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A CALCULATION METHOD OF TRANSIENT MODES OF ELECTRIC SHIPS' PROPELLING ELECTRIC PLANTS.

UDC 629.12-8:629.12.037

Actuality of the problem. In recent years, interest in the use of electromotion on ships of the merchant and navy has noticeably increased in the world shipbuilding industry. This is due to a number of undeniable advantages inherent in such a method of power transfer to propellers: the possibility of using high-speed thermal engines (TEs), using simplified design engines (nonreversible TEs are installed), splitting the total power into several parts and the ability of each heat engine to operate several screws which increases the vitality and flexibility of the power plant), the reduction of shaft length, the wide use of automation systems. The most important advantage of electromotion before the traditional drive of ship propulsors is the ability to provide high maneuverability of vessels equipped with this type of power plant. It is the maneuvering qualities of the propeller system that ensure, above all, the safety of maneuver operations by ships.

Modern rowing electrical installations (REIs) are built on the basis of induction frequency-controlled electric motors and based on valve propulsion motors. At the same time, a large number of vessels are in operation, the power plants of which are constructed using an alternating-direct current system. In this regard, both the design of modern electric trains with high maneuvering properties as well as the tasks of improving the management of rowing power plants of vessels in service are very relevant. Moreover, the need to assess the maneuverability of newly built electric boats is already at the initial stages of their design, and increasing the efficiency of performing maneuver operations of existing vessels is undoubtedly relevant at any stage of their life cycle.

State of the issue under consideration. A distinctive feature of the work of electric boats on maneuvers is that the duration of the transient processes in the ship electric power plant (SEPP) is commensurate with the duration of the transient processes of the ship's movement. Therefore, the traditional assumptions about the constancy of the speed of the propellers during the movement of the ship on maneuvers (as well as vice versa) are not acceptable. All the components of the electric motor complex are in close interconnection. Only when solving particular problems is it permissible to consider any element of a single propulsion complex in isolation from the others. (This, in particular, refers to the steady movement). When analyzing unsteady regimes, propulsion power plants should be considered in unity with all other elements of the propulsion complex, including SEPP, propellers and hull. Only in this setting can a thorough analysis of their maneuvering operating modes and assess the maneuverability of the electric ship.

An analysis of the state of the issue under consideration shows the following. Studies of transient modes of operation of propulsion power plants have a long history. Numerous research works deal with the problems of analytical and machine methods of calculating the basic maneuvering regimes--starting, stopping, reversing propellers. However, even today they all have a serious shortcoming. In practice, only the electric part of the propulsion system (generators, propulsion motors, converters, control panels, control stations) is meant and implied under the electromotive system without primary engines and propulsors. In particular, this is clearly presented in [1-4]. The main distinguishing feature of these and other thorough works is that the speed of the vessel's motion is considered constant throughout the considered maneuvers. Thus, the propulsion system is considered in isolation from a single ship propulsion complex. Of course, as a result of this simplification, it is much easier to analyze the maneuvering modes of the REI operation, but at the same time, the accuracy of the results obtained is reduced and, most importantly, the system principle of the approach to the analysis of operating modes of ship electric power plants is violated. There is no possibility to evaluate the efficiency of the electric power plant operation according to the <<final result>>--in terms of the quality of the maneuver performance of the electric boat as a whole.

The second distinctive feature of earlier studies is that the overwhelming majority of them have been performed with respect to specific electric trains (at best to specific series of electric boats), to specific propulsion systems. The results of the calculations obtained with the help of these methods refer to specific SEPPs and can not be extended to other electric vessels. This does not allow us to carry out wide, valid scientific generalizations. This reduces the scientific value of the results of the investigations carried out.

An attempt to eliminate these shortcomings was undertaken by one of the authors of this paper in [5]. To analyze the maneuvering regimes of rowing electric power plants as part of propulsion complexes of electric ships, he developed a corresponding mathematical model. It describes the transient modes of operation of all components of the complex. The model is universal, covers the vast majority of modern and prospective electric boats with a traditional type of propulsors. At the same time, when describing the processes occurring in thermal engines, a number of assumptions were adopted that do not allow analyzing the dynamic modes of their operation. In addition, a mathematical description of the processes occurring in synchronous generators was built on the basis of a vector diagram of the generator, which also made it impossible to fully appreciate the dynamic modes of their operation. Thus, the mathematical description of the transient modes of operation of diesel generators is in need of improvement, which led to the need for this work.

