Speed control with load sharing capabilities for multi-three phase synchronous motors

The on-going electrification transportation system revolution has already started and the continuous evolution of power electronics allows new and more reliable system layouts and control strategies to be investigated. Mainly, this revolution is driven by increased reliability levels. Since multi-three phase motors are fault tolerant by definition, they are a very good candidate for more reliable transportation systems. In this work, load sharing in multi-three phase synchronous motors is presented. For better explaining the new possibilities introduced in transportation systems by multi-three phase motors, few examples relative to a DC micro-grid for aerospace applications are provided. Analytical equations and experimental validation of speed control with load sharing capabilities are given by means of Matlab/Simulink simulations and by experimental on a 22kW test rig.


I. INTRODUCTION
The electrification transportation system had started at the beginning of the last century with railways [1]. After more than one hundred years, power electronics together with renewable energy sources and storage devices advancements have launched a proper propulsion system revolution. In this context, multiple research projects for transportation systems, i.e. aerospace [2], [3], mining machines [4], [5], ships [6], [7], and road vehicles [8] have been founded by governments and innovation centres around the world. Whenever higher reliability levels are demanded, multi-three phase motors can make the difference [9], [10].
In Fig. 1, an off-the-shelf 2-Level 3-Phase Voltage Source Inverter (2L-3P-VSI) for variable-speed AC drives is shown. Connecting a multi-three phase motor to a multi-drive system like in Fig. 2, new control strategies can be investigated pushing the boundaries in term of performances, operating  ranges, and reliability on both machine and power conversion side. In-fact, if during design process the system is properly over-rated, in case of fault the power delivered to the machine can be kept constant. Keeping into account the mutual interactions within the motor, the system in Fig. 2 can be split into different independent modules (or segments) with one drive (or micro-controller), one 2L-3P-VSI, and one set of windings. Depending on the particular application layout and requirements, while keeping constant the overall torque produced by the motor, every module could deliver to the machine either the same amount of power or not. Similarly, if in generation mode, the machine could generate a different amount of energy for every isolated module. In this work, load sharing capability while controlling the speed in a triple-star synchronous motor with disconnected neutral points (Fig. 2) is provided by mean of Matlab/Simulink simulations and by experimental on a 22kW test rig. In the next section, exploitation of load sharing capability in multithree phase systems will be introduced considering a circular DC micro-grid suitable for aerospace applications. In Sec. III, modelling assumption for both machine and drives are given. Speed control, load sharing, a simplified control design procedure together with their relative simulations are provided in section IV, V, VI, and VII, respectively. After presenting a case study in Sec. VIII, validation of the proposed analysis is provided in Sec. IX. Conclusions are given in Sec. X.
(b) Proposed ZEDS with multi-three phase motors.  Fig. 3a. In Fig. 3b, the proposed ZEDS with multi-three phase motors is shown. In nominal condition each side is provided with 50% of the total power. If during design process every module is properly over-rated, in case of fault the proposed ZEDS in Fig. 3b is able to provide nominal power to both the motors. II. PROPOSED CIRCULAR DC MICRO-GRID FOR

AEROSPACE APPLICATIONS
In this section, a circular DC Zonal Electrical Distribution System (ZEDS) [6], [7] is introduced and further developed for better presenting some of the possibilities enabled by multithree phase motors. In Fig. 3a, a ZEDS with two three-phase motors in red, four generators in green, and for sake of clarity with just two electrical zones in yellow is shown. The switches on the DC ring bus are there to isolate faults that may occur on the buses that distribute power to the zones.
Introducing multi-three phase motors within the ZEDS in Fig. 3a allows higher fault tolerance and reliability levels proper of aerospace applications to be reached (Fig. 3b). Infact, whilst the three-phase motor in Fig. 3a in case of fault is compromised, the multi-three phase motor in magenta in Fig. 3b can still operate either in nominal or sub-optimal conditions. In-fact, full fault compensation is achieved only properly over-rating both the motor and the converters. In Fig.  4a, the case of two unbalanced isolated DC links is shown. Again, assuming the system has been properly over-rated, multi-three phase motors are able to demand different amount of power to every DC power source (i.e. 25% and 75% ) keeping constant the overall power provided to the motors. Finally in Fig. 4b, thanks to the multi-three phase machines, power can be delivered to the DC-link with broken generators.

