The structural scheme of mechanical elastic energy storage (MEES) system served by permanent magnet synchronous motor (PMSM) and bidirectional converters is designed. The aim of the research is to model and control the complex electromechanical system. The mechanical device of the complex system is considered as a node in generalized coordinate system, the terse nonlinear dynamic model of electromechanical coupling for the electromechanical system is constructed through LagrangeMaxwell energy method, and the detailed deduction of the mathematical model is presented in the paper. The theory of direct feedback linearization (DFL) is applied to decouple the nonlinear dynamic model and convert the developed model from nonlinear to linear. The optimal control theory is utilized to accomplish speed tracking control for the linearized system. The simulation results in three different cases show that the proposed nonlinear dynamic model of MEES system is correct; the designed algorithm has a better control performance in contrast with the conventional PI control.
Power balance is a basis for stable operation of power system [
The modeling and control of complex dynamic systems are the rapidly emerging researching topics [
previous researches mainly focus on the motor ontology for the electromechanical issue; the difference in the research is to consider the mechanical device as a node in generalized coordinate system; the terse nonlinear dynamic model for a complex electromechanical system is deduced and constructed through LagrangeMaxwell energy method by the principle of electromechanical coupling.
the generalized coordinates of MEES system are determined. The functions of the kinetic energy, magnetic energy, the potential energy, and the energy dissipation of the whole system are obtained, and the differential equations with electromechanical parameters of MEES system are established by applying the extended LagrangeMaxwell equation.
the theory of direct feedback linearization (DFL) is applied to decouple the nonlinear model and convert the developed model for MEES system from nonlinear to linear. An optimal linear controller is designed to accomplish the speed tracking control for the linearized model.
The paper is organized as the following: it starts with introductions to the ES technologies and the electromechanical characteristics of MEES system in Section
In order to give the readers a better understanding of the operating principle of the research object, the composition scheme of MEES system served by PMSM and bidirectional converters is described in Figure
The composition scheme of MEES system served by PMSM and bidirectional converters.
PMSM based MEES system consists of several elements: ES box (multiple spiral springs wrapped up in it), gear box, PMSM, bidirectional converters, breaker, and controller. Therefore, the electromechanical coupling is caused by the interaction between electromagnetic parameters of PMSM and mechanical parameters of spiral springs. A MEES power station can be built by getting tens or even more units together. Through a central controller, the units in the power station can be controlled orderly to realize the goal of largescale energy storage and power generation. It is important to note that a linkage form structure for ES box is proposed in order to increase the energy storage capacity; for details see [
The basic operation principle of the MEES system is concerned with two processes: energy storage and power generation. The load or drive source of MEES system in energy storage or power generation is spiral springs in ES box. In the process of energy storage, PMSM, which is driven by power grid, tightens the springs through gear box. Then, the electric energy is stored in the form of elastic deformation energy. Once the unit receives a signal of releasing energy, the tightened springs start to release energy and drive the motor to generate electricity.
The operating rotation speed of the spindle in elastic ES box in the process of energy storage or power generation is required to run basically stable owing to the large size and the big inertia of ES box. As mentioned above, the elastic ES box can be considered as an output load of PMSM in ES. Hence, supposing that the rotation speed of the shaft in ES box is a constant, the torque calculation formula of ES box in ES process can be given by
Characteristics curve for ES box in ES.
The servomotor of PMSM consists of two parts, stator and rotor. When threephase alternating current (AC) with mutual phase difference of 120° is flowed into the stator windings of the motor, a rotating magnetic field with a uniform motion in space is induced. The rotation speed of the magnetic field is related to the frequency of sinusoidal wave in stator windings. The driven torque, which is produced by the interaction between the rotating magnetic field of stator and permanent magnetic field of rotor, makes the rotor rotate and achieves the conversion of electric energy to elastic energy.
In the process of ES, the torque of the springs raises with the operation of the system. The power angle between the axis of stator and rotor becomes larger, which augments the magnetic torque to balance the increasing torque of springs. With the rising of energy stored in ES box, the input power and the stator current increase accordingly. Simultaneously, the armature reaction causes the growth of flux linkage of air gap and back electromotive force (e.m.f) of stator and compels the stator voltage to rise. In other words, the interaction between the electromagnetic parameters of electrical system and the mechanical parameters of mechanical system composes the electromechanical coupling in complex MEES system.
Before modeling the system of electromechanical coupling, six assumptions are described as follows:
the saturation of the core is neglected,
the losses of eddy and hysteresis are ignored,
the distribution of air gap is uniform,
the selfinductance and mutual inductance among the windings are independent of the position of the rotor,
the damping winding in the rotor and the damping effect of permanent magnet are both neglected,
back e.m.f. maintains the sinusoidal variation.
Based on the principle of electromechanical coupling dynamics, LagrangeMaxwell energy method is applied to construct the dynamic mathematical model of MEES system as follows.
Six generalized coordinates for MEES system.
Generalized coordinates  Electromagnetic system  Mechanical system  

