Dynamic Modeling and Control of Electromechanical Coupling for Mechanical Elastic Energy Storage System

Abstract

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 Lagrange-Maxwell 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.

1. Introduction

Power balance is a basis for stable operation of power system [1, 2], which is frequently disturbed by some factors such as faults and intermittent power generations. These threatening behaviors may lead to oscillation between the generating units and the interconnected systems, and even worse, which probably cause serious large-scale blackout in power system [3]. Currently, one of the principal methods solving the power imbalance is to reserve adequate spare capacity for the system, which not only increases the investment, but also reduces the utilization of electrical equipments. With the development of energy storage (ES) technologies, a new way of utilizing energy storage technology to solve the power imbalance has been paid more and more attentions [4, 5]. From state-of-the-art technologies of ES, a great variety of the technologies have been developed [6–11]. However, from a viewpoint of power system operation, the technologies of ES at least need to have the following features: large capacity, fast power response speed, high efficiency, long service life, low operation cost and environmental friendship, and so forth. In terms of these requests, most of ES technologies are still immature except pumped-storage. A new method utilizing spiral springs as ES medium is proposed in reference [12], which is called mechanical elastic energy storage (MEES); some of the basic characteristics for the technology are discussed in [13]. Due to the benefits of small size, low loss, and high efficiency [14], permanent magnet synchronous motor (PMSM) is chosen as the servomotor for the MEES system. Thus, the MEES system, which is integrated by the mechanical device (spiral springs and the correspondingly coating shell), the electrical installation (PMSM), and relevant control information elements, is a typically complex nonlinear system of electromechanical coupling. Parameter coupling, which is induced between electromagnetic parameters of PMSM and mechanical parameters of spiral springs, may affect the dynamic performance of the system. The preexisting studies have shown that revealing the law of electromechanical interaction from the point of view of energy is an effective solution to research dynamic characteristics of complex electromechanical system. Lagrange-Maxwell equation of flywheel energy storage is deduced in reference [15] by calculation of integral of magnetic density in the area of magnetic field.

The modeling and control of complex dynamic systems are the rapidly emerging researching topics [16], especially for the nonlinearities in the systems [17]. The paper concentrates on modeling and decoupling control of complex PMSM based MEES system in detail. The mechanical device is taken into account in the analysis of electromechanical coupling, which can make the research become closer to the reality. Especially for the mechanical spiral springs with a big quality and large inertia, neglecting their influence will make the results inaccurate, even unavailable. The overall aim of the paper is to model and control the complex electromechanical system based on the characteristics of mechanical spiral springs in order to improve the operating stability of the system; in particular, the main innovative points of the research are represented as follows:(a)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 Lagrange-Maxwell energy method by the principle of electromechanical coupling. (b)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 Lagrange-Maxwell equation.(c)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 1. The constitution and operating principles of the MEES system are described in Section 2. Mathematical modeling of electromechanical coupling for the complex MEES system is deduced in Section 3. In Section 4, DFL based decoupling control of the dynamic model is proposed, and an optimal linear controller is designed to accomplish the speed tracking control for the linearized model. Section 5 gives the results of numerical simulation. Finally, the conclusions are summarized in Section 6.

2. Preliminaries and Problem Description

2.1. Constitution of MEES System

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 1.

Figure 1 

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 large-scale 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 [13].

2.2. Operating Principle of MEES System

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.

3. Modeling of Electromechanical Coupling for Complex MEES System

3.1. Physical Analysis

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 𝑇=𝐸𝐵𝐻³𝜋6𝐿𝜔𝑡+𝑇₀, (1) where 𝑇 and 𝑇₀ denote the operating torque and initial torque, 𝐸, 𝐵, 𝐻, and 𝐿 are the modulus of elasticity, width, thickness, and length of spring material, and 𝑡 denotes the time. It should be noted that 𝑇 must be ensured between the maximum torque and minimum torque. Additionally, it should be compatible with the electromagnetic torque of PMSM. In terms of formula (1), the characteristic curve for ES box in ES can be shown in Figure 2.

Figure 2 

Characteristics curve for ES box in ES.

The servomotor of PMSM consists of two parts, stator and rotor. When three-phase 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.

