A control method of a diesel-electric-fuel combined propulsion system

Abstract

The application discloses a control method of a diesel-electricity-fuel combined propulsion system, a six-dimensional mathematical model of a six-phase induction motor is established first, then space decoupling transformation is carried out on the mathematical model, a mathematical model of the six-phase induction motor based on a synchronous rotating coordinate system d-q is established, mathematical models of a three-phase voltage type PWM rectifier and a three-phase voltage type PWM inverter are respectively established, and finally, six-phase induction motor double-dq vector control based on SVPWM is carried out; under the motor follow-up power generation condition, the output power of the gas turbine is higher than the total output power of the propulsion system, the redundant power is provided to the motor by the transmission shaft, the motor is driven to generate power, and the requirement of a 10kV level network on harmonics is met.

Description

Technical Field

[0001] This invention belongs to the field of ship propulsion control technology, specifically relating to a control method for a combined diesel-electric and gas turbine propulsion system based on a mathematical model. Background Technology

[0002] With the development of ship propulsion technology, the types of ship propulsion devices have also increased. Currently, ship propulsion systems are mainly divided into two categories: mechanical propulsion and all-electric propulsion. Each type of propulsion has its advantages and disadvantages. To leverage the advantages of both electric and mechanical propulsion, combined diesel-electric / gas turbine propulsion systems emerged at the end of the 20th century. This combined propulsion system uses diesel generator sets to generate electricity, while the propulsion motor and other electrical loads obtain power from the grid through a power conversion device. The propeller is driven by the propulsion motor and the gas turbine, either individually or in combination.

[0003] As more and more large ships convert their power systems to electric propulsion, the power of the motors is increasing, and their voltage levels are also rising. Multiphase motor speed control systems are gradually becoming a better choice. With the increase in the number of phases, the proportion of each phase in the motor system gradually decreases, improving reliability; at the same time, it can reduce the harmonic content of the entire power grid, reducing the impact on the power system. Therefore, in situations with limited power supply voltage, high power output, and high reliability requirements, multiphase systems have more obvious advantages. Summary of the Invention

[0004] To address the shortcomings of the aforementioned technologies, the purpose of this invention is to provide a control method for a combined diesel-electric and gas turbine propulsion system based on a mathematical model.

[0005] The technical solution adopted by this invention to solve its technical problem is: a control method for a diesel-electric-gas combined propulsion system, used to control a diesel-electric-gas combined propulsion system including a transformer, a bidirectional three-phase converter, a six-phase induction motor, a gas turbine, a clutch, a gearbox, and a propeller. The bidirectional three-phase converter includes a three-phase voltage-source PWM rectifier and a three-phase voltage-source PWM inverter. The dual Y-shift 30° six-phase induction motor with power feeding function adopts a squirrel-cage rotor structure and uses a speed outer loop and current inner loop control mode. The steps are as follows:

[0006] Step S1: Establish a mathematical model of the six-phase induction motor based on a natural basis.

[0007] According to KVL's law, for a closed loop, the algebraic sum of the voltages along any direction is always equal to zero. Based on the expression for induced electromotive force ε = dψ / dt, the motor voltage equation can be obtained. In the formula u s For stator voltage, u r For rotor voltage, i s i is the phase current of the stator winding. rψ is the phase current of the rotor winding. s For stator flux linkage, ψ r For rotor flux linkage, R s R is the stator resistance. r Let d be the rotor resistance, and p be the differential operator, replacing the differential symbol d / dt;

[0008] The stator flux linkage is expressed as the sum of the stator self-inductance flux linkage and the rotor-to-stator mutual inductance flux linkage. Similarly, the rotor flux linkage is expressed as the sum of the rotor self-inductance flux linkage and the stator-to-rotor mutual inductance flux linkage. Therefore, the motor flux linkage equation can be obtained. In the formula L ss L is the stator inductance coefficient. sr For the inductance system of the stator corresponding to the rotor, L rs L is the inductance coefficient of the rotor corresponding to the stator. rr The rotor inductance coefficient;

[0009] An electric motor converts input electromagnetic energy into electromagnetic torque and speed, driving the load in the form of mechanical energy. The resulting equation is... That is, the electromagnetic torque output by the motor, where T is the electromagnetic torque output by the motor. em For electromagnetic torque, n p Let θ be the number of pole pairs of the motor, and θ be the electrical angle.

[0010] Obtain the equations of motion for the electric drive system In the formula T e T is the output torque of the motor. L Where ω is the load torque, J is the moment of inertia, D is the torque damping coefficient proportional to the rotational speed, and ω is the torque. r Let D be the mechanical angular velocity of the motor, and K be the torsional elastic torque coefficient. For a constant torque load, we can assume D = 0 and K = 0. Then, the equation of motion for the motor is: This yields a six-dimensional mathematical model of a six-phase induction motor;

[0011] Step S2: Perform spatial decoupling transformation on the mathematical model;

[0012] Using the spatial decoupling matrix T, the six-dimensional motor model in the original space is mapped to three mutually orthogonal two-dimensional coordinate subspaces αβ, z1z2, o1o2, resulting in...

[0013]

[0014] Step S3: Establish a mathematical model of the six-phase induction motor based on the synchronous rotating coordinate system dq;

[0015] Based on spatial decoupling transformation, three motor equations under the harmonic basis of a six-phase induction motor are derived, including voltage, flux linkage, and torque equations. Since vector control of the motor requires a mathematical model of the motor in the dq synchronous rotating coordinate system, and the controller also needs to be designed in the dq synchronous rotating coordinate system, the motor equations related to electromagnetic quantities must undergo Park transformation to obtain the mathematical model needed for deriving and designing vector control strategies. The voltage equation is as follows: The flux linkage equation is In the formula L s L is the inductance coefficient of the stator. r For the rotor's inductance system, L m The inductance coefficient between the rotor and stator is T; the magnetic torque equation is T em =n p L m (i qs i dr -i ds i qr ), where ψ ds , ψ qs , ψ dr , ψ qr Let i be the d-axis and q-axis components of the stator and rotor flux linkages in the dq coordinate system; ds i qs i dr i qr Let ω be the d-axis and q-axis components of the stator and rotor currents in the dq coordinate system; s ω1 is the slip angular velocity; ω2 is the synchronization angular velocity.