Stator  Rotor  Gear  ES box  








—  —  — 







— 






— 


For the MEES system, the magnetic energy
For the distributed winding with a phase belt of 60°, due to the number
The potential energy of the system
The Lagrange function can be written as
The dissipation function of magnetic system
The dissipation function of mechanical system
Then, the dissipation function of the whole system
Nonconservative generalized force of mechanical system (
External e.m.f. of electromagnetic system is given by
The LagrangeMaxwell equation of MEES system can be expressed as
For electromagnetic system, firstly, the stator winding of Aphase (
Similarly, the voltage equation of stator winding of Bphase (
For permanent magnet of rotor, the effective flux
For mechanical system, when
For mechanical system, when
The electromechanical dynamic model of MEES system in static ABC coordinates can be obtained through simultaneous equations of (
Define
Similarly, the motion equations of mechanical system in rotating frame of dq0 are given as
For an affine nonlinear system in the standard form,
The state variables
Taking the derivative of
Taking the derivative of
The comprehensive order of the system equals
Optimal linear control block diagram of speed tracking for linearized system (
Control block diagram of speed tracking for linearized system.
For the linearized system (
Based on a 0.16 kWh/0.8 kW MEES system being developed, the operation parameters of the system are given in Table
Operation parameters of a 0.16 kWh/0.8 kW MEES system.
Rated work voltage 
380 V 
Rated work torque  2.1 N 
Rated rotating speed of ES box  15 r/min 
Number of pole pairs for rotor 
4 
Rated rotating speed of rotor  600 r/min 
Transformation ratio of the gear 
40 : 1 
Moment of inertia for rotor 
0.8 × 10^{−3} kg m^{2} 
Equivalent moment of inertia for the gear 
0.0013 kg m^{2} 
Equivalent moment of inertia for the ES box 
4.3264 kg m^{2} 
Torsion constant of spring 
5.0178 
Effective flux of permanent magnet 
0.1442 Wb 
Resistance of stator 
1.95 Ω 
Equivalent inductance of direct axis 
0.2541 H 
Equivalent inductance of quadrature axis 
0.2541 H 
Viscous damping coefficient of the motor 
0.015 N/rad/s 
Viscous damping coefficient of the gear 
0.013 N/rad/s 
Viscous damping coefficient of the spring 
0.005 N/rad/s 
In terms of the operation parameters in Table
In the paper, the matrices of weight coefficient
In order to evaluate the effectiveness of the proposed algorithm properly and adequately, the simulation schemes are selected as three different cases, which are case I (always cutting off the energy storage device), case II (from the initial state of storing no energy to the final state of storing full of energy), and case III (from the initial state of storing half of full of energy to the final state of storing full of energy). The simulation results are shown in Figures
The simulation results of case I (always cutting off the energy storage device).
Electric angular speed of rotor
Current of
The simulation results of case II (from the initial state of storing no energy to the final state of storing full of energy).
Electric angular speed of rotor
Current of
The simulation results of case III (from the initial state of storing half of full of energy to the final state of storing full of energy).
Electric angular speed of rotor
Current of
In terms of the characteristics of electromechanical coupling for PMSM based MEES system presented in the paper, dynamic modeling and control of the system are proposed from an electromechanical point of view. The conclusions were made as follows:
dynamic model for PMSM based MEES system is nonlinear with significant electromechanical coupling. The nonlinear model constructed by LagrangeMaxwell energy method can reflect the characteristics of electromechanical coupling for the system;
DFL can perform the dynamic decoupling for the complex electromechanical system completely;
the speed tracking controller proposed in the paper in terms of optimal linear control theory for linearized system has a better performance in contrast with conventional PI control algorithm, and the simulation results in three different cases prove the validity and feasibility of the designed method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20120036130001, the Fundamental Research Funds for the Central Universities of China under Grant no. 11MG40, and the Independent Research Funds of State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources of China under Grant no. 201209.