The goal of the paper is development of an improved mathematical model of transitional modes of propulsion power plants of electric boats and a method of calculating on its basis maneuverable modes of operation of ships.

The solution of the problem was carried out with reference to the most common electric-motive system on the basis of frequency-controlled induction propulsion motors.

A method for solving the problem. The structural diagram of the SEPP <<thermal engine--synchronous generator--frequency converter--induction motor>> of the propulsion complex of the electric ship is shown in Fig. 1. It corresponds to the generally accepted in the theory of electromotion the layout of the electric power plant [14, 16, 20]. At the same time, in accordance with the goal, namely, the need to improve the design and management of electric power plants in terms of the quality of maneuver operations performed by the ship, it additionally includes propellers, the steering wheel and the body of the electric ship.

The propulsion complex consists of two <<power>> circuits. Here: thermal motors--D, synchronous generators (SG)--G, frequency converters of electricity SE, induction propulsion motors (IPM)--M, propellers P, rudder--H and hull.

Elements of the automatic control system and the main parameters linking the power units and control signals: speed regulators of the primary engine--DR; automatic voltage regulators of the generator--GR; MD and [[omega].sub.D]--the torque and angular speed of rotation of the heat engine; [M.sub.G]--electromagnetic torque of the generator; [U.sub.d] and [U.sub.q]--the generator voltage along the longitudinal and transverse axes (internal coordinates); [I.sub.d] and [I.sub.q]--generator currents along the longitudinal and transverse axes (internal coordinates); [U.sub.G]--voltage at the generator output; [[omega].sub.Set]--set the angular velocity of the speed controller; [[xi].sub.D]--the course of the fuel pump rail; [DELTA][[xi].sub.D]--the increment in the stroke of the fuel pump rail; 1/[T.sub.sp]--the link of the servomotor; [K.sub.Fb] and [Ki.sub.s]--the gain factors for the links of rigid and flexible (isodromic) feedbacks; [U.sub.f] and [I.sub.f]--excitation voltage and current of the synchronous generator; [I.sub.G], [I.sub.M]--currents of SG and IPM; [[alpha].sub.Set] and [[gamma].sub.Set]--relative frequency and voltage of the converter (setting values); [alpha] and [gamma]--the relative frequency and voltage at the output of the converter (taking into account the feedbacks); FC--the functional converter forming the law of frequency control [gamma] = f([alpha]); [M.sub.P] and [P.sub.P] --the torque and emphasis of propellers; [M.sub.m] and [[omega].sub.M]--the torque and the angular velocity of the IPM rotation.

Isodromic all-regulators of indirect action are used as regulators of the rotation speed of the heat engine [8, 9]. As regulators of the voltage of synchronous generators [9, 10], combined (based on the control action and on the deviation of the regulated variable) regulators are used.

The system of equations describing the transient modes of operation of the electric power plant as part of the propulsion complex of the electric motor is presented below.

To make the results of the analysis more general, the model is reduced to relative units. As a result, the criteria for the dynamic similarity of propulsion complexes were revealed. These are the dimensionless parameters of the system <<thermal engines--propulsion system propulsors --hull>>. The ranges of the values of these parameters are found, covering the majority of series of electric boats with a traditional propeller drive. It is these parameters that determine the current values of the relative regime parameters of all components of the complex and determine the numerical values of the main quality indicators of maneuvering. This approach allows us to give a generality to the results obtained--electric vessels with equal values of dimensionless parameters will have the same values (in relative units) of the maneuvering quality parameters, respectively. There is an opportunity to generalize the results of research.

Relative values of performance indicators will be indicated by a symbol with a bar. (The index <<0>> corresponds to the values of the modes indicators when the electric motor operates in the nominal steady-state mode.)

For example, the relative electromagnetic torque of the generator:

[bar.[M.sub.G]] = [M.sub.G]/[M.sub.G0].

To simplify the perception of the material, the term <<relative>> is omitted from the text below.

The relative time is determined differently:

T = [v.sub.0]/L t,

where [v.sub.0] is the vessel speed, L is the vessel length, t is the current time.