III. MODELLING
In the previous section, some of the features enabled by multi-three phase machines were shown. Modelling of the mutual interactions within the motor is here introduced in order to better understand how the parallel current control loops can be tuned on their relative plants.
Multi-three phase electrical motors are a particular group of split-phase winding machines. Defining m the number of phases per set of windings and defining N the number of sets of windings, the total number of phases is equal to n = N m. The motor modelled in this paper and shown in Fig. 2 is composed by nine phases (m = 3, N = 3, n = 9) with phase progression α = π/n.

A. Machine modelling assumptions
The work presented in this paper is based on the assumption that stator inductances are constant. Therefore, it applies to electric machines with negligible saturation effects. In addition it is assumed that: • all phases are geometrically identical; • each phase is symmetrical around its magnetic axis; • the spatial displacement between two whatever phases is an integer multiple of the phase progression α; • within the air-gap, only the fundamental component of magneto-motive force is considered.
No restrictive assumption is made, instead, about whether the winding is distributed or concentrated and no leakage flux component is ignored [11]- [13]. Three-phase machine stator variables (i.e. voltage, current, etc., denoted with subscript abc) can be transformed within the rotor-attached orthogonal dq0 reference frame (denoted with subscript dq) thanks to the Park's transformation [14], [15]. Distributed current control of the machine can be achieved thanks to the following equation [16]: where v dq and i dq are voltage and current vectors nx1, respectively. R dq and L dq are resistance and inductance matrices nxn, respectively. Whilst R dq matrix is diagonal, due to the mutual electro-magnetic interactions among different axes of different sets of windings within the machine, L dq is not diagonal. Full de-coupled three-phase Field Oriented Control (FOC) can be used in multi-three phase applications transforming rotor-attached orthogonal dq0 reference frame variables into the Vector Space Decomposition frame (denoted with subscript vsd). In-fact, the transformed nxn L vsd = T T vsd L dq T vsd inductance matrix is diagonal (T vsd can be found in [12], [13]).

B. Drive modelling assumptions
Distributed current control is based on previous machine modelling assumptions. Current Proportional Integral (P I I ) controllers are tuned on the first harmonic inductances d 1 and q 1 in (2). A simplified control diagram not considering actuation nor filtering delays is shown in Fig. 5, where Λ represents d or q axis and s is the Laplacian operator.
Once the PI controller of the simplified current loop in Fig.  5 is tuned, the closed current loop can be modelled like a low- 5. Current control diagram within the synchronous reference frame without axes decoupling with first harmonic inductor Λ 1 (Λ identifies d or q axis) and phase resistor rs. K pIΛ and K iIΛ are the PI gains.
pass filter with bandwidth ω c and phase φ c described by the following transfer function: IV. SPEED CONTROL In this section, speed control loop set-up for multi-three phase machines is introduced. Since the current loops are in parallel, the design procedure has to take into account the overall torque produced by the all N segments (Fig. 7). After the equivalent speed loop is designed (Fig. 8), the same PI parameters can be used in the actual paralleled speed loops in Fig. 9.
Defining the angular speed of the shaft ω, the machine constant K t , the inertia J and the friction F , the simplified control diagram of the machine configured in torque mode is shown in Fig. 6. T L is the load torque. Provided that torque and i q current are directly proportional (T = K t i q ), the final speed of the shaft at steady state depends on the balance between the i q currents flowing within the motor and the load torque T L [17]. The parallel of the three current loops can be further simplified with control diagram in Fig. 7. In general, speed control is set by the outer speed loop governed by a speed P I S regulator. In multi-three phase application, regulators can be computed considering the loop in Fig. 8, where the (a) Speed is not affected by US transients.
(c) Multi-three phase rig. Fig. 10. In Fig. 10a, the equivalence between the equivalent (EQ) diagram in Fig. 8 and the Common Speed Reference (CSR) control schematic in Fig. 9 either in ES and US operation is shown. In Fig. 10b, iq current transients not affecting the speed in Fig. 10a are highlighted. equivalent (EQ) closed speed loop is shown. Once the speed P I S parameters have been computed, the same values can be used in the Common Speed Reference (CSR) simplified control schematic in Fig. 9 where the mechanical part is not shown for simplicity.