3.2. Mathematical Modeling

Before modeling the system of electromechanical coupling, six assumptions are described as follows:(a)the saturation of the core is neglected,(b)the losses of eddy and hysteresis are ignored,(c)the distribution of air gap is uniform,(d)the self-inductance and mutual inductance among the windings are independent of the position of the rotor,(e)the damping winding in the rotor and the damping effect of permanent magnet are both neglected,(f)back e.m.f. maintains the sinusoidal variation.

Based on the principle of electromechanical coupling dynamics, Lagrange-Maxwell energy method is applied to construct the dynamic mathematical model of MEES system as follows.

(1) Determining Generalized Coordinates of the System. Owing to the large size, big quality, and high inertia of the mechanical spiral springs, neglecting their influence will make the results inaccurate, even unavailable. Hence, the mechanical device is taken into account and considered as a node in generalized coordinate system. Six generalized coordinates shown in Table 1 are constructed for the MEES system, four of which belong to electromagnetic system (𝑘=1,2,3,4), and the remaining two are mechanical system (𝑗=1,2). The parametric descriptions of six generalized coordinates shown in Table 1 are as follows: 𝑖_𝑎, 𝑖_𝑏, and 𝑖_𝑐  are three-phase AC flowed in stator, 𝑢_𝑎, 𝑢_𝑏, and 𝑢_𝑐  indicate three-phase voltage applied in stator, 𝜓_𝑓 represents the magnetic flux of permanent magnet in rotor, 𝜗₁ and 𝜗₂ are the angular displacement of rotor and the spindle of ES box, respectively, Ω₁ and Ω₂ are the mechanical angular velocity of rotor and spindle of ES box, respectively, and 𝑇_𝑔 and 𝑇_𝐿 represent the torque of gear and load. 𝑇_𝐿 is equal to the torque 𝑇 of the springs in the ES box in energy storage.

Table 1 

Six generalized coordinates for MEES system.

(2) Finding Out the Energy Relationship for the MEES System. The kinetic energy of the MEES system 𝑊_𝑘 can be given as 𝑊_𝑘=12𝐽₁Ω²₁+12𝐽_𝑒Ω²₁+12𝐽_spΩ²₂=12𝐽₁Ω²₁+12𝐽_𝑒Ω²₁+12𝐽_spΩ₁𝑖², (2) where 𝐽₁ is the moment of inertia of PMSM, 𝐽_𝑒 and 𝐽_sp are the equivalent moment of inertia of the gear and ES box with respect to input shaft, and 𝑖 is the transformation ratio of gear.

For the MEES system, the magnetic energy 𝑊_𝑚, which is jointly produced by the stator current, permanent magnet of rotor, and interaction with each other, consists of three parts. The calculation of the magnetic energy 𝑊_𝑚1 produced by the current of stator is given as 𝑊_𝑚1=12𝐿_𝑎𝑖²_𝑎+12𝐿_𝑏𝑖²_𝑏+12𝐿_𝑐𝑖²_𝑐+𝑀𝑖_𝑎𝑖_𝑏+𝑀𝑖_𝑏𝑖_𝑐+𝑀𝑖_𝑐𝑖_𝑎, (3) where 𝐿 and 𝑀 indicate the self-inductance and mutual inductance of three-phase windings in stator, for an exterior-gibbose PMSM, 𝑀_𝑎𝑎=𝑀_𝑏𝑏=𝑀_𝑐𝑐=𝑀_𝑎𝑏=𝑀_𝑏𝑐=𝑀_𝑐𝑎=𝑀, 𝐿_𝑎=𝐿_𝑏=𝐿_𝑐.

For the distributed winding with a phase belt of 60°, due to the number 𝑞 of the slots for per pole per phase is often more than two, the magnetic energy of air gap 𝑊_𝑚2 can be considered not changed with the rotation of rotor, namely, the magnetic energy produced by permanent magnet of rotor is a constant, 𝑊_𝑚2=𝐶₁ (𝐶₁ is a constant).

𝑊_𝑚12 is the magnetic energy produced by the interaction between the stator current and stator flux generated by the rotor; the calculation formula of 𝑊_𝑚12 can be written as 𝑊_𝑚12=𝑖_𝑎𝜓_𝑓cos𝜃₁+𝑖_𝑏𝜓_𝑓cos𝜃₁−23𝜋+𝑖_𝑐𝜓_𝑓cos𝜃₁+23𝜋. (4)

The potential energy of the system 𝑉 is given by 1𝑉=2𝛼_sp𝜃²₂, (5) where 𝛼_sp is a torsion constant of the springs.

