A Path to Configurable Solid State Transformers and Energy Routers: Introduction to Modular Active Cell Control

1 A Path to Configurable Solid State Transformers and Energy Routers: Introduction to Modular Active Cell Control Raffael Schwanninger, Niklas St¨ ocklein, Nikolai Weitz, Xiaotian Yang and Martin M¨ arz Abstract—This paper introduces a control strategy for a Modular Multi-Active Bridge (MMAB) architecture that extends the single phase shift as a novel approach to enable fully decentralized control of the Modular Active Cell (MAC) on a single, arbitrarily large multi-winding transformer. The MAC aims to provide a fully modular, configurable, and scalable solution for Solid State Transformers (SSTs) and Energy Routers, enabling full power and voltage variability in isolated interconnections of diverse DC networks. A novel modeling approach based on discrete-time state-space is introduced. The model is verified through measurements and simulation, showing high accuracy and prediction capability. Index Terms—Multi-input-multi-output (MIMO), multiport energy router, multicell, multi-active bridge (MAB), power electronics building block, solid state transformer (SST). I. INTRODUCTION The global energy landscape is undergoing a significant transformation, marked by the increasing adoption of direct current (DC) networks across various sectors. Applications such as electric vehicle (EV) fast-charging [1]–[3], data centers [4]–[6], naval electrification [7]–[9], renewable energy integration [10]–[12], industrial automation [13]–[15], and intelligent medium voltage direct current (MVDC) distribution grids [16]–[18] are driving the demand for efficient and flexible DC power distribution systems. These DC networks offer advantages in terms of efficiency and compatibility with modern electronic loads as well as higher cost efficiency in distribution [19]–[21]. Although all of these applications are similar, they vary greatly in terms of voltage and power requirements. To enable a swift transition to renewable energies, modular solutions are therefore necessary to limit the required engineering effort. Thus, there is a pressing need for advanced power electronic architectures that can provide configurable, efficient, and isolated capabilities to facilitate the seamless integration and operation of diverse DC networks [22], [23]. The simplest case of this modularity can be found in power electronic building blocks (PEB), which were introduced to focus on the optimization of only a few identical building This work was conducted within the project ECS4DRES. ECS4DRES is supported by the Chips Joint Undertaking under grant agreement number 101139790 and its members, including the top-up funding by Germany, Italy, Slovakia, Spain and The Netherlands. Raffael Schwannninger, Niklas St¨ ocklein, Nikolai Weitz and Martin M¨ arz are with Institute of Power Electronics at the at the Friedrich-Alexander-University, 90429 Nuremberg, Germany (e-mail: [email protected]; [email protected]; [email protected], [email protected]). Xiaotian Yang is with the Fraunhofer Institute for Integrated Systems and Device Technology IISB, 91058 Erlangen, Germany (e-mail: [email protected]). blocks, enabling faster and more efficient design of power systems [24], [25]. An early example is [26], where a PEB is developed to simplify the need for external communication and synchronization in bidirectional switches in matrix converters. The concept of modularity is extended in [27] with the modular multilevel converter (MMC). For the MMC, modularity allows for voltage and power scalability in conversion from AC to DC through identical sub-modules. Depending on configuration, these sub-modules allow high blocking voltages through serialization, or high currents through parallelization. To allow for galvanic isolation and high conversion ratios, this modular sub-module approach has been applied to Solid State Transformers (SST) as well [28]–[32]. To enable the direct connection of Low Voltage Direct Current (LVDC) to MVDC, a connection to a transformer can be added to each MMC module. The modules can either be connected to LVDC with multiple transformers, or with a single large transformer connecting all modules to multiple LVDC ports. As shown in [33], the benefits of a shared transformer are better voltage balancing, higher power density and a generally higher efficiency. On the other side of the power and voltage spectrum, multiport converters or energy routers are investigated in [34]– [39], these allow for Multiple Input Multiple Output (MIMO) connection of renewable energy sources and storage systems to DC microgrids. The cells of this MIMO converter are reconfigurable, allowing for serialization and parallelization to realize different voltage and current specifications. This allows for a single optimized power electronics design to be applied to diverse applications such as server racks [40], modular