V. LOAD SHARING
Either in case of module fault or whenever needed by the particular application, power can be split among the drives varying the internal set-points i * q1,2,3 . Introducing a sharing coefficient W j per module, the internal current set-point and therefore the torque produced by each module can be set up. The new current set-points i * q1,2,3 are defined by the following equation: i * qj = i * qj W j with j = 1..N . In nominal condition, power is equally split (ES) and loop gains are assumed to be W 1,2,3 = 1 = W (ES) j . Depending on the particular application like previously discussed in Sec. II, unbalanced sharing (US) can be obtained varying the sharing coefficients. Assuming all the modules are producing torque, the following equations define the global sharing coefficient W T and the power P j in p.u. produced by the j-th module.
Equations (4) and (5) are valid under both ES and US operation. However, in ES condition with W (ES) j = 1, W T = N and P j = 1/N . It will be later shown that as long as W T is kept constant, the speed is not affected by internal current set-point step variations.

VI. CONTROL DESIGN PROCEDURE
Based on the previous discussions, a design procedure for a system with N modules here is presented. Current P I are tuned with (3) and speed P I S are calculated considering the equivalent control scheme (EQ) in Fig. 8. The same speed P I parameters can be put into control schematic in Fig. 9 where W 1,2,3 are initially set to one for equal sharing (ES) operation. Load sharing, or unbalanced sharing (US), can be further achieved with (4) and (5) keeping constant the global sharing coefficient W T .

VII. SIMULATIONS
In Fig. 10a, speed dynamic equivalence between the EQ control schematic in Fig. 8 and the CSR simplified control schematic in Fig. 9, either in ES and US operation, is shown. In Fig. 10b, the relative i q currents, are shown. Since After six seconds, the load has been unbalanced with the following power ratios P 1 = 2/3, P 2 = 1/12, and P 3 = 1/4. At second nine, first and second power ratios have been swapped. In Table I, sharing coefficients computed with (4) and (5)  (c) Phase current transients. Fig. 11. In Fig. 11a, constant speed during sharing and swapping operation is highlighted. In Fig. 11b, iq current transients not affecting the speed in Fig.  11a are highlighted. In Fig. 11c, phase current transients during swapping operation are shown. Signals within the dotted circle are zoomed in Fig. 12. As previously discussed in Sec. III, the first harmonic inductances d 1 and q 1 in (2) must be obtained before tuning the current control loops. Both d and q current PI gains (K pId , K iId , K pIq , K iIq ) in Fig. 5 have been calculated considering a control loop bandwidth BW C = 211[rad/sec] and a phase margin P M C = 65 • . The speed loop regulator parameters (K pS and K iS ) have been designed with a control loop bandwidth BW S = 6[rad/sec] and a phase margin P M S = 60 • considering the equivalent control diagram in Fig. 8.

IX. EXPERIMENTAL
The load sharing previously detailed has been validated on a test bench with a multi-three phase two poles synchronous generator with three sets of windings shown in Fig. 10c. The machine has been wired to three off-the-shelf converters controlled by a custom control platform named uCube [18]. The encoder signal has been wired to the uCube where three independent synchronised Field Oriented Controllers (FOC) have been coded. The whole control schematic within the reference frame for speed control with load sharing capabilities is shown in Fig. 13. DC link voltage and switching frequency have been set up to 350[V ] and 10[kHz], respectively. In Table  II, electrical and mechanical machine parameters are reported.  Fig. 11a, constant rotor speed under load sharing transients in Fig. 11b is highlighted. In Fig. 11c, phase currents while swapping the first power ratio with the second one are shown. Signals within the dotted circle are zoomed in Fig. 12. Due to the presence of mutual electro-magnetic couplings among different sets of windings, currents of the third set of windings are affected by current transients within the other two sets of windings (for sake of clarity only i a currents are shown). Controlled current transients during load sharing operations could be achieved with the speed-drooped control strategy described in [19].

X. CONCLUSIONS
This work is focused on load sharing for multi-three phase synchronous electrical motors. The control design procedure with a real case study has been introduced and validated. A simplified design procedure for controlling the speed of a multi-three phase motor has been presented. The load sharing feature has been introduced and described by theoretical equations. Both speed control and load sharing have been validated in Matlab/Simulink environment first, and then on a 22kW experimental rig showing good agreement with the expected results. The proposed system appears to be a good candidate for aerospace applications with load sharing capabilities.