The Lagrange function can be written as 𝐿=𝑊_𝑘+𝑊_𝑚1−𝑉=2𝐽₁Ω²₁+12𝐽_𝑒Ω²₁+12𝐽_spΩ²₂+12𝐿_𝑎𝑖²_𝑎+12𝐿_𝑏𝑖²_𝑏+12𝐿_𝑐𝑖²_𝑐+𝑀𝑖_𝑎𝑖_𝑏+𝑀𝑖_𝑏𝑖_𝑐+𝑀𝑖_𝑐𝑖_𝑎+𝐶₁+𝑖_𝑎𝜓_𝑓cos𝜃₁+𝑖_𝑏𝜓_𝑓cos𝜃₁−23𝜋+𝑖_𝑏𝜓_𝑓cos𝜃₁+23𝜋−12𝛼_sp𝜃²₂. (6)

The dissipation function of magnetic system 𝐹_𝑒 can be expressed as 𝐹_𝑒=12𝑅_𝑎𝑖²_𝑎+12𝑅_𝑏𝑖²_𝑏+12𝑅_𝑐𝑖²_𝑐, (7) where 𝑅_𝑎, 𝑅_𝑏, and 𝑅_𝑐 indicate three-phase resistance of the stator.

The dissipation function of mechanical system 𝐹_𝑚 can be written as 𝐹_𝑚=12𝐵₁+𝐵_𝑒Ω²₁+12𝐵_spΩ₁𝑖², (8) where 𝐵₁, 𝐵_𝑒, and 𝐵_sp are the equivalent viscous damping coefficients of the motor, the gear, and the spring, respectively.

Then, the dissipation function of the whole system 𝐹 can be given as 𝐹=𝐹_𝑒+𝐹_𝑚=12𝑅_𝑎𝑖²_𝑎+12𝑅_𝑏𝑖²_𝑏+12𝑅_𝑐𝑖²_𝑐+12𝐵₁+𝐵_𝑒Ω²₁+12𝐵_spΩ₁𝑖². (9)

Nonconservative generalized force of mechanical system (𝑗=1,2) can be expressed as 𝑄₁=−𝑇_𝑔𝑄,𝑗=1,₂=−𝑇_𝐿,𝑗=2. (10)

External e.m.f. of electromagnetic system is given by 𝑈_𝑘=−𝑢_𝑘. (11)

The Lagrange-Maxwell equation of MEES system can be expressed as 𝑑𝑑𝑡𝜕𝐿𝜕𝑒_𝑘̇−𝜕𝐿𝜕𝑒_𝑘+𝜕𝐹𝜕𝑒_𝑘̇=𝑈_𝑘,𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇_𝑗−𝜕𝐿𝜕𝑞_𝑗+𝜕𝐹𝜕𝑞̇_𝑗=𝑄_𝑗. (12)

For electromagnetic system, firstly, the stator winding of A-phase (𝑘=1) is considered as 𝜕𝐿𝜕𝑒₁=0,𝜕𝐿𝜕𝑒₁̇=𝜕𝐿𝜕𝑖_𝑎=𝐿𝑖_𝑎+𝑀𝑖_𝑏+𝑀𝑖_𝑐+𝜓_𝑓cos𝜃₁,𝑑𝑑𝑡𝜕𝐿𝜕𝑒₁̇⟶𝑑𝑑𝑡𝜕𝐿𝜕𝑖_𝑎=𝐿𝑑𝑖_𝑎𝑑𝑡+𝑀𝑑𝑖_𝑏𝑑𝑡+𝑀𝑑𝑖_𝑐+𝑑𝜓𝑑𝑡_𝑓cos𝜃₁,𝑑𝑡𝜕𝐹𝜕𝑒₁̇⟶𝜕𝐹𝜕𝑖_𝑎=𝑅_𝑎𝑖_𝑎,𝑈₁=𝑑𝑑𝑡𝜕𝐿𝜕𝑒₁̇−𝜕𝐿𝜕𝑒₁+𝜕𝐹𝜕𝑒₁̇,𝑢_𝑎=𝑈₁; (13) the voltage equation of stator winding of A-phase can be written as 𝑢_𝑎=𝑈₁=𝑑𝑑𝑡𝜕𝐿𝜕𝑒₁̇−𝜕𝐿𝜕𝑒₁+𝜕𝐹𝜕𝑒₁̇=𝐿𝑑𝑖_𝑎𝑑𝑡+𝑀𝑑𝑖_𝑏𝑑𝑡+𝑀𝑑𝑖_𝑐+𝑑𝜓𝑑𝑡_𝑓cos𝜃₁𝑑𝑡+𝑅_𝑎𝑖_𝑎. (14)