battery storage [41], vehicular applications [35], and renewable energy systems [42]. Im im Vmvm Cell m Isolation MCU im _ Q1 vm,Q1 Cell m+1 Cell m-1 im _ * (a) Conventional: Centralized Im im Vmvm MCU Cell m im _ Q1 vm,Q1 im _ * (b) Proposed: Decentralized Fig. 1: Control strategy of an MAB: Placement of the MCU Although many different topologies for use in more or less modular concepts exist [34], Dual Active Bridge (DAB) based solutions, as initially introduced in [43], are prime This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 2 candidates due to their bidirectionality and ease of control. If more than two active bridges are connected to a multi-winding transformer, the DAB becomes a Multi-Active Bridge (MAB). One cell of such an MAB is shown in Fig. 1. Whether the MAB is used on Energy Routers [37], [38] or SSTs [33], all cells require a central control unit (MCU), which sets the phase-shifts between all connected cells. As indicated in Fig. 1a, this MCU therefore needs to communicate control signals to the cells. These need to be calculated accordingly from the voltage and possibly even current measurements within each cell. Therefore, measurement signals need to be measured locally and fed back to the central MCU. In terms of modularity, this centralized control is unfavorable, since each added cell increases the load on the MCU. Additionally, the necessary wiring to connect all modules to one controller scales likewise with the number of modules. In MVDC applications, costly signal transmission over MV isolation barriers needs to be implemented to maintain the galvanic isolation, adding additional cost for the low latency communication required [44], [45]. Ideally, each cell would be controlled by its own MCU as shown in Fig. 1b. This MCU controls the cell locally, generating gate signals based on the cells measured voltage and current. The only communication to other cells or a central controller would be through the exchange of slow moving set-values where the increased delay of cost effective solutions [46] does not matter for primary control stability. While distributed control with primary local controllers have been investigated [47], [48], SSTs with DABs still need to break the primary to secondary side isolation barrier with low latency. 20 µs/div 20 µs/div Fig. 2: Experimentally measured DAB transformer current with implemented single phase shift without a central control unit This necessity can easily be showcased by two cells, each controlled by its own MCU connected to form a DAB. Fig. 2 shows the measured transformer current of a DAB in the case of a decentralized control strategy with an initially fixed phase shift of approx. 20◦. While this phase shift might be initially constant between switching cycles, even minute differences in the MCUs local oscillators will lead to the two cells eventually de-synchronizing. In consequence, the phase shift increases linearly with no stable operating point. Furthermore, even setting an initial phase shift is not easily possible, as there is no information shared regarding the phase of the two cells. This effect is worsened in an MAB, as all cells will de-synchronize, requiring some external master-signal to keep all cells in lockstep. To reduce or even fully mitigate the need for any external communication, this paper introduces a control strategy that extends the single phase shift as a novel approach to enable a fully Modular Multi-Active Bridge (MMAB). The MMAB consists of Modular Active Cells (MAC), which can operate independently on a shared transformer. The MAC aims to provide a modular, configurable, and scalable solution for SSTs and energy routers, enabling efficient and isolated interconnection of diverse DC networks. Instead of controlling the current flow from cell to cell by setting a phase shift, the switching frequency of each MAC is varied slightly based on its local control law. This frequency variation then induces a phase shift between the MAC and all other MAC on the transformer. To allow designers to design stable local control, we develop a discrete-time state-space model for the MAC on a multi-winding transformer. Although discrete-time models have been developed before for general power converters [49]– [54] as well as DAB [55]–[60], the model developed in this paper does not use averaging of continuous-time models, but rather develops the discrete-time model ab initio, as it directly results from the digital control method. The main contribution of this paper can be summarized as: •The introduction of MAC control •The description of the MMAB with a linear discrete-time state-space model •Introduction of a stability criterion for an MMAB under MAC control •Identification of causes for instability As many readers might not be familiar with the discretetime state-space approach, this paper introduces the basic and simplified control strategy independent of the modeling