The final version of the generalized mathematical model of transient and steady-state operating modes of propulsion complexes of electric motors is presented below.

Heat engine and speed controller of its rotation.

Equation of motion of the heat engine [5]:

d[bar.[[omega].sub.D]]/dT = [N.sub.D] ([[bar.[M.sub.D]] - [bar.[M.sub.G]]), (1)

where

[N.sub.D] = [M.sub.D0]L/[J.sub.D][[omega].sub.D0][v.sub.0] (2)

is the criterion of dynamic similarity, [J.sub.D] is the moment of inertia of the engine and the generator brought to the shaft of the heat engine.

The torque of the heat engine can be represented as the relative displacement of the fuel pump rail [8]:

[bar.[M.sub.D]] = [bar.[[xi].sub.D]]. (3)

The heat engine power:

[bar.[P.sub.D]] = [bar.[M.sub.D][[omega].sub.D]]. (4)

The equations of transient processes in the regulator of the speed of rotation of a thermal engine, taking into account rigid and flexible feedbacks and taking into account the operation of the active power distribution system (with parallel running generating sets (GS)), can be represented as [9, 10].

Increment of the stroke of the fuel pump rail:

d[bar.[[xi].sub.D]]/dT = [K.sub.p] (l - (([bar.[[omega].sub.D]] - [DELTA][bar.[[omega].sub.H]])) - [K.sub.Fb] [bar.[DELTA][[xi].sub.D]] - [K.sub.is] [bar.[DELTA][[xi].sub.D]], (5)

where [K.sub.P], [K.sub.Fb], [K.sub.is] are the gain factors for the regulated value (change in the speed of rotation of the heat engine), rigid feedback and flexible (isodromic) feedback, respectively;

[mathematical expression not reproducible] (6)

is the difference in the angular frequencies of rotation of the generators;

[bar.[U.sub.s1]] = [bar.[I.sub.ae1]] - [bar.[I.sub.ae2]]/[k.sub.ael] (7)

is the voltage on the servomotor; [k.sub.ae] is the gain factor of the automatic control loop of the active load distribution system;

[bar.[I.sub.ae1]] = [bar.[U.sub.d1][I.sub.d1]] + [bar.[U.sub.q1] [I.sub.q1]], (8)

is the active component of the SG current.

For the heat engine of the second generator set, the equations are written similarly.

A synchronous generator and an automatic voltage regulation system. In contrast to the <<classical>> [11] description of the SG, processes that are incommensurable with the time constants of the main components of the propulsion complex of the electric drive are not taken into account here, namely, the transformer EMF in the stator windings [12]. In view of the smallness, we neglect the active resistance of the armature [13], the mutual inductance, which is incommensurably small in comparison with the inductance of the excitation winding, as well as the flux linkages of the damper windings.

The combined system of automatic voltage regulation of synchronous generators includes a control loop for reactive power distribution (with parallel GSs). Switching angle:

[mathematical expression not reproducible], (9)

where

[mathematical expression not reproducible], (10)

[x".sub.d] and [x".sub.q] are the supertransitive inductive resistance along the axes d and q.

Angle of phase shift between vectors [[bar.I].sub.G] and [[bar.U].sub.G]:

[[phi].sub.G] = [[gamma].sub.G]/2. (11)

Angle of phase shift between vectors [[bar.E].sub.G] H [[bar.I].sub.G]:

[[psi].sub.G] = arctg (sin [[phi].sub.G]/cos [[phi].sub.G] + [x.sub.q] [[bar.I].sub.G]/[[bar.U].sub.G] cos [[phi].sub.G]), (12)

where [x.sub.q] is the synchronous inductive resistance along the axis q.

Angle of phase shift (load angle) between vectors [[bar.E].sub.G] and [[bar.U].sub.G]:

[[theta].sub.G] = [[psi].sub.G] - [[phi].sub.G]. (13)

Generator currents (internal coordinates d-q):

[[bar.I].sub.d] = - [[bar.I].sub.G] sin [[psi].sub.G], (14)

[[bar.I].sub.q] = [[bar.I].sub.G] cos [[psi].sub.G]. (15)

Generator voltages (internal coordinates d-q):

[[bar.U].sub.d] = - [K.sub.d1] [[bar.I].sub.G] cos [[psi].sub.G], (16)

[[bar.U].sub.q] = - [K.sub.q1] [[bar.I].sub.G] sin [[psi].sub.G] + [K.sub.q2] [[bar.I].sub.f], (17)

where [[bar.I].sub.f] is the current of the field winding of the generator.

