On the Unification of Optimal Current Reference Theory for Wound Rotor Synchronous Machines

1. Problem Statement

The paper addresses the challenge of generating optimal current references for Wound Rotor Synchronous Machines (WRSMs). Unlike Permanent Magnet Synchronous Machines (PMSMs), WRSMs possess an additional control degree-of-freedom (the rotor current, $i_r$) in addition to the stator d-axis and q-axis currents ($i_d, i_q$).

The core problem is to determine the current vector $\mathbf{i} = [i_r, i_d, i_q]^\top$ that:

1. Satisfies a requested torque ($T_p$).
2. Respects physical constraints (stator current limit, rotor current limit, and maximum phase voltage).
3. Minimizes total electrical losses (copper + core losses).

Key Challenges:

• Nonlinearity: Magnetic saturation and cross-saturation make the flux-current relationship ($\lambda = \phi(\mathbf{i})$) highly nonlinear.
• Complexity: The addition of the rotor current creates a 3-degree-of-freedom (3-DOF) optimization problem, making the feasible set larger and the solution structure more complex than the 2-DOF PMSM case.
• Real-time Constraints: Existing analytical methods often neglect saturation or cross-coupling, while numerical optimization methods (like generic nonlinear solvers) are often too computationally expensive for real-time embedded implementation.

2. Methodology

The authors propose a unified framework that formulates the optimal current reference problem as a Quadratically Constrained Quadratic Program (QCQP) within specific regions of the operating space.

A. Piecewise-Affine (PWA) Flux Modeling

To handle magnetic saturation and cross-saturation without losing the quadratic structure required for efficient optimization:

• The nonlinear flux map $\phi(\mathbf{i})$ is approximated using a Piecewise-Affine (PWA) function derived from Finite Element Analysis (FEA) data.
• The current space is partitioned into simplices via Delaunay triangulation. Within each simplex, the flux is modeled as $\lambda \approx L_j \mathbf{i} + \psi_j$, where $L_j$ is the local inductance matrix and $\psi_j$ is the flux offset.
• This approach preserves the quadratic nature of the cost and constraint functions, allowing the problem to remain a QCQP within each region.

B. Loss and Constraint Modeling

• Objective Function: Total loss is modeled as a quadratic form including copper loss ($\mathbf{i}^\top R \mathbf{i}$) and speed-dependent core loss ($\omega_e^2 \lambda^\top G \lambda$).
• Constraints:
  • Current limits (stator and rotor).
  • Voltage limit: A quadratic constraint derived from the back-EMF and inductance matrices.
  • Torque equality: A non-convex quadratic equality constraint.

C. Regime-Wise Analytical Solution

The paper decomposes the problem based on which constraints are "active" (binding) at the optimal solution. This yields distinct solution strategies for five operating regimes:

1. Cruise (Torque-Only): No inequality constraints are active.
   • Solution: Closed-form solution derived from KKT conditions, requiring the root finding of a 6th-degree polynomial.
2. Launch Control (Stator Current Constrained): Stator current limit is active.
   • Solution: Reduced to a univariate optimization in the angle $\theta$ of the stator current vector, solvable via a 10th-degree polynomial (using Weierstrass substitution).
3. Fast Driving (Voltage Constrained): Voltage limit is active.
   • Solution: Formulated as an inhomogeneous QCQP. The authors use a Semidefinite Programming (SDP) relaxation. They prove computationally that the relaxation is exact (rank-1 solution) across the tested envelope, recovering the global minimizer.
4. Forceful Fast Driving (Fully Constrained): Both stator current and voltage limits are active.
   • Solution: The feasible set becomes a finite set of points (intersection of quadrics). The problem reduces to algebraic root-finding of a 4th-degree polynomial.
5. PMSM Case (Rotor Current Constrained): Rotor current hits its bound.
   • Solution: Reduces to the standard 2-DOF PMSM problem, solvable via existing closed-form methods.

3. Key Contributions

1. Unified WRSM Formulation: The first general analytical treatment of WRSM current reference generation that simultaneously accounts for magnetic saturation, cross-saturation, and speed-dependent core losses.
2. Regime-Wise Analytical Structure: Derivation of closed-form or low-dimensional polynomial solutions for four of the five operating regimes, avoiding the need for iterative nonlinear solvers in most cases.
3. Exact SDP Relaxation: Demonstration that the voltage-constrained regime can be solved via a small SDP relaxation that is provably exact (rank-1) for the tested operating envelope.
4. Computational Efficiency: A strategy that exploits problem structure to achieve real-time performance, significantly outperforming generic solvers.

4. Results

The framework was validated on a physical WRSM prototype (65 kW, 12,000 rpm) using FEA-derived data.

• Optimality: The proposed method matched or slightly improved upon (by up to 1 W) the solutions found by the state-of-the-art nonlinear solver Gurobi across 101 operating points.
• Runtime Performance:
  • Median Solve Time: 0.99 ms (proposed) vs. 22.22 ms (Gurobi baseline).
  • Reduction: A 95.5% reduction in median solve time.
  • Worst Case: Even the worst-case SDP solve (4.14 ms) was more than 5x faster than the Gurobi median.
• Physical Insights: The results confirmed that the 3-DOF formulation captures unique WRSM behaviors, such as the torque-dependent boundary between cruise and voltage-constrained regimes (due to active flux control), which 2-DOF PMSM models cannot represent.

5. Significance

This work bridges the gap between theoretical optimality and real-time implementation for advanced motor drives.

• Efficiency: By enabling real-time calculation of optimal currents that account for saturation and cross-coupling, the method maximizes the efficiency of WRSMs across a wide torque-speed envelope, which is critical for electric vehicle traction applications.
• Scalability: The regime-aware approach demonstrates that complex, non-convex motor control problems can be solved analytically or via small SDPs, making them viable for embedded hardware with strict timing constraints (typically 10–50 $\mu$s).
• Generalization: It extends unified optimal control theory beyond PMSMs to the increasingly popular WRSM topology, offering a path toward reducing reliance on rare-earth permanent magnets while maintaining high performance.