in Section II. The individual subsections of Section III will introduce the intended control scheme alongside the discretetime state-space model step by step with increasing complexity. Measurement results are shown for each step to act as verifications, as well as examples. Section IV finally acts as a full verification of the developed model. As this paper is intended to introduce the fundamentals of MAC control, topics concerning efficiency optimization, ideal transformer design, or even regular operation in SST or energy router applications are not extensively studied in this paper. Section V therefore discusses some of the many questions that remain open before the MMAB can be used effectively in any of its intended applications. II. FUNDAMENTALS OF MAC CONTROL Before developing the complete control for the MAC in Section III, this section should act as a simplified introduction to the control scheme. Therefore, the basic control strategy is explained using a DAB, which is shown in Fig. 3 as a simplified and lossless equivalent circuit. The phase shift This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 3 between both full bridges is given by (1) with t1and t2being the points in time of the rising edge of v1and v2, respectively. Tsis the time of a switching period. I1 i1 V1v1vL N:1 I2 V2 v2 L i1 _ i2 _ Fig. 3: Lossless equivalent circuit of a DAB with definitions of voltages and currents. φ12 =φ1−φ2= (t2−t1)·ωs(1) ωs=2π Ts = 2πfs(2) The power transfer from cell 1to 2and the current ¯ i1 is determined by (3) and (4), respectively, both having a parabolic dependency on φ12. P12 =V1·NV2 ωs·L·φ12 ·(1 −|φ12| π)(3) ¯ i1=NV2 ωs·L·φ12 ·(1 −|φ12| π)(4) v1 i2 uL ~~ L i1 v2 Fig. 4: First harmonic approximation of a DAB with N= 1 To explain the control principle, Fig. 3 can be simplified using the first harmonic approximation (FHA) as shown in Fig. 4. To simplify the equations, Nshould be equal to 1. The fundamental frequency equals the initial switching frequency ω0. Furthermore, using complex values in a coordinate system that rotates with ω0defined by Re′(v) = Re(v)·ejω0t(5) and Im′(v) = Im(v)·ejω0t, (6) oscillations with ω0can be described as a phasor at a fixed phase angle. An arbitrary state of the DAB with power transfer from v1to v2is shown in Fig. 5a. The voltages are depicted as complex phasors v1and v2. If v1and v2are inequal in phase or length, the complex currents i1and i2are non-zero. For the proposed control scheme, the active current ¯ imat each cell is of interest. In the phasor diagram, the active currents Im Re v1 v2 vL i1 𝜑2 𝜑1 𝜑12 ' ' (a) Voltage and current phasors Im Re v1 i1 i2 ' ' v2 i1 _ i2 _ (b) Obtaining the active current Fig. 5: Voltage-phasors v1and v2of a DAB in the rotating complex plane Re′and Im′using the FHA at an arbitrary phase angle φ12. The measured active currents are projections of the complex i1and i2onto their respective voltages. ¯ i1and ¯ i2can therefore be determined as the projection of the complex currents i1and i2onto their respective voltages, as depicted in Fig. 5b. The fundamental principle of the proposed control strategy is based on changing the phase angle of the phasors by adjusting ωs,m of cell m. The assumption for this is, that a current ¯ i∗ mcan be measured, which represents ¯ imsufficiently as explained later in Sec. III-A. Because of ˙φm=dφm dt=ωs,m (7) the desired phase shift and therefore the state of the DAB can be achieved by ˙φ12 =dφ12 dt=ωs,1−ωs,2. (8) Thus, a change in phase is obtained by integration of the difference in frequency. A control loop with a P controller with a constant gain KPcan be formed as shown in Fig. 6, which here is referred to as the source control. This controller adjusts the ωs,m of its own cell mfrom an offset value ω0 based on the measured current ¯ i∗ mand a set value iset,m: ωs,m =ω0+KP·(iset,m −¯ i∗ m)(9) _ iset,1 P𝜑12 𝜑1f( ) 𝜑12 DAB i1 _ 𝜑2 𝜑12 _ i1 _ * 1 s Fig. 6: Source control loop with a P controller. The DAB can be modeled as a frequency integrator leading to a change in phase and a nonlinear function translating phase to active current. ˙φ1and ˙φ2represent the change in phase due to the switching frequency of cells 1and 2. The DAB itself can be modeled as a subtraction of angular frequencies followed by an integration to obtain the phaseshift and the DAB transfer function (4). The DAB transfer This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 4 function outputs the transformer current ¯ i1as a change in the operating point of the DAB. Implementing this control on both sides of the DAB in Fig. 4 with a setpoint of 0 A results in decreasing φ1due to the negative error (Fig. 7a). φ2is increased, respectively. This concludes in a reduction of φ12 until the steady state (Fig. 7b) is reached with ˙φ12 =φ12 = 0. (10) Both full bridges are switching synchronously with fs. In this case, no active power is transmitted. Im Re v1 vL i1 ' ' |𝜑1| |𝜑2| v2 (a) Initial