[0016] Step S4: Establish the three-phase bridge, filter inductor L, and DC bus support capacitor C based on fully controlled devices. dc Mathematical models of three-phase voltage-source PWM rectifiers and three-phase voltage-source PWM inverters: Bidirectional three-phase converters, also known as back-to-back converters, can be mainly divided into three-phase AC power supply, three-phase voltage-source PWM rectifier (including filtering) and three-phase voltage-source PWM inverter (including filtering). When the motor is working, the power grid is responsible for providing the motor with the required active and reactive power. When the motor is working in the power supply state, it absorbs the electrical energy fed back by the motor. Both the controllable rectifier and the inverter can work in the rectification or inversion state.

[0017] Step S5, Dual dq vector control of a six-phase induction motor based on SVPWM:

[0018] The winding structure of a six-phase induction motor with a 30° Y-shift can be considered as two three-phase windings spatially separated by 30° (with independent neutral points). After decoupling the six-dimensional strongly coupled system through coordinate transformation, a mathematical model that is easy to analyze and control is obtained. Since the two sets of three-phase windings and the six-phase winding are unified in terms of physical quantities in the αβ two-phase stationary coordinate system, it is proven that their fundamental components are equivalent. After transforming each physical quantity to the dq rotating coordinate system, only the angle of the vector changes, while the magnitude of the vector remains unchanged. Therefore, the double three-phase winding and the six-phase winding are still equivalent in the dq rotating coordinate system. Therefore, a single set of three-phase windings can be controlled as a three-phase induction motor. In order to transform the two sets of three-phase windings to the same rotating coordinate system, assuming that the angle between the first set of three-phase windings and the d-axis is θ, then the angle between the second set of three-phase windings and the d-axis is θ-30°.

[0019] By using two coordinate transformations with a 30° difference in angle, the mathematical model of a six-phase induction motor can be simplified into two sets of mathematical models for three-phase windings, i.e.

[0020]

[0021] In the formula, x = 1, 2, representing the first set of windings and the second set of windings.

[0022]

[0023]

[0024] Combining the above three equations, we can obtain the formula required for dual dq vector control:

[0025]

[0026] The neutral points of the two sets of three-phase windings have common-mode voltages with the midpoints of their corresponding inverter DC bus lines, denoted as u. mg and u m’g’ And expressed as In the formula u Ag u Bg u Cg u Dg u Eg u Fg These are the bridge arm voltages of the two inverters (phases A, B, C, D, E, and F).

[0027] Furthermore, 000 to 111 represent the eight switching combinations of the six switching devices in the three-phase bridge arm of the three-phase voltage-source PWM inverter circuit topology. The six non-zero vectors U1 to U6 and the two zero vectors U0 and U7 output under the eight switching states are space vectors respectively. The six space vector planes of U1 to U6 are divided into six sectors with an angle of 60°. The common-mode voltage values ​​of the eight basic voltage vectors are shown in the table below:

[0028]

[0029] As shown in the table above, the common-mode voltages of the two zero vectors U0 and U7 are the largest. Therefore, the effect of the zero vectors needs to be eliminated as much as possible in the SVPWM algorithm to reduce the common-mode voltage. Since the common-mode voltages of U1, U2, and U4 are all -U... dc The common-mode voltages of / 6, U3, U5, and U6 are all U dc / 6, these two sets of basic voltage vectors can be synthesized separately, but doing so will result in a very small maximum linear output voltage. Therefore, a common-mode rejection SVPWM algorithm is designed by combining U1, U2, U4 with U3, U5, U6 to improve the maximum linear output voltage. All non-zero basic voltage vectors used in the conventional SVPWM algorithm are rotated 30° clockwise to obtain new sectors and renumbered. Their indices are the same as the indices of their corresponding basic voltage vectors. U1, U2, U3, U4 ... ref Combine U1, U2, and U4; U falls into sectors S3, S5, and S6. ref Using U3, U5, and U6 as a composite, and taking U1, U2, and U4 as examples, let the durations of the three basic voltage vectors be T1, T2, and T4 respectively. Then the volt-second balance equation is: The expressions for T4, T2, and T1 can be obtained as follows:

[0030] Using formula U ref T = U x T x +U y T y +U0T0 synthesizes the two non-zero vectors and two zero vectors of each sector to obtain all voltage vectors on the plane, where U ref For the desired voltage vector, U x U y U0 is the basic voltage vector of the sector's beginning and ending sides; T is the zero vector, and T is the sampling period. x T y T0 represents the duration of the basic voltage vectors acting on the beginning and end sides of the sector, and T0 represents the duration of the zero vector.

[0031] After determining the action sequence and timing of U0 to U7, the U0-U7 value is determined by sector analysis. ref A three-phase voltage-source PWM inverter can be obtained from the sector in question.

[0032] After adopting common-mode rejection SVPWM, the common-mode voltage is in U dc / 6 and -U dc The voltage varies periodically between the two values ​​of / 6, forming a square wave. In contrast, the common-mode voltage of traditional SVPWM varies between U and U. dc / 2 and -U dc / 2 shows a U-shaped relationship between the two values. dc / 2、U dc / 6、-U dc / 6 and -U dc The voltage changes in a stepwise manner, which is 2 / 2. Therefore, the common-mode rejection SVPWM algorithm can significantly suppress the common-mode voltage.

[0033] Furthermore, in step S4, the switching function for each phase of the three-phase voltage-source PWM rectifier is defined as... Let v NO Capacitor C for DC bus support dc The voltage difference between the positive N-point and the neutral O-point, when applied to the three-phase input circuit of a three-phase voltage-source PWM rectifier using Kirchhoff's voltage law, yields the following three-phase voltage loop equation: From Kirchhoff's current law, the DC bus support capacitor C can be derived. dc Current relationship at negative terminal P

[0034] Furthermore, the feature is that, in order to facilitate the design of the rectifier controller, step S4 requires establishing a mathematical model of the circuit in the dq rotating coordinate system, that is, coordinate transformation is required. First, the variables of the three-phase stationary coordinate system abc are transformed to the αβ two-phase stationary coordinate system through Clark transformation, and then the variables of the αβ two-phase stationary coordinate system are transformed to the dq two-phase synchronous rotating coordinate system through Park transformation, thereby obtaining the low-frequency mathematical model of the three-phase voltage-source PWM inverter. Where v a v b v c i is the three-phase output voltage on the inverter side. La i Lb i Lc For the three-phase AC inductor current of the inverter, v oa v ob v oc The voltages relative to the neutral point O are the AC three-phase capacitor voltages of the inverter, and also the three-phase load voltages, i oa i ob i oc For the three-phase load current, U dc C is the DC bus voltage. dc For DC bus support capacitor, r is the equivalent resistance of AC side line, and L is the filter inductance;

[0035] Select the three-phase inductor current (i) La i Lb i Lc ), capacitor voltage (v) oa v ob voc ) and load current (i oa i ob i oc Using ) as state variables, we can obtain the low-frequency mathematical model of the three-phase voltage-source PWM inverter.