Similarly, the voltage equation of stator winding of B-phase (𝑘=2) can be written as 𝑢_𝑏=𝑈₂=𝑑𝑑𝑡𝜕𝐿𝜕𝑒₂̇−𝜕𝐿𝜕𝑒₂+𝜕𝐹𝜕𝑒₂̇=𝐿𝑑𝑖_𝑏𝑑𝑡+𝑀𝑑𝑖_𝑐𝑑𝑡+𝑀𝑑𝑖_𝑎+𝑑𝜓𝑑𝑡_𝑓cos𝜃₁−(2/3)𝜋𝑑𝑡+𝑅_𝑏𝑖_𝑏; (15) the voltage equation of stator winding of C-phase (𝑘=3) can be written as 𝑢_𝑐=𝑈₃=𝑑𝑑𝑡𝜕𝐿𝜕𝑒₃̇−𝜕𝐿𝜕𝑒₃+𝜕𝐹𝜕𝑒₃̇=𝐿𝑑𝑖_𝑐𝑑𝑡+𝑀𝑑𝑖_𝑎𝑑𝑡+𝑀𝑑𝑖_𝑏+𝑑𝜓𝑑𝑡_𝑓cos𝜃₁+(2/3)𝜋𝑑𝑡+𝑅_𝑐𝑖_𝑐. (16)

For permanent magnet of rotor, the effective flux 𝜓_𝑓 of permanent magnet is generally considered to be a constant, namely, 𝜓_𝑓 = 𝐶₂(𝐶₂ is a constant). Thus 𝜕𝐿𝜕𝜓_𝑓=0,𝜕𝐿𝜕𝜓̇_𝑓=0. (17)

For mechanical system, when 𝑗=1 has 𝜕𝐿𝜕𝑞₁⟶𝜕𝐿𝜕𝜃₁=−𝑖_𝑎𝜓_𝑓sin𝜃₁−𝑖_𝑏𝜓_𝑓sin𝜃₁−23𝜋−𝑖_𝑐𝜓_𝑓sin𝜃₁+23𝜋−𝛼_sp𝜃₁𝑖²,𝜕𝐿𝜕𝑞̇₁⟶𝜕𝐿𝜕Ω₁=𝐽₁+𝐽_𝑒Ω₁+𝐽_spΩ₁𝑖²,𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇₁⟶𝑑𝑑𝑡𝜕𝐿𝜕Ω₁=𝐽₁+𝐽_𝑒+𝐽_sp𝑖²𝑑Ω₁,𝑑𝑡𝜕𝐹𝜕𝑞̇₁⟶𝜕𝐹𝜕Ω₁=𝐵₁+𝐵_𝑒+𝐵_sp𝑖²Ω₁,𝑄₁=𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇₁−𝜕𝐿𝜕𝑞₁+𝜕𝐹𝜕𝑞̇₁,𝑄₁=−𝑇_𝑔, (18) the motion equation of mechanical system 𝑗=1 can be expressed as 𝐽₁+𝐽_𝑒+𝐽_sp𝑖²𝑑Ω₁𝑑𝑡=−𝑖_𝑎𝜓_𝑓sin𝜃₁−𝑖_𝑏𝜓_𝑓sin𝜃₁−23𝜋−𝑖_𝑐𝜓_𝑓sin𝜃₁+23𝜋−𝛼_sp𝜃₁𝑖²−𝐵₁+𝐵_𝑒+𝐵_sp𝑖²Ω₁−𝑇_𝑔. (19) Let 𝐽=𝐽₁+𝐽_𝑒+(𝐽_sp/𝑖²) and 𝐵=𝐵₁+𝐵_𝑒+(𝐵_sp/𝑖²); the motion equation of mechanical system 𝑗=1 can be rewritten as 𝐽𝑑Ω₁𝑑𝑡=−𝑖_𝑎𝜓_𝑓sin𝜃₁−𝑖_𝑏𝜓_𝑓sin𝜃₁−23𝜋−𝑖_𝑐𝜓_𝑓sin𝜃₁+23𝜋−𝛼_sp𝜃₁𝑖²−𝐵Ω₁−𝑇_𝑔. (20)