state Im Re v1 vL i1 ' ' v2 (b) Steady state Fig. 7: Phasors of two cells in source control. The measured currents will lead to frequency adjustments (gray arrows). The change in frequency moves v1and v2closer, reducing φ12. In steady state, both v1and v2are in phase and no active current is transferred. To achieve stationary accuracy with one full bridge at a defined current, it is necessary to implement a PI controller with an additional integrator gain KI. ωs,m =ω0+KP·(iset,m −¯ i∗ m) + KIZ(iset,m −¯ i∗ m)dt(11) This control loop can be interpreted as load control and shown in Fig. 8. PI _ iset,2 𝜑21 𝜑2f( ) 𝜑21 DAB i2 _ 𝜑1 𝜑21 _ i2 _ * 1 s Fig. 8: Load control loop consisting of a PI controller and DAB model. Implementing this controller on the v2side of Fig. 4 with a setpoint of ¯ iset,2<0 A and the starting point of synchronous full bridges (Fig. 7b) results in the response shown in Fig. 9a. The full bridge represented by v1remains with source control. Therefore, φ2is decreased by the PI controller. Because of the integral part the PI controller ensures stationary accuracy. ωs,2will therefore stay different from ω0even if no remaining error is present. At i1the error remains such that φ1is reduced constantly. All phasors rotate with a fixed frequency within the itself rotating coordinate system. Fig. 9b shows this rotating steady state at an arbitrary point in time. While the switching Im Re v1 vL i1 ' ' |𝜑1|=0 |𝜑2| v2 (a) Reaction of controller Im Re v1 vL i1 ' ' |𝜑| |𝜑| 𝜑12 i2 v2 i2=i2,set _ i1=i1,set _ (b) Steady state: Constant rotation with |˙φ| Fig. 9: Principle of load control at full bridge 2 (source control at full bridge 1). Load control will force a constant φ12 to maintain a ¯ i2. In steady state (b), all phasors rotate at a fixed frequency in the itself rotating coordinate system, resulting in a change of switching frequency depending on the load current. frequency fsis lower than the initial switching frequency fs,0, the phase shift φ12 remains constant. In contrast to (10), with load control, the phase shift is φ12 = 0. (12) However, since ˙φ1= ˙φ2= ˙φ, (13) following equation must hold: ˙φ12 = 0. (14) Fig. 10 shows an exemplary experimentally measured behavior of a DAB in source and load control. Cell 1 and cell 2 are configured with a source (setpoint to 0 A) and load control, respectively. To the zero point in time, the set point of the load steps from 0 A to −1 A. Due to the control error, cell 2 adapts its phase with the change of frequency (7). Therefore, a difference in the frequencies is present as shown in Fig. 10, which leads to a change in the phase shift (8) and in turn also in the transferred current (4). In steady state, (12), (13) and (14) are valid. tin µs fs,min kHz i1in A 𝜑12 in ° Fig. 10: Experimentally measured current of a DAB configuration with cell 1 and cell 2 as source and load control, respectively. Frequencies and phase shift are calculated from the measured gate signals v1,Q1 and v2,Q1 This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 5 As the control scheme is based on both cells constantly changing their frequency to control the current, a slight offset in clock frequency in the respective controllers has little effect. Therefore, the control can be fully decentralized. Communication of the state of the DAB is transmitted by the transformer current. The control strategy can also be extended on multi-winding transformers, as explained in the next section. The closest analogy to MAC control is the operation of synchronous generators (SG) in an AC grid. With SGs the phase shifts and power transfer are induced by slight variations in rotational frequency as a result of loading. For the MMAB, the switching frequency therefore plays a role very similar to the rotational frequency of an SG, changing phase shifts and again power flow based on measured current. III. DISCRETE-TIME MODELING OF AN MMAB SYSTEM L11 L14 L34 L12 L13 L24 L23 1:N2 N4:1 N3:1 v1 v2 v4 v3 i4 i3 i2 i1 imag Fig. 11: ECM for a transformer with four windings/cells. Each interconnection of two cells is characterized by a stray inductance Lab. The magnetizing inductance of all cells are included in L11. To determine stable control parameters for each cell, a statespace model of the MMAB is required. In this paper, the transformer is described by the Extended Cantilever Model (ECM) as described in [61], as it is often used in multi port transformers [37], [38], and shown for a four cell transformer in Fig. 11. In the ECM, each cell ais connected to all other cells bthrough an inductance Lab. Transformation ratios are included as Nain each cell, apart from cell 1, where N1can be set to 1. The magnetizing inductance of all cells is included in L11. The magnetizing inductance Laa as seen from any other cell aneeds to be calculated from L11,Naand the series parallel connection of all paths from cell ato cell 1. The total magnetization current imag is