[0036]

[0037] Where v a v b v c This refers to the three-phase output voltage on the inverter's machine side.

[0038] To simplify the inverter controller design, Clark and Park transformations are required to obtain the mathematical model of the three-phase voltage-source PWM inverter in a two-phase rotating dq coordinate system.

[0039]

[0040] Where v a v b v c The inverter's three-phase output voltage on the machine side, v oa v ob v oc The voltages relative to point O are the inverter's three-phase AC capacitor voltages and three-phase load voltages, where C is the filter capacitor, which together with the inductor L forms an LC filter.

[0041] The beneficial effects of this invention are as follows: This invention adopts a motor speed control strategy, simplifies the gas turbine and propeller by constructing a mathematical model, analyzes four typical operating conditions: motor-driven propulsion alone, motor-driven propulsion in parallel and in series, and motor-driven power generation, and sets the propulsion motor operating mode and gear switching method by dividing the propulsion system into gears based on the system output shaft speed and power.

[0042] This invention analyzes the definition of the number of phases in a multiphase motor and the winding structure of a six-phase induction motor, and elucidates the equivalent relationship between the mathematical model of a three-phase induction motor in the αβ coordinate system and the mathematical model of a six-phase induction motor with a double Y-shift of 30°, thus deriving the mathematical model of a six-phase winding induction motor with a double Y-shift of 30°. This invention also utilizes the Matlab / Simulink simulation platform to simulate and analyze the vector control and multi-modal regulation strategies of the six-phase induction motor and the coordinated control method for power flow direction during switching between various operating conditions.

[0043] To control a traditional six-phase induction motor, this invention optimizes the performance of conventional direct torque control (DTC) and proposes an improved rotor flux closed-loop SVM-DTC (Space Vector Modulation and Direct Torque Control) control scheme based on a backpropagation algorithm. Compared with the improved DTC method using PI control, it has advantages such as faster response speed, smaller torque ripple, and fewer design parameters. Direct torque control does not require equating the multiphase induction motor to a DC motor, thus avoiding the complex coordinate transformations and spatial decoupling inherent in vector control systems. Attached Figure Description

[0044] Figure 1 This is a block diagram of the existing diesel-electric-gas combined propulsion system;

[0045] Figure 2 This is a winding structure diagram of a six-phase induction motor with a dual Y-shift of 30° in the propulsion system of this invention;

[0046] Figure 3 This is a structural diagram of the bidirectional three-phase converter system in the propulsion system of this invention;

[0047] Figure 4 This is the topology of a three-phase voltage-source PWM rectifier in a bidirectional three-phase converter;

[0048] Figure 5 This is the topology diagram of a three-phase voltage-source PWM inverter in a bidirectional three-phase converter;

[0049] Figure 6 This is a block diagram of a dual three-phase induction motor rotor magnetic field orientation dual dq vector control system;

[0050] Figure 7 It is the SVPWM voltage space vector;

[0051] Figure 8 yes Figure 7 The 7-segment SVPWM switching status of sector I;

[0052] Figure 9 It is a dual three-phase motor system topology powered by a two-level inverter;

[0053] Figure 10 It is a common-mode suppression SVPWM algorithm sector partitioning;

[0054] Figure 11 It is a simulation model of a combined gas turbine and electric propulsion system;

[0055] Figure 12 This is the output waveform of the motor;

[0056] Figure 13This is the simulated output waveform when the motor is disconnected:

[0057] Figure 14 It is the power curve waveform when the motor is disconnected;

[0058] Figure 15 This is the simulated output waveform of the motor during parallel driving.

[0059] Figure 16 It is the power simulation curve waveform during parallel driving;

[0060] Figure 17 The simulated output waveform of the motor-driven generator;

[0061] Figure 18 The simulated power curve waveform of the motor-driven generator;

[0062] Figure 19 The waveforms are the grid-side voltage and current.

[0063] Figure 20 The results are from the FFT analysis of the grid-side voltage.

[0064] Figure 21 The results are from the FFT analysis of the grid-side current. Detailed Implementation

[0065] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0066] The combined diesel-electric and gas turbine propulsion system mainly consists of important modules such as a transformer, a bidirectional three-phase converter, a six-phase induction motor, a gas turbine, a clutch, a gearbox, and a propeller. Figure 1 As shown, in a combined gas turbine and electric propulsion system, when used for speed regulation, the majority of the motor's power is provided by the gas turbine, which operates with an open-loop throttle input. The bidirectional three-phase converter includes a three-phase voltage-source PWM rectifier and a three-phase voltage-source PWM inverter. The dual Y-shift 30° six-phase induction motor with power feeding function employs a squirrel-cage rotor structure and a speed-outer-current-inner-loop control mode, ensuring high power output only in the final gear. Furthermore, in the combined gas turbine and electric propulsion system, the six-phase induction motor also has a power feeding function. In servo-generation mode, energy is provided by the gas turbine to drive shaft power generation, which is then fed into the ship's electrical grid via the bidirectional three-phase converter.

[0067] The winding structure of a six-phase induction motor with a squirrel-cage rotor structure and a double Y-shift 30° is as follows: Figure 2As shown, the stator winding has 6 output terminals (i.e., A1 and A2, B1 and B2, C1 and C2), and the phase angle is 30°. It can also be called a half-twelve-phase motor or a phase-shifted double three-phase motor. Its stator winding can be regarded as consisting of two sets of three-phase windings. A1, B1, and C1 are the first set of three-phase stator windings, and a1, b1, and c1 are the corresponding rotor windings. A2, B2, and C2 are the second set of three-phase windings, and a2, b2, and c2 are the corresponding rotor windings. Their neutral points are not connected.

[0068] The present invention relates to a control method for a combined diesel-electric and gas turbine propulsion system based on a mathematical model, the steps of which are as follows.

[0069] S1. Establish a mathematical model of a six-phase induction motor in the natural coordinate system, including voltage equation, flux linkage equation, torque equation, and motion equation.