Increment of the excitation current:

[mathematical expression not reproducible], (18)

where

[DELTA][bar.[U.sub.H1]] = [bar.[I.sub.rel]] - [bar.[I.sub.re2]]/[k.sub.re1] (19)

generators' voltages difference;

[bar.[I.sub.re1]] = [bar.[U.sub.q1] [I.sub.d1]] - [bar.[U.sub.d1] [I.sub.q1]] (20)

s the reactive component of the SG current;

[K.sub.d1] = sin [[theta].sub.0]/cos [[psi].sub.0], (21)

[K.sub.q1] = 1 - cos [[theta].sub.0]/sin [[psi].sub.0] (22)

[K.sub.q2] = 1, (23)

[K.sub.Uq] = [U.sub.G0]/[U.sub.f0], (24)

[K.sub.Id] = [I.sub.G0] [x.sub.d]/[U.sub.f0], (25)

[K.sub.U] = [U.sub.G0]/[U.sub.f0] (26)

are the dimensionless parameters;

[N.sub.f] = L[U.sub.f0]/[L.sub.f][I.sub.f0][v.sub.0] (27)

is the criterion for dynamic similarity; [L.sub.f] is the inductance of self-induction of the excitation winding; [U.sub.f] and [I.sub.f] are the voltage and current of the field winding; [K.sub.f1], [K.sub.f2], [K.sub.f3] are the gain factors for the main signal, the disturbance and the deviation of the controlled variable, respectively; [k.sub.re] is the gain factor of the automatic control loop of the reactive load distribution system.

Electromagnetic torque of the generator:

[bar.[M.sub.G]] = -[K.sub.G1] [[bar.I].sup.2.sub.G] sin [[psi].sub.G] cos [[psi].sub.G] + [K.sub.G2] [bar.[I.sub.f]] [bar.[I.sub.G]] cos [[psi].sub.G], (28)

where

[K.sub.G1] = ([L.sub.d] - [L.sub.q]) [I.sup.2.sub.G0]/[M.sub.G0], (29)

[K.sub.G2] = [M.sub.ad] [I.sub.f0] [I.sub.G0]/[M.sub.G0] (30)

are the dimensionless parameters; [L.sub.d] and [L.sub.q] are the inductances of self-induction of the armature winding along the d- and q- axes; [M.sub.ad] is the inductance of mutual induction along the d-axis.

The relations (9) - (30) were obtained in [6, 7].

Voltage on the generator output:

[bar.[U.sub.G]] = [square root of ([[bar.U].sup.2.sub.q] + [[bar.U].sup.2.sub.q]). (31)

Active power of the generator:

[bar.[P.sub.G]] = [bar.[U.sub.G] [I.sub.G]] cos [[psi].sub.G]. (32)

A converter of the electric power. Considering the frequency converter as a non-inertial <<quantizer>> of electricity with ideal gates, we do not take into account the electromagnetic processes taking place in it, and we consider that the current at the output of the converter is continuous, and the converter itself does not go beyond the region of normal loads [14]. The converter type is a frequency converter with an autonomous voltage inverter (FC with AVI).

Output voltage versus input one:

[bar.[U.sub.M]] - [gamma][bar.[U.sub.Q]], (33)

where [U.sub.M] is the IPM voltage

Rowing electric motor. In the mathematical model [5], a mathematical description of the generalized propulsion motor is given. As a special case from it, a mathematical model of an induction electric motor (IM) follows with frequency control. It is based on the exact classical scheme for the replacement of blood pressure, which determines the necessary assumptions and simplifications. The voltage at the stator terminals is considered to be sinusoidal, the saturation of the steel of the machine is not taken into account, the distribution of the flow along the arc of the air gap is assumed to be sinusoidal, the steel losses in the stator are taken into account approximately, and in the rotor are not taken into account [15, 16].