therefore the sum of the individual magnetization currents imm of all cells on an M-winding transformer. imag = M X m=1 imm (15) An M-winding transformer can be described as a simple linear time-invariant (LTI) system, with interconnections between each winding. The following subsections will therefore deal with modeling the non-linear time-variant active bridges on the LTI transformer. As the analysis behind the control scheme might not be trivial to every reader, we will introduce it step by step during the next subsections. The final control scheme and accompanying model are given in Subsection III-E. A. State-Space Model of Active Currents in Trapezoidal Mode As explained in Section II, each cell of the MMAB should be controlled by its active input current ¯ i. As ¯ iais not directly measurable by cell a, the MAC control instead relies on sampling the transformer current iaat least once every switching period, as indicated in Fig. 12. For the following subsections, each switching period will be referenced by an index k. Sampling the current in each half period will be introduced in Subsection III-E. Time t Voltage Current Δt2,k/2 0 0 0 Δt2,k T2,k δt12,k i2,k _ *i2,k+1 _ * Fig. 12: Fundamental waveforms of a two cell MMAB. The sampling in cell 2 is indicated by the vertical black arrows. The phase shift δt12,k is adjusted by this sampled current through prolonging both half-waves by ∆t2,k/2. The resulting current ¯ i∗ a,k measured, is the sum of the currents from ato every other cell: ¯ i∗ a,k = M X m=1 ¯ i∗ am,k (16) The magnetization inductance is not included in the considerations for this subsection. Therefore, ¯ i∗ aa = 0 (17) is applied. For the following subsection, all cells are assumed to be operating in trapezoidal mode, which is defined as the absolute values of the phase angles between all cells are lower than π/2. If iais therefore sampled exactly at the center of the first half-cycle in the period k– as indicated by the black arrow in Fig. 12 for cell 2 – the individual currents between the cells aand bcan be written as: ¯ i∗ ab,k =NaNbVb Lab ·δtab,k (18) We call ¯ i∗ ab,k the dressed active currents, as they are a representation of the bare active currents ¯ iab,k. Compared to ¯ iab,k in (4), ¯ i∗ ab,k is calculated with the absolute time lag / lead δtab,k between two phases, as phase angles φab and angular frequencies ωsare ill defined in variable frequency operation. Unlike centralized control schemes δtab,k is adjusted by cells aand bthrough each cell changing its own cycle duration Ta,k This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 6 and Tb,k by ∆ta,k and ∆tb,k, respectively. This results in a changed ¯ i∗ ab,k+1 at the next sampling: ¯ i∗ ab,k+1 =¯ i∗ ab,k −NaNbVb Lab ·(∆ta,k −∆tb,k)(19) This effect is depicted by the dashed lines in Fig. 12. Assuming a simple proportional controller with gain KP,a in cell a, ∆ta,k will be determined from the sum of all partial currents and the cells set current value iset,a,k. ∆ta,k =−KP,a ·(iset,a,k − M X m=1 ¯ i∗ am,k)(20) The equations (16), (19) and (20) can therefore be reformulated as a discrete-time state-space representation consisting the state transition matrix Afor the state vector and the input matrix Bfor the set-point vector.     ¯ i∗ 1,k+1 ¯ i∗ 2,k+1 ... ¯ i∗ M,k+1     =A    ¯ i∗ 1,k ¯ i∗ 2,k ... ¯ i∗ M,k     +B    iset,1,k iset,2,k ... iset,M,k     (21) The B-matrix elements are given as: bij =−KP,j ·NiNjVj Lij (22a) bii = M X m=1 KP,i ·NiNmVm Lim (22b) The A-matrix elements can be calculated from: aij =KP,j ·NiNjVj Lij (23a) aii = 1 − M X m=1 KP,i ·NiNmVm Lim (23b) As the magnetization inductance Laa does not pay a role in active power transfer and as assumed in (17) its current ¯ i∗ aa is – assuming no DC bias – ideally 0 at each sampling, we will initially assert: Laa =∞(24) The effect of DC bias and its mitigation will be discussed in Subsection III-E. The dynamics of the system can then be quantified by the eigenvalues λiof A. If only stability is concerned, the necessary stability condition must hold: |λi|<1(25) When using (23) and (24) to construct A, there will always be at least one λiviolating (25) by being equal to 1. λ1= 1 (26) This boundary stable eigenvalue indicates a neutral mode state component. In this case, the neutral mode is an equivalence of the sum of energy flow into the transformer. Different from the physical transformer, the description used above allows for the sum of all applied and drawn power not being equal to 0 in a lossless transformer. Therefore, to model the transformer physically correct, the following condition must hold: M X m=1 ¯ i∗ m,k ·Vm·Nm= 0 (27) To bring the model closer to reality, a reduced system ARS can be defined, so that the state transitions are still correct, when replacing Awith ARS.     ¯ i∗ 1,k+1 ¯ i∗ 2,k+1 ... ¯ i∗ M,k+1     =ARS     ¯ i∗ 1,k ¯ i∗ 2,k ... ¯ i∗ M,k     +B    iset,1,k iset,2,k ... iset,M,k     (28) Aside from the last row, the ARS-matrix elements are still defined the same way as for the A: aRS,ij =aij (29) To obtain physical accuracy, (27) can be used to describe the current in the last cell M: ¯ i∗ M,k ·NM·VM=− M−1 X m=1 ¯ i∗ m,k ·Nm·Vm(30) The state progression from ¯ i∗ M,k to ¯ i∗ M,k+1 therefore needs to be adjusted. The ARS-matrix elements of the last row aRS,Mj can be calculated from the sum of all other elements in the same column: aRS,Mj =− M−1 X m=1 amj (31) For a M×MARS, the Mth row is therefore given as: aRS,Mj =−NjVj NMVM +KP,j ·NMNjVj LMj (32a) aRS,MM =− M X m=1 KP,i ·NMNMVm Lim (32b) Since this reduces the degree of freedom by 1, there will always be an eigenvalue λRS,1of ARS in vanishing mode: λRS,1= 0 (33) The asymmetry in currents will accordingly vanish in a single cycle. For stability analysis, the relevant eigenvalues of Aand ARS are identical. Only for steady state analysis, as discussed in Section III-F, the difference in the first eigenvalue becomes relevant. Accordingly, a deeper discussion can be found there. If individual cells operate as constant current loads, their controllers integrator behavior can be modeled just as easily by including an integrated error εIwhich propagates as: εIa,k+1 =εIa,k + (iset,a,k −¯ i∗ a,k)(34) εIa,k therefore behaves like an additional state. Weighted with an integrator gain KI,a, its effect on ∆ta,k can be included into the control equation. ∆ta,k =−KP,a ·(iset,a,k −¯ i∗ a,k)−KI,a ·εIa,k (35) This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 7 In general, Aand Bcan now be extended to include all εI, with KI,a being set to 0, if the integrator is not in effect.         ¯ i∗ 1,k+1 ... ¯ i∗ M,k+1 εI1,k+1 ... εM,k+1         =A         ¯ i∗ 1,k ... ¯ i∗ M,k εI1,k ... εM,k         +B    iset,1,k iset,2,k ... iset,M,k     (36) For an Mcell transformer, Awill be a 2M×2Mand Ba 2M×Mmatrix. For sake of simplicity, the integrator states will be omitted in the following sub-sections. B. Example: Stability-Analysis of a Dual Active Bridge To test the approach in practice, the MAC control is implemented on two cells forming a DAB as shown in Fig. 13. It is important to note, that the full control scheme used to generate the results in this subsection will be explained later in Subsection III-E. The results shown in this subsection should therefore be seen as qualitative examples. A full comparison of model and measurement is shown in Subsection IV-A. Cell 1 1 2 Cell 2 Fig. 13: Two MACs forming a DAB. Each MAC is connected to the shared transformer and controlled by its own MCU (red board). For the DAB, only a single coupling inductance L12 as well as the transformation ratio N2must to be considered for the transformer model. For the in-lab example, L12 was determined to be 63 µHwith N2≈1. Each cells input voltages V1and V2were set to 30 V. Both cells are controlled with their own MCUs without data connection between them. The cell controllers are configured with only a proportional component KP,1and KP,2. The Amatrix is therefore given as: A= 1−KP,1·N2V2 L12 KP,2·N2V2 L12 KP,1·N2V1 L12 1−KP,2·N2V1 L12 !(37) Solving for the eigenvalues λleads to: λ1= 1 (38a) λ2= 1 −K1·N2V2+K2·N2V1 L12 (38b) As mentioned before, λ1is boundary stable, but not physically relevant for the system. For λ2to remain stable, the following inequality must hold: 0<K1·N2V2+K2·N1V1 L12 <2(39) If the reduced state equations are used, ARS is written as: ARS = 1−K1·N2V2 L12 K2·N2V2 L12 −V1 N2V2+K1·N2V1 L12 −K2·N2V1 L12 !(40) In this case, λRS is given as follows: λRS,1= 0 (41a) λRS,2= 1 −K1·N2V2+K2·N2V1 L12 (41b) As intended, λRS,1is 0 and the eigenvalues λ2and λRS,2are equal. To investigate the system, KP,1and KP,2were adjusted to result in different λ2of 0.6,0,−0.6and −1.2while applying a fixed step response of 0.5 A. To obtain a stable initial OP, KP,2was set to a fixed stable value and KP,1,iset,1, and iset,2 were set to 0. The step response was then triggered through setting iset,1to 1 A and KP,1equal to KP,2simultaneously. This allows for the same operating point (OP) current iOP of ±0.5A for each measurement, while also being able to change from stable to unstable OP in the case of λ2=−1.2. The parameters for the step response comparison are summarized in Tab. I. TABLE I: Step response comparison: Setting iset,1to 1 A and KP,1equal to KP,2simultaneously resulting in differently damped systems KP,1in ns/AKP,2in ns/Aλ2State Fig. 0→466 466 0.6 overdamped 14 0→1166 1166 0 critically damped 15 0→1866 1866 -0.6 underdamped 16 0→2799 2799 -1.2 unstable 17 The step response for λ2of 0.6is shown in Fig. 14. As expected, the MMAB finds a stable OP of ±0.5A. As λ2 is greater than 0, the OP is reached monotonously without oscillation. The phase shift φ12 therefore takes multiple cycles to increase to its final value. tin µs imin A vm,Q1 in V 𝜑12 in ° Fig. 14: Experimental measurement results for λ2= 0.6on a step response of 0.5 A. In Fig. 15, KP,1and