note that in this case, the proposed consensus algorithms autonomously calculate a new operating point in the cells of the STATCOM to achieve the consensus aims.
It should be noted that in this scenario (step 4), the STATCOM continues injecting the reactive power references (Q*=10MVar) to the system, as shown in Fig. 7. It should be pointed out that in step 4, it was assumed that the STATCOM has cell redundancy in its clusters.
Finally, Fig. 8 and Fig. 9 show the voltage at the output of the STATCOM for step 3 and step 4, respectively. It worth remembering that during step 4, the cluster "a" is composed just by three cells since a failure in cell a 4 is emulated (it is bypassed from that cluster). By comparing the output voltage depicted in Fig. 8 (before the failure), it is seen that voltages of the three phases are composed of nine levels. On the other hand, from Fig. 9, it is appreciated that the voltage of cluster "a" (where the failure occurs), the voltage is composed just by seven levels. From these figures, it is concluded that even this critical situation, the distributed control scheme ensure a balanced output voltage for the STATCOM.

B. Test scenario 2: Performance of the distributed controller in front o f communication time delays in the communication network
To analyse the performance of the distributed controllers (13) and (14) in front of delays in the communication network, a communication time-delay r is introduced, as shown in (17) and (18) respectively. This means, that the consensus algorithm (17) in cell " ¿" receives the information from the rest of cells with a time-delay equal to r. The similar occurs for the inter-cluster consensus algorithm (18). The performance of controllers is analysed for the following cases: r = 0.1s a n d r = 0.7s. ;=i At the beginning of this test, just the overall control system (see Fig. 5) is working to regulate at 3750V and Q* to 5MVA. Then, at t=5s, the inter-cell (17) and inter-cluster consensus algorithm (18) are simultaneously enabled. Finally, at the t=10s, the reactive power reference Q* is changed to 10MVA. Fig. 10(a) shows the cell voltage and Fig. 10(b) shows the mean DC voltage in the clusters; both traces are getting for a time-delay equal to 0.1s. Fig. 11 shows the same information but for a time-delay of 0.7s. From these figures, it is concluded that the consensus of voltages is achieved for the two cases considered in this test ( r = 0.1s and r = 0.7s). Note that, for the largest time-delay (t = 0.7s), the consensus is achieved after some oscillations. From this test, an excellent performance of the consensus algorithms (13) and (14) is concluded, in terms of time delays.
Finally, Fig. 12, shows both the active and reactive power for the case r = 0.7s. From this figure, it is concluded that even though the communication time-delay, the STATCOM is able to feed the system with reactive power.

C. Test scenario 3: Performance o f the distributed controller in front offailures on the communication network links
In this test, the performance of the proposed consensusbased distributed control system is evaluated in front of failures in the communication links. At the beginning of this test, just the overall control system (see Fig. 5) is working to regulate at 3750V and Q* 5MVA. Then, at t=5s, the inter cell (13) and inter-cluster consensus algorithm (14) are simultaneously enabled. It is worth remembering that in this case, the inter-cell communication topology is that given by the graph (a) in Fig. 5, and the inter-cluster communication topology is that shown by graph (d) on the top of Fig. 5. Afterwards, at t=10s, a communication link failure in both graphs is emulated. Fig. 13 and Fig. 14 show, respectively, the inter-cell and inter-cluster communication topologies before and after the communication link failure. Finally, at the t=15s, the reactive power reference Q* is changed to 10MVA.   V. Co n c l u s io n s Two consensus-based distributed controllers have been proposed and validated by simulation, one for balancing the inter-cell voltages, and the other one to control the inter clusters mean-voltage in an MMCC STATCOM. The proposal was verified by considering the following cases: (i) performance of the control system front of a cell failure, (ii) the effects of communication time delays in the communication network, and (iii) performance of the controller in front of failures on the communication links. The performance of the consensus-based distributed controller was excellent. As future work, the experimental validation of the proposal along with its extension to be used in more complex control systems.