Equation of motion of the IPM:

d[bar.[[omega].sub.M]] = [N.sub.M] ([bar.[M.sub.M]] - [bar.[M.sub.P]]), (34)

where

[N.sub.M] = [M.sub.M0]L/[J.sub.M][[omega].sub.M0][v.sub.0] (35)

is the criterion of dynamic similarity;

[bar.[M.sub.M]] = [K.sub.M] [bar.[I.sub.M]] [bar.[[PHI].sub.M]] cos [[phi].sub.M] (36)

is the IPM torque;

[mathematical expression not reproducible] (37)

is the motor current;

[mathematical expression not reproducible] (38)

is the IPM magnetic flux;

[mathematical expression not reproducible] (39)

is the motor power factor;

[C.sub.M17] = [b.sup.2.sub.M]/[r'.sup.2.sub.2M], (40)

[C.sub.M18] = [c.sup.2.sub.M]/[r'.sup.2.sub.2M] [[alpha].sup.2.sub.0], (41)

[C.sub.M19] = [[alpha].sub.0], (42)

[C.sub.M20] = [[omega].sub.M0]/[[omega].sub.1Mn], (43)

[C.sub.M21] = [d.sup.2.sub.M], (44)

[C.sub.M22] = [e.sup.2.sub.M][[alpha].sup.2.sub.0], (45)

[C.sub.M23] = 2 [r.sub.1M]/[r'.sub.2M] [[alpha].sub.0], (46)

[mathematical expression not reproducible], (47)

[mathematical expression not reproducible], (48)

[C.sub.M26] = [x'.sup.2.sub.2M]/[r'.sup.2.sub.2M] (49)

are the dimensionless parameters;

[b.sub.M] = [r.sub.1M](1 + [[tau].sub.2M]), (50)

[c.sub.M] = [x.sub.0M] [[tau].sub.M], (51)

[d.sub.M] = [r.sub.1M]/[x.sub.0M], (52)

[e.sub.M] = 1 + [[tau].sub.1M] (53)

are the constant coefficients of asynchronous frequencycontrolled IPM;

[[tau].sub.1M] = [x.sub.1M]/[x.sub.0M], (54)

[[tau].sub.2M] = [x'.sub.2M]/[x.sub.0M], (55)

[[tau].sub.M] = [[tau].sub.IM] + [[tau].sub.2M] + [[tau].sub.1M] [[tau].sub.2M] (56)

are the scattering coefficients; [J.sub.M] is the moment of inertia of the motor; [K.sub.M] is the constant constructive coefficient; [[omega].sub.1Mn] is the rotational frequency of the stator magnetic field; r1M and [r'.sub.2M] are the active resistances of stator and rotor (reduced); [x.sub.1M] and [x'.sub.2M] are the inductive resistance of the stator and rotor (reduced) of the IM; [x.sub.0M] is the inductive magnetization resistance; [[beta].sub.M0] is the absolute slip of the IM rotor. The relations (34) - (49) were obtained in [5].

Electric power plant control. Management is carried out from the control station (CS) on the bridge. The main task is to control the movement of the vessel in real time. The output signal of the CS forms two control actions on the frequency converter (with respect to the frequency a and the voltage [gamma]).

Relative frequency of the output voltage of the frequency converter SE [5]:

[mathematical expression not reproducible], (57)

where [[alpha].sub.Set] is the reference value of the relative frequency; [[alpha].sub.P] is the corrective communication for IPM power; [[alpha].sub.[omega]] is the corrective coupling for the angular velocity of rotation of the IPM; [[alpha].sub.f] is the cut-off according to the output frequency of the frequency converter; [[alpha].sub.M] is the cutoff on the torque of the IPM; [[alpha].sub.I] is the cutoff of the stator current of the IPM; [[alpha].sub.PD] is the cut-off according to the power consumed by the electromotive system; [K.sub.[alpha]P], [K.sub.[alpha][omega]], [K.sub.[alpha]f] [K.sub.[alpha]M], [K.sub.[alpha]I], [K.sub.[alpha]PD] are the gain factors that are determined by the particular control system.