[0070] According to KVL's law, for a closed loop, the algebraic sum of the voltages along any direction is always equal to zero. Furthermore, based on the expression for induced electromotive force: ε = dψ / dt, the motor voltage equation can be obtained as follows:

[0071]

[0072] In the formula u s For stator voltage, u r For rotor voltage, i s i is the phase current of the stator winding. r ψ is the phase current of the rotor winding. s For stator flux linkage, ψ r For rotor flux linkage, R s R is the stator resistance. r d is the rotor resistance. p is the differential operator, replacing the differential symbol d / dt.

[0073] The stator flux linkage is expressed as the sum of the stator self-inductance flux linkage and the rotor-to-stator mutual inductance flux linkage. Similarly, the rotor flux linkage is expressed as the sum of the rotor self-inductance flux linkage and the stator-to-rotor mutual inductance flux linkage. Therefore, the motor flux linkage equation can be obtained.

[0074]

[0075] In the formula L ss L is the stator inductance coefficient. sr For the inductance system of the stator corresponding to the rotor, L rs L is the inductance coefficient of the rotor corresponding to the stator. rr is the rotor inductance coefficient.

[0076] An electric motor converts input electromagnetic energy into electromagnetic torque and speed, driving the load in the form of mechanical energy. Based on the interconversion of energy, the output electromagnetic torque T of the motor can be obtained. em The equation is:

[0077]

[0078] In the formula n p Let θ be the number of pole pairs of the motor, and θ be the electrical angle.

[0079] The equations of motion for an electric drive system can be expressed as follows:

[0080]

[0081] In the formula T e T is the output torque of the motor. L For the load torque, T e Where J is the motor output torque, D is the moment of inertia, and ω is the torque damping coefficient proportional to the rotational speed. r Let D be the mechanical angular velocity of the motor, and K be the torsional elastic torque coefficient. For a constant torque load, we can assume D = 0 and K = 0, then the equation of motion for the motor is:

[0082]

[0083] The present invention proposes a dq pure hybrid model that incorporates mutual leakage flux and digital field-oriented control. It uses a two-level six-phase inverter to control a dual-Y asynchronous motor. This scheme has the advantages of DFOC (Digital Field-Oriented Control) schemes, such as simple structure, fast response speed, and reduced torque ripple. However, the number of space voltage vectors used in six-phase SVPWM modulation is much greater than that in three-phase SVPWM. Due to the equivalent relationship between the mathematical models of a three-phase induction motor and a dual-Y-shifted 30° six-phase induction motor, two three-phase inverters can be used to drive the dual-Y-shifted 30° six-phase induction motor.

[0084] Step S2: Each phase winding of the six-phase induction motor can generate independent stator voltage and current. Therefore, the six-phase induction motor can be regarded as a six-dimensional mathematical system. Generally speaking, such a six-dimensional system has strong coupling, making direct analysis extremely difficult. Decoupling is necessary to simplify the mathematical model.

[0085] Therefore, using the spatial decoupling matrix T, the six-dimensional motor model in the original space is mapped to three mutually orthogonal two-dimensional coordinate subspaces αβ, z1z2, o1o2, i.e.

[0086]

[0087] After spatial decoupling transformation, the αβ subplane contains only the fundamental wave and 12m±1 (m=1,2,3,…) harmonic components, and overlaps with the plane containing the magnetic flux. Therefore, the fundamental wave component of the motor can induce a rotating magnetomotive force on the rotor winding, driving the rotor to rotate. The o1o2 subplane contains only 3m (m=1,2,3,…) harmonic components, and the z1z2 subplane contains only 6m±1 (m=1,2,3,…) harmonic components, which are unrelated to electromechanical energy conversion.

[0088] Step S3: Based on the spatial decoupling transformation, derive the three motor equations under the harmonic basis of the six-phase induction motor, including voltage, flux linkage, and torque equations. Since the vector control of the motor requires a mathematical model of the motor in the dq synchronous rotating coordinate system, and the controller also needs to be designed in the dq synchronous rotating coordinate system, the motor equations related to electromagnetic quantities need to be transformed using Park transformation to obtain the mathematical model required for deriving and designing the vector control strategy.

[0089] The voltage equation is

[0090]

[0091] In the formula ψ ds , ψ qs , ψ dr , ψ qr Let i be the d-axis and q-axis components of the stator and rotor flux linkages in the dq coordinate system; ds i qs i dr i qr Let ω be the d-axis and q-axis components of the stator and rotor currents in the dq coordinate system; s ω is the slip angular velocity; ω1 is the synchronous angular velocity; the flux linkage equation is:

[0092]

[0093] In the formula L s L is the inductance coefficient of the stator. r For the rotor's inductance system, L m The inductance coefficient between the rotor and stator; electromagnetic torque T em The equation is T em =n p L m (i qs i dr -i ds i qr (9)

[0094] Step S4, the bidirectional three-phase converter, also known as a back-to-back converter, has the following structure: Figure 3As shown, its main structure can be divided into a three-phase AC power supply, a PWM rectifier (including filtering), and a PWM inverter (including filtering). The power grid is responsible for providing the motor with the required active and reactive power when the motor is running, and absorbs the electrical energy fed back by the motor when the motor is in a feed-powered state. Both the controllable rectifier and the inverter can operate in either rectification or inversion mode.

[0095] Three-phase voltage-source PWM rectifier topology, such as Figure 4 As shown, it includes a three-phase bridge composed of fully controlled components, a filter inductor L, and a DC bus support capacitor C. dc R is the equivalent resistance of the line.

[0096] exist Figure 4 In the middle, e a e b e c For the three-phase AC voltage of a three-phase power grid, i a i b i c It is the three-phase inductor current on the grid side, u a u b u c U is the three-phase input voltage on the rectifier side. dc i is the DC-side output bus voltage. dc This is the DC output current, with a resistive load R. L i L This is the load current.

[0097] For the switching state of each phase of the PWM rectifier, the switching function is defined as follows:

[0098]

[0099] Let v NO Capacitor C for DC bus support dc The voltage difference between the positive terminal N and the neutral terminal O. Applying Kirchhoff's voltage law to the three-phase input circuit of the rectifier, the three-phase voltage loop equation is obtained as follows:

[0100]

[0101] From Kirchhoff's current law, the DC bus support capacitor C can be derived. dc The current relationship at the negative terminal P is:

[0102] To facilitate the design of the rectifier controller, a mathematical model of the circuit in the dq rotating coordinate system needs to be established, which requires coordinate transformation. First, the three-phase stationary coordinate system variables abc are transformed to the αβ two-phase stationary coordinate system using Clark transformation. Then, the αβ two-phase stationary coordinate system variables are transformed to the dq two-phase synchronous rotating coordinate system using Park transformation. This yields the low-frequency mathematical model of the PWM rectifier.