KP,2were selected to realize a λ2of 0. The OP is now reached after a single switching period T. In terms of dynamics, this can be seen as optimal MAC control. In terms of susceptibility to noise, higher dynamics will lead to more oscillations during normal operation. In terms of phase This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 8 tin µs imin A vm,Q1 in V 𝜑12 in ° Fig. 15: Experimental measurement results for a calculated λ2= 0 on a step response of 0.5 A. shift, it can be observed that KPhas no impact on the steadystate value. Increasing KP,1and KP,2further leads to a negative λ2. The resulting step response for λ2=−0.6is shown in Fig. 16. As expected, the step response now leads to overshoot. Still, the MMAB will find a stable OP. In practice, λbelow 0 are not advised. tin µs imin A vm,Q1 in V 𝜑12 in ° Fig. 16: Experimental measurement results for λ2=−0.6on a step response of 0.5 A. The current shows overshoot due to the negative eigenvalue. Once λ2<−1, the system becomes unstable. In Fig. 17, λ2is increased from −0.1to −1.2. The MMAB therefore no longer maintains a stable OP. According to the model, the currents would oscillate with increasing amplitude. As the state equations are accurate only for absolute phase shifts below π/2, the model cannot accurately describe the physical system anymore. For larger absolute phase shifts, the MMAB enters triangular mode, which will be explained in the next subsection. Instead of ever increasing amplitude, the MMAB continuously shifts between trapezoidal and triangular modes in a nonlinear oscillation pattern. Aside from describing the effects different eigenvalues have tin µs imin A vm,Q1 in V 𝜑12 in ° Fig. 17: Experimental measurement results for λ2=−1.2on a step response of 0.5 A from a system with λ2=−0.1. The system oscillates unstably after the step response. on the system, the findings also show the consequence of parameter mismatch. As given in (41), the eigenvalues depend on the control parameters, the cell voltages, and the coupling inductance. Instabilities due to control parameter mismatch are caused when the eigenvalue crosses −1. Assuming one set of control parameters for the whole operating range, the most critical point occurs at maximum cell voltages and the lowest inductance. While the maximum cell voltages should be known for any application, the coupling inductance may be misestimated during the design process, or shift in value due to core-temperature. As long as the expected eigenvalues stay positive, even larger variations in coupling inductance, e.g. L12 being halved, will not cause instability. It is still advised to properly characterize the transformer in all operating conditions before setting final control parameters. C. State-Space Model of Active Currents in Triangular Mode As observed in the unstable DAB example, the developed model is only accurate while the absolute phase shift of two cells is below π/2. For higher absolute phase shifts, the two cells enter triangular mode. In triangular mode – using the same sampling point in the center of the positive half-wave as in trapezoidal mode – the state ¯ i∗ ab,k can be written as: ¯ i∗ ab,k =NaNbVb Lab ·(−δtk)(42) Compared to the trapezoidal mode (Eq. 18), the sign of δtk is flipped. Similarly, the state transition equation also differs only in sign. ¯ i∗ ab,k+1 =¯ i∗ ab,k +NaNbVb Lab ·(∆ta,k −∆tb,k)(43) To integrate the change in sign, an additional viable ζab can be added for each port combination aand b.ζab changes sign, once the absolute value of φab crosses the π/2threshold. ζab = sign(cos(φab)) (44) Different from the linear discrete time system introduced above, the inclusion of ζab requires knowledge of the current This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 9 OP of the system. A method for finding ζab will be discussed in Section III-F. For a generalized solution, Aand Bcan be adjusted for the possible mode change: aij =ζij Kj·NjVj Lij (45a) aii = 1 − M X m=1 ζim Ki·NmVm Lim (45b) bij =−ζij Kj·NjVj Lij (45c) bii = M X m=1 ζim Ki·NmVm Lim (45d) When only considering a DAB, MAC control can be set up to operate only in triangular mode by selecting negative values for KP,1and KP,2. As shown in Fig. 18, the control behaves exactly as in trapezoidal mode, albeit with the frequency and, therefore, phase shift decreasing in a positive step response. tin µs imin A vm,Q1 in V 𝜑12 in ° Fig. 18: Experimental measurement results of the operation and step response of two MAC with negative KPvalues operating in triangular mode. In general, the MMAB cannot operate in triangular mode between all cells. Even for three cells connected to one transformer, a phase shift of πbetween cells 1 and 2 as well as πbetween 1 and 3 would already lead to a phase shift of 0 between cells 2 and 3. Therefore, a control scheme based only on negative KPto force triangular mode operation would not be possible in multi-cell transformers. In