For mechanical system, when 𝑗=2 has 𝜕𝐿𝜕𝑞₂⟶𝜕𝐿𝜕𝜃₂=−𝑖𝑖_𝑎𝜓_𝑓sin𝜃₂−𝑖𝑖_𝑏𝜓_𝑓sin𝜃₂−23𝜋−𝑖𝑖_𝑐𝜓_𝑓sin𝜃₁+23𝜋−𝛼_sp𝜃₂,𝜕𝐿𝜕𝑞̇₂⟶𝜕𝐿𝜕Ω₂=𝑖²𝐽₁+𝐽_𝑒Ω₂+𝐽_spΩ₂,𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇₂⟶𝑑𝑑𝑡𝜕𝐿𝜕Ω₂=𝑖²𝐽₁+𝐽_𝑒+𝐽_sp𝑑Ω₂,𝑑𝑡𝜕𝐹𝜕𝑞̇₂⟶𝜕𝐹𝜕Ω₂=𝑖²𝐵₁+𝐵_𝑒+𝐵_spΩ₂,𝑄₂=𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇₂−𝜕𝐿𝜕𝑞₂+𝜕𝐹𝜕𝑞̇₂,𝑄₂=−𝑇_𝐿, (21) the motion equation of mechanical system 𝑗=2 can be written as 𝑖²𝐽₁+𝐽_𝑒+𝐽_sp𝑖²𝑑Ω₂𝑑𝑡=−𝑖𝑖_𝑎𝜓_𝑓sin𝜃₂−𝑖𝑖_𝑏𝜓_𝑓sin𝜃₂−23𝜋−𝑖𝑖_𝑐𝜓_𝑓sin𝜃₂+23𝜋−𝛼_sp𝜃₂−𝑖²𝐵₁+𝐵_𝑒+𝐵_sp𝑖²Ω₂−𝑇_𝐿. (22) In terms of 𝐽=𝐽₁+𝐽_𝑒+(𝐽_sp/𝑖²), 𝐵=𝐵₁+𝐵_𝑒+(𝐵_sp/𝑖²), that is, (22) can be rewritten as 𝑖²𝐽𝑑Ω₂𝑑𝑡=−𝑖𝑖_𝑎𝜓_𝑓sin𝜃₂−𝑖𝑖_𝑏𝜓_𝑓sin𝜃₂−23𝜋−𝑖𝑖_𝑐𝜓_𝑓sin𝜃₂+23𝜋−𝛼_sp𝜃₂−𝑖²𝐵Ω₂−𝑇_𝐿. (23)

The electromechanical dynamic model of MEES system in static ABC coordinates can be obtained through simultaneous equations of (14), (15), (16), (20), and (23).

Define 𝐶_abc/dq0 as the transformation matrix from static coordinates of ABC to rotating frame of dq0 as follows: 𝐶_abc/dq0=232cos𝜃cos𝜃−3𝜋2cos𝜃+3𝜋2−sin𝜃−sin𝜃−3𝜋2−sin𝜃+3𝜋121212; (24) in terms of the transformation matrix 𝐶_abc/dq0, the relationship between current and voltage in ABC frame and dq0 frame can be written as 𝑖_𝑑𝑖_𝑞𝑖₀=𝐶_abc/dq0𝑖_𝑎𝑖_𝑏𝑖_𝑐,𝑢_𝑑𝑢_𝑞𝑢₀=𝐶_abc/dq0𝑢_𝑎𝑢_𝑏𝑢_𝑐. (25) Substituting (25) into equations (14), (15), and (16), respectively, the voltage equations of the electromagnetic system in dq0 frame can be obtained as 𝑢_𝑑𝑢_𝑞=𝐿_𝑑00𝐿_𝑞𝑑𝑖_𝑑𝑑𝑡𝑑𝑖_𝑞+𝑅𝑑𝑡^𝑠𝑖_𝑑−𝐿_𝑞𝜔_𝑒𝑖_𝑞𝑅_𝑠𝑖_𝑞+𝐿_𝑑𝜔_𝑒𝑖_𝑑+𝜔_𝑒𝜓_𝑓, (26) where 𝜔_𝑒=𝑝_𝑛Ω₁=𝑝_𝑛𝜃̇₁ is the electric angular speed, 𝑝_𝑛 is the number of pole-pairs, 𝑅_𝑠 is the resistance of the stator, 𝑅_𝑠=𝑅_𝑎=𝑅_𝑏=𝑅_𝑐, 𝐿_𝑑,𝐿_𝑞 are the equivalent inductance of direct axis and quadrature axis, and 𝐿_𝑑=𝐿_𝑞=𝐿_𝑠=3(𝐿−𝑀)/2.