[0103]

[0104] Three-phase voltage-source PWM inverter topology as follows: Figure 5 As shown in the figure. v a v b v c The inverter's three-phase output voltage on the machine side; i La i Lb i Lc For the inverter's three-phase AC inductor current; v oa v ob v oc The voltages relative to point O are the AC three-phase capacitor voltages of the inverter, and also the three-phase load voltages; oa i ob i oc This refers to the three-phase load current; U dc C is the DC bus voltage. dc is the DC bus support capacitor; r is the equivalent resistance of the AC side line; L is the filter inductance.

[0105] This patent application selects three-phase inductor current (i La i Lb i Lc ), capacitor voltage (v) oa v ob v oc ) and load current (i oa i ob i oc If ) is used as a state variable, then the low-frequency mathematical model of the PWM inverter can be obtained, that is

[0106]

[0107] To simplify the inverter controller design, Clark and Park transformations are required to obtain the mathematical model of the PWM inverter in a two-phase rotating dq coordinate system, i.e.

[0108]

[0109] The winding structure of a double Y-shifted 30° six-phase induction motor can be considered as two three-phase windings spatially separated by 30° (with independent neutral points). After decoupling the six-dimensional strongly coupled system through coordinate transformation, a mathematical model that is easy to analyze and control is obtained. Since the two sets of three-phase windings and the six-phase winding are unified in physical quantities in the αβ two-phase stationary coordinate system, it is proven that their fundamental components are equivalent. After transforming to the dq rotating coordinate system, only the angle of the vector changes, while the magnitude of the vector remains unchanged. Therefore, the double three-phase winding and the six-phase winding are still equivalent in the dq rotating coordinate system. Therefore, a single set of three-phase windings can be controlled as a three-phase induction motor. In order to transform the two sets of three-phase windings to the same rotating coordinate system, assuming that the angle between the first set of three-phase windings and the d-axis is θ, then the angle between the second set of three-phase windings and the d-axis is θ-30°.

[0110] By using two coordinate transformations with a 30° difference in angle, the mathematical model of a six-phase induction motor can be simplified into two sets of mathematical models for three-phase windings, i.e.

[0111]

[0112] In the formula, x = 1, 2, representing the first set of windings and the second set of windings.

[0113]

[0114]

[0115] Combining the above three equations, we can obtain the formula required for dual dq vector control:

[0116]

[0117] When the motor is running stably, the rotor flux linkage remains constant, and its partial derivative is zero. At this time, only the excitation component of the stator current (i) exists. ds1 +i ds2 The rotor flux linkage is determined by the torque component (i). qs1 +i qs2 This is irrelevant, simplifying the model, facilitating control, and achieving good control characteristics. The block diagram of a dual three-phase induction motor rotor field-oriented dual dq vector control system is shown below. Figure 6 As shown.

[0118] Traditional Voltage Space Vector Pulse Width Modulation (SVPWM) technology: Unlike the standard voltage pulse (SPWM) algorithm, the SVPWM algorithm does not use a directly given voltage waveform. Instead, it synthesizes a given voltage vector using different basic voltage vectors. When the motor is running, the synthesized voltage vector rotates with the motor's motion. The changing voltage vector causes the current to change over time, thus generating a nearly circular rotating magnetic flux in the air gap.

[0119] In a three-phase voltage-source PWM inverter circuit topology, there are six switching devices in the three-phase bridge arms, resulting in eight possible switching combinations, which can be represented by 000 to 111. The inverter outputs eight basic voltage space vectors U0 to U7 in these eight switching states, including six non-zero vectors U1 to U6 and two zero vectors U0 and U7, as shown in the spatial distribution diagram. Figure 7 As shown.

[0120] All voltage vectors located in this plane, regardless of their sector, can be synthesized from the two non-zero vectors and two zero vectors of that sector, i.e., U ref T = U x T x +U y T y +U0T0 (20)

[0121] In the formula U ref U is the desired voltage vector; x U y U0 is the basic voltage vector of the sector's beginning and ending sides; T is the zero vector; T is the sampling period; T x T y T0 represents the duration of the basic voltage vectors acting on the beginning and end sides of the sector; T0 represents the duration of the zero vector.

[0122] Using a voltage synthesis method, any desired synthesized voltage vector in the coordinate system can be represented by calculating the duration of action of eight basic voltage vectors with fixed magnitudes and spatial orientations. Once the action sequence and duration of U0 to U7 are determined, and then U is determined through sector analysis... ref The sector in which the inverter is located determines the switching state and timing of all its switching devices, thus enabling SVPWM modulation. Taking sector I as an example, using 7-segment SVPWM, the switching states of the inverter's three-phase bridge arms are as follows: Figure 8 As shown.

[0123] The topology of a dual three-phase motor system powered by a two-level inverter is as follows: Figure 9 As shown. Analysis Figure 9 It can be seen that the neutral points of the two sets of windings in the dual three-phase induction motor are not connected to each other, and are also independent of the midpoint of the inverter's DC bus. The neutral points of the two sets of three-phase windings each have a common-mode voltage with their corresponding midpoints of the inverter's DC bus, denoted as u. mg and u m’g’ And expressed as

[0124] In the formula u Ag u Bg u Cg u Dg u Eg u FgThese are the bridge arm voltages of the two inverters (phases A, B, C, D, E, and F).

[0125] Step S5: The SVPWM algorithm uses eight basic voltage vectors, U0 to U7. Table 1 shows the common-mode voltage values ​​of the eight basic voltage vectors.

[0126] Table 1 Common-mode voltage values ​​of the eight basic voltage vectors

[0127]

[0128] Analysis of Table 1 shows that the common-mode voltages of the two zero vectors, U0 and U7, are the largest. Therefore, the effect of the zero vectors needs to be eliminated as much as possible in the SVPWM algorithm to reduce the common-mode voltage. Since the common-mode voltages of U1, U2, and U4 are all -U... dc The common-mode voltages of / 6, U3, U5, and U6 are all U dc / 6. These two sets of basic voltage vectors can be synthesized separately, but doing so will result in a very small maximum linear output voltage. Therefore, a common-mode rejection SVPWM algorithm is designed by combining U1, U2, U4 with U3, U5, and U6 to improve the maximum linear output voltage. All non-zero basic voltage vectors used in the conventional SVPWM algorithm are rotated 30° clockwise, as follows: Figure 10 As shown.