contrast, individual cells of the MMAB entering triangular mode during regular operation is possible. This can occur, if the load current between two cells is increased above a limit. If the system remains stable needs to be investigated depending on the new state transition matrix. The effect will be shown exemplarily in the next subsection. D. Example: 4-Port Stability under Different Phase Angles To investigate the effect of mode changing, a 4-cell MMAB is tested on a single transformer as shown in Fig. 19. Each cell is still controlled by its own MCU with no data connection between any of the cells. The cells are controlled with proportional controllers with 167 ns/Aeach and connected to the same 30 V voltage source. Cell 1 1 2 3 4 Cell 2 Cell 3 Cell 4 Fig. 19: 4 cells of an MMAB on a shared transformer. Each cell is controlled by its own MCU. The transformer is modeled with the equivalent circuit shown in Fig. 11, with the measured values given in Tab.II. TABLE II: Measured inductance values, transformation ratios, and coupling factors Variable Value L11 in µH670 N21.012 N31.047 N41.047 Variable Value L12 in µH39.6 L13 in µH-380.0 L14 in µH90.7 L23 in µH90.4 L24 in µH-391.0 L34 in µH40.1 Variable Value k12 0.9743 k13 0.9442 k14 0.9496 k23 0.9504 k24 0.9443 k34 0.9758 The negative inductance for L13 and L24 is due to the modeling approach. In practice, every measurable inductance – every inductance that can be measured from any cell a to cell b– is positive, as every negative inductance in the model has smaller positive inductances in parallel. Readers too uncomfortable with negative inductance can use the coupling factors kab in Tab. II. The resulting system stays the same. The transformer was not designed for optimal operation for a 4-cell MMAB, but rather to showcase operation between differently coupled cells. °° ° Fig. 20: Eigenvalues of ARS under different phase shifts. While φ13 greater 90° still leads to a stable system, both φ13 and φ23 greater 90° leads to a single unstable λgreater 1. Initially, the set values for all cells are 0 A. The system eigenvalues using ARS are shown as blue circles in Fig. 20. A step response is now applied to the set value of cell 3. Due to the lowest coupling between cell 1 and cell 3, φ13 will be This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 16 clock frequency between individual controllers renders setting fixed phase shifts between cells impossible, MAC control uses slight variations in switching frequency to continuously adjust phase shift between cells. Power flow is therefore controlled by the ensemble of individual cells changing their individual switching frequency according to their control law. An MMAB with MAC control therefore behaves very closely like a centrally controlled MAB with single phase shift in terms of efficiency and voltage gain. To allow for stable readjustment of frequencies, a digital control is developed. The control scheme is introduced step by step with increasing complexity. In the simplest case, each cells MCU samples the transformer current once every switching period, with all cells operating in trapezoidal mode. If the current is sampled at the center of either the positive or negative half-wave in each cell, the system can be represented as a linear discrete time system. This fact can be used to select control parameters based on the eigenvalues of the system. The model is then extended to allow for individual cells to operate with absolute phase shifts larger than π/2. To also allow for active DC-bias compensation, the control is finally extended to the controller sampling each half-wave. The resulting effects are integrated in the model. To test the prediction capabilities of the model, MAC control is verified experimentally. The model was able to predict instabilities caused by badly selected control parameters as shown in Section III-B, or specific combinations of cells increasing their phase shift past π/2(Section III-D). As investigated in Section III-E, MAC control allows for the semi-independent control of positive and negative half-waves, thereby eliminating DC-offsets that typically occur in traditional single phaseshift control. For dynamic events, high model accuracy was shown in Section IV for DAB and MAB applications, thereby validating the description of multiple active cells with variable switching frequencies by a simple linear system. Since this work only introduces the basic control scheme, many open questions remain regarding implementation in various applications. The potential benefits of MAC control along with remaining challenges are therefore discussed for different possible applications like energy routers, solid state transformers or wireless power transfer. REFERENCES [1] P. 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This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2025.3621885 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/