Similarly, the motion equations of mechanical system in rotating frame of dq0 are given as 𝐽00𝑖²𝐽𝜃̈₁𝜃̈₂+𝐵00𝑖²𝐵𝜃̇₁𝜃̇₂+𝛼_sp𝑖²00𝛼_sp𝜃₁𝜃₂+−32𝑖_𝑞𝜓_𝑓+𝐿_𝑑−𝐿_𝑞𝑖_𝑑𝑖_𝑞−32𝑖𝑖_𝑞𝜓_𝑓+𝐿_𝑑−𝐿_𝑞𝑖_𝑑𝑖_𝑞=−𝑇_𝑔−𝑇_𝐿. (27) Due to 𝜃₂=𝜃₁/𝑖 and 𝑇_𝑔=𝑇_𝐿/𝑖, two formulas in (27) express the same meaning essentially. Considering 𝜔_𝑒=𝑝_𝑛Ω₁=𝑝_𝑛𝜃̇₁, namely, 𝜃̈₁=𝜔̇_𝑒/𝑝_𝑛, (27) can be transformed into a new form: 𝐽𝑝_𝑛𝜔̇_𝑒+𝐵𝑝_𝑛𝜔_𝑒−32𝑖_𝑞𝜓_𝑓+𝐿_𝑑−𝐿_𝑞𝑖_𝑑𝑖_𝑞+𝛼_sp𝑖²𝜃₁=−𝑇_𝑔, (28) The electromechanical dynamic model of MEES system in rotating frame of dq0 can be obtained through simultaneous equations of (26) and (28).

4. Decoupling Control of the Dynamic Model

4.1. Fundamental Principle of Direct Feedback Linearization

For an affine nonlinear system in the standard form, 𝑥=𝑓(𝑥)+^𝑝_𝑖=1𝑔_𝑖(𝑥)𝑢_𝑖,𝑦_𝑖=ℎ_𝑖(𝑥),𝑖=1,2,…,𝑝, (29) where 𝑥∈𝑅^𝑛 are state variables, 𝑢∈𝑅^𝑝 are input variables, 𝑦∈𝑅^𝑝 are output variables, 𝑓,𝑔 are smooth vectors fields defined on 𝑅^𝑛, and ℎ are smooth scalar functions defined on 𝑅^𝑛. The nonlinearity of MEES dynamic model in rotating frame of dq0 is caused by product terms of 𝜔_𝑒𝑖_𝑑 and 𝜔_𝑒𝑖_𝑞, which will weaken the performance of conventional PI controller. Linearization and decoupling control of the system are effective solutions to improve the servo capability. The theory of direct feedback linearization (DFL) is known intimately up to now and is applied widespread to the varieties of nonlinear domains [18, 19]. The basic principle of DFL is to differentiate the output function repeatedly, until the 𝑟th derivatives 𝑦^(𝑟) of output variable explicitly contain control variable 𝑢, where 𝑟 is the relatively degree of the system.

4.2. DFL for MEES System

The state variables 𝑥∈𝑅³, output variables 𝑦∈𝑅², and control variables 𝑢∈𝑅² are chosen as 𝑖_𝑑, 𝑖_𝑞, 𝜔_𝑒,   𝑖_𝑑, 𝜔_𝑒, and 𝑢_𝑑, 𝑢_𝑞, respectively. The output variables are differentiated with respect to time until at least an input variable explicitly appears by DFL theory.

Taking the derivative of 𝑖_𝑑, we can obtain 𝑖_𝑑̇=−𝑅_𝑠𝐿_𝑑𝑖_𝑑+𝜔_𝑒𝑖_𝑞+1𝐿_𝑑𝑢_𝑑; (30) (30) contains the input 𝑢_𝑑 and the relatively degree 𝑟₁=1. Introducing a new variable 𝑣₁ makes 𝑖_𝑑̇=𝑣₁.