[0129] like Figure 10 As shown, the new sectors are renumbered, and their indices are the same as the indices of their corresponding basic voltage vectors for easier analysis. U values ​​falling into sectors S1, S2, and S4... ref Combine U1, U2, and U4; U falls into sectors S3, S5, and S6. ref Synthesize using U3, U5, and U6. Taking U1, U2, and U4 as examples, and setting the durations of the three fundamental voltage vectors as T1, T2, and T4 respectively, the volt-second balance equation is:

[0130]

[0131] From equation (22), the expressions for the action times T4, T2, and T1 corresponding to U1, U2, and U4 are respectively...

[0132]

[0133] After adopting common-mode rejection SVPWM, the common-mode voltage is in U dc / 6 and -U dc The voltage varies periodically between the two values ​​of / 6, forming a square wave. In contrast, the common-mode voltage of traditional SVPWM varies between U and U. dc / 2 and -U dc / 2 shows a U-shaped relationship between the two values. dc / 2、U dc / 6、-U dc / 6 and -U dc The voltage varies in steps of 2 / 2, therefore, the SVPWM (Simultaneous Mode Suppression Pulse Width Modulation) algorithm can significantly suppress the common-mode voltage. This invention constructed a simulation system for a diesel-electric-gasoline combined propulsion system and designed a control strategy employing motor speed control. Experimental results demonstrate its excellent operating characteristics. This invention also studied the operating characteristics of a six-phase induction motor speed control system under various operating conditions and during switching in a gas-electric combined propulsion system, providing a theoretical basis for subsequent research and design.

[0134] Modeling and simulation analysis of typical operating conditions of gas-fired power combined propulsion system.

[0135] Ship operating systems are mainly divided into four typical operating conditions: PTH (propulsion by electric motor alone), GT (propulsion by gas turbine alone), PTI (propulsion by parallel operation), and PTO (propulsion by electric motor following the generator). Since this patent application focuses on the six-phase induction motor in a combined gas turbine and electric propulsion system, it only analyzes PTH, PTI, and PTO, including the switching analysis between the above modes.

[0136] The combined propulsion system employs two operating modes: PTH and PTI. To leverage the strong speed regulation capability of the electric motor, the system utilizes motor speed control. The electric motor acts as a speed governor, with most power supplied by the gas turbine. The gas turbine uses an open-loop throttle setting, while the electric motor uses an outer speed loop and an inner current loop control. The motor only outputs high power in the final gear. The system's speed-up and gear-shifting process is briefly described as follows: 1) In the low gear, the combined propulsion system is driven solely by the electric motor (PTH); 2) As the gear shifts, if the load exceeds the electric motor's capacity, the electric motor disconnects, and the load is transferred to the gas turbine, which then drives the ship solely (GT). The electric motor continuously coordinates with the gas turbine, actively adjusting the drive shaft speed and compensating for power; 3) When the gear shifts to the highest gear, the electric motor rejoins, working with the gas turbine to propel the ship at full speed (PTI). The combined propulsion system is divided into 10 levels, of which levels 1-5 are the motor-only propulsion mode, levels 6-9 are the gas turbine-only propulsion mode, and level 10 is the motor and gas turbine combined propulsion mode.

[0137] PTO (Power Toll Collection) mode: When the ship is in economic cruising mode (gas turbine driven alone), the spare capacity of the main engine (gas turbine) is utilized to drive the shaft-driven motor to generate electricity while driving the propeller. This generates electricity in parallel with the ship's power station according to the ship's power demand. In PTO mode, there are two operating modes: 1) Grid-connected mode: the shaft-driven generator inverter and the auxiliary engine (ship's generator) work simultaneously in parallel to form the ship's power station; 2) Standalone grid mode: the auxiliary engine (ship's generator) is shut down, and the shaft-driven generator system forms the ship's power station independently.

[0138] Considering that the excitation current of the asynchronous motor needs to be provided by the three-phase converter during startup, this simulation only considers the grid-connected mode. A simulation model of the combined gas turbine and electric propulsion system is built in the MATLAB / Simulink environment, as follows: Figure 11 As shown in the diagram, this system uses an electric motor as a speed governor. The gas turbine speed follows the electric motor speed, and the gas turbine is given a throttle according to the gear position, i.e., a given output power. The electric motor torque is obtained by calculating the difference between the total output power and the gas turbine power, and then calculating the torque difference based on the speed of each gear.

[0139] The total power of the combined gas turbine and electric propulsion system simulation model is 50MW, of which the rated power of the propulsion motor is 15MW and the full power of the gas turbine is 36MW. The key parameters of the propulsion motor used in this simulation are summarized in Table 2.

[0140] Table 2. Key parameters of the propulsion motor used in the simulation of this embodiment.

[0141]

[0142] The division of gears 1-5 for motor-driven operation is shown in Table 3.

[0143] Table 3 shows the division of gears 1-5 for motor-only propulsion operation.

[0144]

[0145] Based on the system output power and speed of gears 1-5, the motor speed and torque in each of the first 5 gears can be obtained. The motor speed is [249 403 540 738 813], in r / min; the motor torque is [9703 2497 5444 408 1907 99076], in N*m. Since the motor has a propeller, the speed cannot change abruptly; therefore, a gradual increase in speed is required when changing gears.

[0146] The simulation time was set to 50 seconds, with a 5-second rise time and a 5-second settling time for each gear. The simulation results are as follows. Figure 12 As shown. Figure 12 This represents the motor output waveform, including single-phase stator current, motor torque, and motor rotor speed. Analysis Figure 12 It can be seen that when the propulsion system shifts up, the load torque suddenly increases, causing a sudden increase in motor torque and overshoot, while the motor speed suddenly drops. The speed overshoot increases with the increase of load, with a maximum speed overshoot of 13.6%; the torque overshoot decreases with the increase of load, with a maximum torque overshoot of 11.3%. Both track the given value or given curve after 0.5s, indicating that the propulsion system has good speed regulation characteristics and load-carrying capacity.