Taking the derivative of 𝜔_𝑒, we can obtain 𝜔̇_𝑒𝐵=−𝐽𝜔_𝑒+3𝑝_𝑛𝑖2𝐽^𝑞𝜓_𝑓−𝑝_𝑛𝛼_sp𝐽𝑖²𝜃₁+𝑝_𝑛𝐽𝑇_𝑔; (31) as (31) does not contain any input variable, second derivation calculus to 𝜔_𝑒 can be obtained as follows: 𝜔̈_𝑒=𝐵𝐽𝜔̇_𝑒+3𝑝_𝑛𝑖̇2𝐽^𝑞𝜓_𝑓−𝑝_𝑛𝛼_sp𝐽𝑖²𝜃̇₁+𝑝_𝑛𝐽𝑇̇_𝑔=𝐵𝐽−𝐵𝐽𝜔_𝑒+3𝑝_𝑛𝑖2𝐽^𝑞𝜓_𝑓−𝑝_𝑛𝛼_sp𝐽𝑖²𝜃₁+𝑝_𝑛𝐽𝑇_𝑔+3𝑝_𝑛𝜓_𝑓−12𝐽𝐿_𝑞𝑅_𝑠𝑖_𝑞+𝐿_𝑑𝜔_𝑒𝑖_𝑑+𝜔_𝑒𝜓_𝑓+1𝐿_𝑞𝑢_𝑞−𝑝_𝑛𝛼_sp𝐽𝑖²𝜔_𝑒𝑝_𝑛+𝑝_𝑛𝐽𝑇̇_𝑔; (32) (32) contains the input 𝑢_𝑞 and the relatively degree 𝑟₂=2. Introducing another new variable 𝑣₂ makes 𝜔̇_𝑒=𝑣₂.

The comprehensive order of the system equals 𝑟₁+𝑟₂=3, which is exactly equal to the degree of the original system. Hence, a nonlinear state feedback control law is devised as follows: 𝑢=𝐸(𝑥)+𝐷(𝑥)𝑣, (33) where 𝑢=[𝑢_𝑑,𝑢_𝑞]^𝑇, 𝑣=[𝑣₁,𝑣₂]^𝑇, 𝐷(𝑥)=^𝐿𝑑0_{02𝐽𝐿𝑞/3𝑝𝑛𝜓𝑓}, 𝐸(𝑥)=𝑅_𝑠𝑖_𝑑−𝐿_𝑑𝜔_𝑒𝑖_𝑞2𝐿_𝑞𝐵3𝑝_𝑛𝜓_𝑓−𝐵𝐽𝜔_𝑒+3𝑝_𝑛𝑖2𝐽^𝑞𝜓_𝑓−𝑝_𝑛𝛼_sp𝐽𝑖²𝜃₁+𝑝_𝑛𝐽𝑇_𝑔+𝑅_𝑠𝑖_𝑞+𝐿_𝑑𝜔_𝑒𝑖_𝑑+𝜔_𝑒𝜓_𝑓+𝛼_sp𝐽𝑖²𝜔_𝑒−𝑝_𝑛𝐽𝑇̇_𝑔. (34) Substituting (33) into the electromechanical dynamic model in rotating frame of dq0, the ultimate linearization equation of the MEES system can be obtained as follows: 𝑖_𝑑̇𝑖_𝑞̇𝜔̇_𝑒=0𝐵000𝑝_𝑛𝐽−2𝐵²3𝑝_𝑛𝜓_𝑓𝐽+2𝛼_sp3𝜓_𝑓𝑖²03𝑝_𝑛𝜓_𝑓−𝐵2𝐽𝐽𝑖_𝑑𝑖_𝑞𝜔_𝑒+0102𝐽3𝑝_𝑛𝜓_𝑓𝑣00¹𝑣₂+02𝐵3𝜓_𝑓𝐽𝑇_𝑔+23𝜓_𝑓𝑇̇_𝑔−2𝐵𝑝_𝑛𝛼_sp3𝜓_𝑓𝐽𝑖²𝜃₁𝑝_𝑛𝐽𝑇_𝑔−𝑝_𝑛𝛼_sp𝐽𝑖²𝜃₁=𝐴𝑥+𝐸𝑣+𝐶, (35) where 𝐴, 𝐸, and 𝐶 are three matrices.

4.3. Design of Optimal Linear Control

Optimal linear control block diagram of speed tracking for linearized system (35) is designed in Figure 3.

Figure 3 

Control block diagram of speed tracking for linearized system.

For the linearized system (35), the theory of linear control can be applied to determine 𝑣. In order to improve the control performance, in terms of optimal control theory of linear system, optimal control inputs are chosen as 𝑣=−𝐾^∗⋅𝑥, (36) where 𝐾^∗=𝑅⁻¹𝐸^𝑇𝑃^∗, 𝑅=𝑅^𝑇 is the matrix of weight coefficient.

𝑃^∗ is the solution of Riccati equation as follows: 𝐴^𝑇𝑃+𝑃𝐴−𝑃𝐸𝑅⁻¹𝐸^𝑇𝑃+𝑄=0, (37) where 𝑄 is a positive definite matrix.

5. Simulation and Results

Based on a 0.16 kWh/0.8 kW MEES system being developed, the operation parameters of the system are given in Table 2. In terms of the description above, the control objectives in the paper are determined as the speed of rotor and the current of 𝑑-axis, namely, the speed of rotor tracking the reference speed and the current of 𝑑-axis being zero. The reference speed of the motor is set at 600 r/min. The work of modeling and control is implemented in Matlab environment.

Table 2 

Operation parameters of a 0.16 kWh/0.8 kW MEES system.

In terms of the operation parameters in Table 2, the matrix parameters 𝐴 and 𝐸 in (35) are obtained as follows: 0𝐵𝐴=000𝑝_𝑛𝐽2𝐵²3𝑝_𝑛𝜓_𝑓𝐽+2𝛼_sp3𝜓_𝑓𝑖²03𝑝_𝑛𝜓_𝑓−𝐵2𝐽𝐽=000000.0019−0.232100.19990.0076𝐸=102𝐽3𝑝_𝑛𝜓_𝑓=001005.002900. (38)(39)

In the paper, the matrices of weight coefficient 𝑅 and positive definite 𝑄 are taken as 1 and a 3×3 identity matrix, respectively. Then, the matrix of 𝑃^∗ can be calculated as 𝑃^∗=1000−0.000228−0.0173180−4.985561−0.000265. (40) Considering (36), the inputs of the optimal control can be expressed as 𝑣=−𝐾^∗⋅𝑥=−𝑅⁻¹𝐸^𝑇𝑃^∗=⋅𝑥−𝑖_𝑑0.001141𝑖_𝑞+0.086640𝜔_𝑒. (41) Substituting (41) into (33), the nonlinear state feedback controller is given as 𝑢=1.6969𝑖_𝑑−0.2541𝑖_𝑞𝜔_𝑒1.953387𝑖_𝑞+0.265858𝜔_𝑒+0.2541𝑖_𝑑𝜔_𝑒+00.008956𝑇_𝑔−0.924108𝑇̇_𝑔−0.000449𝜃₁. (42)

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 4, 5, and 6, respectively. In order to better display the control performances, the proposed algorithm is in contrast with the conventional PI control. The proportional and integral coefficients for 𝑑-axis PI current controller are set as 2.6 and 50, respectively, and the proportional and integral coefficients for PI speed tracking controller are set as 0.24 and 10, respectively. Case I simulates PMSM operating in no load state, which is to test the no-load performance of the servomotor. The results in Figure 4 show that the proposed algorithm makes the controlled objective become stable more quickly and almost has no oscillation in stabilization process in comparison with the conventional PI control. Figure 5 shows the results of case II, which essentially represents the process from point 𝐽 to point 𝐸 in Figure 2. Figure 6 displays the results of case III, and it should be noted that the running time in the condition will be half of the whole time of energy storage, which is 5 s. The simulation results in case II and case III are similar to case I. Therefore, the simulation results reveal that in contrast with conventional PI control, the presented algorithm not only has a preferable ability of speed tracking in all of the three cases, but also makes the electromechanical system have a better dynamic performance in energy storage process.

(a) Electric angular speed of rotor

(b) Current of 𝑑-axis

Figure 4 

The simulation results of case I (always cutting off the energy storage device).

(a) Electric angular speed of rotor

(b) Current of 𝑑-axis

Figure 5 

The simulation results of case II (from the initial state of storing no energy to the final state of storing full of energy).

(a) Electric angular speed of rotor

(b) Current of 𝑑-axis

Figure 6 

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).

6. Conclusions

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:(a)dynamic model for PMSM based MEES system is nonlinear with significant electromechanical coupling. The nonlinear model constructed by Lagrange-Maxwell energy method can reflect the characteristics of electromechanical coupling for the system;(b)DFL can perform the dynamic decoupling for the complex electromechanical system completely;(c)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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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.