[0147] Parallel propulsion (PTI) and disconnection mode: According to the working condition switching process of the combined propulsion system, when shifting from 5th gear to 6th gear, the motor needs to be disconnected and the load is transferred to the gas turbine; when shifting from 9th gear to 10th gear, the motor needs to be paralleled with the gas turbine to propel the ship forward at full speed.

[0148] 1) The disconnection requires the motor load to be transferred to the gas turbine, but the propulsion system speed is still controlled by the motor. Therefore, the load torque of the motor must be transferred to the gas turbine during disconnection to ensure that the total output power remains unchanged before and after disconnection.

[0149] The simulation is set up so that the motor operates alone for the first 50 seconds. At 58 seconds, the turbine is disconnected, switching from an inertial state to operation at a given throttle. The motor's load torque changes from a setpoint to the difference between the total output torque and the gas turbine torque. The motor speed remains under closed-loop control. The simulation results are as follows: Figure 13 and Figure 14 As shown. Figure 13 This represents the output waveform when the motor is disconnected, including single-phase stator current, motor torque, and motor rotor speed; Figure 14 This represents the power waveform during motor disconnection. Analysis. Figure 13 and Figure 14 It can be seen that within 1 second after the disconnection command was issued, the motor's electromagnetic torque returned to zero, and the stator current amplitude decreased significantly, indicating successful unloading. During the sudden load disconnection, the motor speed jumped from 817 r / min to 950 r / min, stabilizing at 817 r / min after 1.3 seconds. The motor power decreased from 8.47 MW to 0, while the gas turbine power increased from 3 MW in an inertial operating state to 8.47 MW. Before 58 seconds, the gas turbine was not under load, and the propulsion system's output power was solely provided by the motor; therefore, the total output power remained unchanged before and after disconnection.

[0150] 2) Because the power required for the final gear of the combined propulsion system exceeds the maximum power that the gas turbine can provide, a motor is needed to compensate for the power shortfall. The simulation is set to perform parallel operation for 90 seconds while simultaneously upshifting; therefore, the motor speed needs to increase during the parallel operation and upshifting process, and a 5-second increase time is also set here. The simulation results are as follows: Figure 15 and Figure 16 As shown in the diagram. Analysis reveals that when the motor is connected in parallel, the motor load gradually increases from 0 to 87 kN·m, with a sudden drop in speed at the moment of load transfer. Since the motor directly loads 12 MW during a single upshift, both its speed and torque fluctuate during the upshift, eventually stabilizing in about 15 seconds. During the upshift, the increased motor torque causes a sudden drop in the gas turbine torque, resulting in a sudden drop in output power. Once the motor power stabilizes, the gas turbine output power recovers, reaching full power output. The combined propulsion system, with the motor and gas turbine operating in parallel, ultimately outputs 43 MW at 10th gear.

[0151] Table 4. System speed and power parameters under PTO mode

[0152]

[0153] According to the gear division in Table 4, the motor speed is set to [54867681794810151217] r / min, corresponding to a time setting of [0406080100120] s, the same as the previous motor speed setting. A 5-second upshift time is set when shifting gears. The given torque of the motor is the difference between the gas turbine torque and the total output torque.

[0154] During the first five gears, the motor feedback power will be designed at the maximum value in the range of Table 4, that is, a given generating torque of 95500 N·m. Only in the last gear, due to the increase in the total output power of the system, the residual power provided by the gas turbine to the motor for generating electricity decreases, the generating power decreases, and the absolute value of the electromagnetic torque decreases relatively.

[0155] The simulation time was set to 130 seconds. The first 30 seconds of the simulation were in the third gear condition where the motor drove alone, and the remaining 30 seconds were in the motor-driven generator condition. The simulation results are as follows: Figures 17-21 As shown. By Figures 17-21 Simulation results show that when the motor switches from motoring to generating mode, the electromagnetic torque abruptly changes from positive to negative. The speed suddenly increases to 740 r / min, and the speed and torque stabilize to the given value after 2 seconds. Subsequently, when increasing the generating mode, the actual motor speed tracks the given speed well, although the electromagnetic torque fluctuates during the speed increase. In the motor-driven generating mode, the gas turbine output power is higher than the total output power of the propulsion system; the excess power is supplied to the motor via the drive shaft, driving the motor to generate electricity. The grid-side voltage and current waveforms are out of phase, indicating that the grid-side three-phase converter is in inverter mode, demonstrating successful grid-connected power generation. Figure 17 The FFT analysis results of the grid-side voltage and current show that the voltage THD is 0.18% and the current THD is 3.81%, which meets the national requirements for harmonics in 10kV networks.

Claims

1. A control method for a diesel-electric-gas combined propulsion system, used to control a diesel-electric-gas combined propulsion system comprising a transformer, a bidirectional three-phase converter, a six-phase induction motor, a gas turbine, a clutch, a gearbox, and a propeller, wherein the bidirectional three-phase converter comprises a three-phase voltage-source PWM rectifier and a three-phase voltage-source PWM inverter, the dual Y-shift 30° six-phase induction motor with power feeding function adopts a squirrel-cage rotor structure, and a control mode of speed outer loop and current inner loop is adopted, characterized in that: Includes the following steps Step S1: Establish a mathematical model of a six-phase induction motor. According to the expression of induced electromotive force ε = dψ / dt The motor voltage equation is obtained. , In the formula u s Stator voltage, u r For rotor voltage, i s This refers to the phase current of the stator winding. i r This refers to the phase current of the rotor winding. ψ s For stator flux linkage, ψ r For rotor flux linkage, R s For stator resistance, R r For rotor resistance, p For differential operators, replace the differential symbol d / dt ; The stator flux linkage is expressed as the sum of the stator self-inductance flux linkage and the rotor-to-stator mutual inductance flux linkage. The rotor flux linkage is expressed as the sum of the rotor self-inductance flux linkage and the stator-to-rotor mutual inductance flux linkage, thus yielding the motor flux linkage equation. , In the formula L ss The stator inductance coefficient, L sr The inductance coefficient of the stator corresponding to the rotor. L rs The inductance coefficient of the rotor corresponding to that of the stator. L rr The rotor inductance coefficient; The electric motor drives the load in the form of mechanical energy, thus yielding the equation for the motor's output electromagnetic torque. ， In the formula T em For electromagnetic torque, n p This represents the number of pole pairs of the motor. θ For electrical angle, , ; Obtain the equations of motion for the electric drive system , In the formula T e For the motor output torque, T L For load torque, J For rotational inertia, D The torque damping coefficient is proportional to the rotational speed. ω r Let be the mechanical angular velocity of the motor. K The torsional elastic torque coefficient is used for constant torque loads. D =0, K =0, thus obtaining the equation of motion for the motor. , This yields a six-dimensional mathematical model of a six-phase induction motor; Step S2: Perform spatial decoupling transformation on the mathematical model; Using spatial decoupling matrix T The six-dimensional motor model in the original space is mapped to three mutually orthogonal two-dimensional coordinate subspaces. αβ , z 1 z 2, o 1 o In 2, we obtained ； Step S3: Establish a mathematical model of the six-phase induction motor based on the synchronous rotating coordinate system dq; Based on the voltage, flux linkage, and torque equations derived from the spatial decoupling transformation, the motor equations related to electromagnetic quantities are subjected to the Park transformation to obtain the mathematical model required for deriving and designing the vector control strategy; among which the voltage equation is... , In the formula ψ ds , ψ qs , ψ dr , ψ qr for dq Stator and rotor flux linkages in coordinate system d shaft and q Axial components, i ds , i qs , i dr , i qr for dq Stator and rotor currents in coordinate system d shaft and q Axial components, ω 1 represents the synchronous angular velocity. ω s It is the slip angular velocity; The flux linkage equation is In the formula L s The inductance of the stator. L r Let be the inductance coefficient of the rotor. L m The inductance coefficient between the rotor and stator is given by: The electromagnetic torque equation is: ; Step S4: Establish the three-phase bridge and filter inductor based on fully controlled devices respectively. L and DC bus support capacitor C dc Mathematical models of three-phase voltage-source PWM rectifiers and three-phase voltage-source PWM inverters: The switching function of each phase of the three-phase voltage-source PWM rectifier is defined as follows: ,set up v NO DC bus support capacitor C dc positive electrode N Point and Neutral Point O The voltage difference between the points is used to apply Kirchhoff's voltage law to the three-phase input circuit of a three-phase voltage-source PWM rectifier, resulting in the three-phase voltage loop equation. In the formula e a , e b , e c For three-phase AC voltage of a three-phase power grid, i a , i b , i c It is the three-phase inductor current on the grid side. U dc This is the DC bus voltage. i dc This is the DC-side output current. R L For resistive load, i L Given the load current, the DC bus support capacitance is derived using Kirchhoff's current law. C dc Current relationship at negative terminal P ; Establish a three-phase voltage-source PWM inverter in dq The mathematical modeling steps in a rotating coordinate system are as follows: The three-phase stationary coordinate system is transformed using Clark transformation. abc Variable transformation to αβ Two stationary coordinate systems, then transformed by Park. αβ Two-phase stationary coordinate system variable transformation to dq A two-phase synchronous rotating coordinate system is used to obtain the low-frequency mathematical model of the three-phase voltage-source PWM inverter. , Select three-phase inductor current i La , i Lb , i Lc capacitor voltage v oa , v ob , v oc and load current i oa , i ob , i oc As a state variable, U dc This is the DC bus voltage. C dc The DC bus support capacitor is used; the low-frequency mathematical model of the three-phase voltage-source PWM inverter is obtained as follows: ， in v a , v b , v c This refers to the three-phase output voltage on the inverter side. v oa , v ob , v oc Relative to O The voltage at the point is the inverter's three-phase AC capacitor voltage, which also serves as the three-phase load voltage. i La , i Lb , i Lc This refers to the AC three-phase inductor current of the inverter. L For filtering inductors, C For filtering capacitors; Clark and Park transformations are performed on a three-phase voltage-source PWM inverter to obtain a two-phase rotating inverter. dq Mathematical model of a three-phase voltage-source PWM inverter in coordinate system , In the formula i oa , i ob , i oc This refers to the three-phase load current. C For filtering capacitors; r The equivalent resistance of the AC side line; Step S5, Dual dq vector control of a six-phase induction motor based on SVPWM: The winding structure of a double Y-shifted 30° six-phase induction motor is considered as two three-phase windings spatially separated by 30°. Coordinate transformation is used to decouple the strongly coupled six-dimensional system. A single set of three-phase windings is controlled as a single three-phase induction motor. The first set of three-phase windings and... d The included angle of the axis is θ The second set of three-phase windings is with d Axis angle θ -30°; By using two coordinate transformations with a 30° difference in angle, the mathematical model of the six-phase induction motor is simplified into two sets of mathematical models for three-phase windings. , In the formula, x=1,2 represents the first set of windings and the second set of windings. ， ， Combining the above three equations, we can obtain the formula required for dual dq vector control: , The neutral points of the two sets of three-phase windings have common-mode voltages with the midpoints of their corresponding inverter DC bus lines, denoted as . u mg and u m’g’ , represented as In the formula u Ag , u Bg , u Cg , u Dg , u Eg , u Fg These are the bridge arm voltages of the two inverters, respectively. Let 000~111 represent the eight switching combinations of the six switching devices in the three-phase bridge arm of a three-phase voltage-source PWM inverter circuit topology, and the six non-zero vectors output in each of the eight switching states. U 1~ U 6 and two zero vectors U 0、 U 7 spatial vectors, U 1~ U The six space vector planes are divided into six sectors at an angle of 60°. The common-mode voltage values ​​of the eight basic voltage vectors are shown in the table below: ， Using formula The two non-zero vectors and two zero vectors of each sector are combined to obtain all voltage vectors on the plane, where... U ref For the desired voltage vector, U x , U y For the basic voltage vectors of the beginning and end sides of the sector, U 0 is the zero vector. T The sampling period is T x , T y The duration of the basic voltage vectors at the beginning and end of the sector. T 0 represents the zero vector action time; Based on combination U 1, U 2, U 4 and U 3, U 5, U The SVPWM algorithm with common-mode rejection in version 6 rotates all non-zero basic voltage vectors 30° clockwise to obtain new sectors and renumbers them, placing them into sectors S1, S2, and S4. U ref use U 1, U 2, U 4. Synthesis; falling into sectors S3, S5, and S6 U ref use U 3 ，U 5 ，U 6. Combining, let the duration of action of the three basic voltage vectors be... T 1. T 2 and T 4, then U 1, U 2, U The 4-volt-second equilibrium equation is , achievable T 4. T 2 and T The expression for 1 is .

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