Relative output voltage of the converter SE [5]:

[gamma] = [[gamma].sub.Set] - [K.sub.[gamma]I] ([[gamma].sub.I] - [[gamma].sub.Imax]) - [K.sub.[gamma]U] [[gamma].sub.U], (58)

where [[gamma].sub.Set] is the reference value of the relative voltage of the frequency converter--a signal that is a function of the relative frequency and the adopted voltage control law; [[gamma].sub.U] is the corrective coupling for the IPM voltage; [[gamma].sub.I] is the current stator cutoff of the IPM; [K.sub.[gamma]I] [K.sub.[gamma]U] are the gain factors. In each specific case, the automatic control system has its own <<set>> of control signals for each control channel. It is these control actions that directly affect the performance indicators of ship maneuver operations.

Propellers. The hydrodynamic reversal characteristic of the propeller, taking into account the ship's motion along a curvilinear trajectory, is represented in the form of a parabolic polynomial [17, 18].

Propeller torque:

[mathematical expression not reproducible], (59)

where [a.sub.21], [b.sub.21], [c.sub.21] are the coefficients of the universal characteristic of the screw (are constant in certain ranges of measurement of [bar.[[omega].sub.P]] and [bar.[v.sub.e]]); [bar.[[omega].sub.P]] is the angular velocity of rotation of the propeller; [bar.[v.sub.e]] is the rate of water leakage onto the propeller; [bar.v] is the speed of the vessel; [[alpha].sub.M] is the constant coefficient; [[alpha].sub.Bev] is the angle of the bevel of the water flow.

Propeller stop:

[mathematical expression not reproducible], (60)

where [a.sub.11], [b.sub.11], [c.sub.11] are the coefficients of the universal characteristic of the propeller; [a.sub.P] is the constant coefficient.

The body of the electric boat. The movement of the vessel along the free water surface in the associated coordinate system GXYZ whose origin coincides with the center of gravity of the vessel G is considered. The GXY plane is parallel to the ship's main plane, the GX axis is located in the center plane and directed to the bow, the ZY axis to the starboard, GZ axis--vertically up. The associated moment X26 is neglected [19].

The components of the ship's speed along the axes X, Y and the speed of rotation around the Z axis:

[mathematical expression not reproducible], (61)

[mathematical expression not reproducible], (62)

[mathematical expression not reproducible], (63)

where

[mathematical expression not reproducible] (64)

is the longitudinal force of the rudder;

[mathematical expression not reproducible] (65)

is the transverse power of the rudder;

[mathematical expression not reproducible] (66)

is the turning torque;

[C.sub.[lambda]2] = m + [[lambda].sub.22]/m + [[lambda].sub.11]. (67)

[C.sub.[lambda]21] = 2([[lambda].sub.22] - [[lambda].sub.11])/m + [[lambda].sub.11], (68)

[C.sub.RX] = [[mu].sub.rx] [rho]/2[v.sup.2.sub.0][S.sub.C][(1 - [psi]).sup.2]/[summation] [K.sub.Pj][P.sub.ej0], (69)

[C.sub.RY] = [[mu].sub.K][rho]/2[v.sup.2.sub.0][S.sub.C][(1 - [psi]).sup.2]/[summation][K.sub.Pj][P.sub.ej0], (70)

[C.sub.11] = [R.sub.X0]/[summation][K.sub.Pj][P.sub.ej0], (71)

[C.sub.12] = 0.07 [rho]/2 [v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (72)

[C.sub.13] = [c.sub.4][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (73)

[C.sub.21] = 0.5[C.sup.[beta].sub.Y][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (74)

[C.sub.22] = [c.sub.2][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (75)

[C.sub.23] = [c.sub.3][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (76)

[C.sub.61] = 2[m.sub.1][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (77)

[C.sub.62] = 2[m.sub.2][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (78)

[C.sub.63] = 2[m.sub.3][rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (79)

[C.sub.64] = 2[m.sub.4] [rho]/2[v.sup.2.sub.0][F.sub.D]/[summation][K.sub.Pj][P.sub.ej0], (80)

[mathematical expression not reproducible] (81)

are the dimensionless parameters;

[N.sub.X] = L[summation][K.sub.Pj][P.sub.ej0]/(m + [[lambda].sub.11])[v.sup.2.sub.0], (82)

[N.sub.[OMEGA]] = [L.sup.3][summation][K.sub.Pj][P.sub.ej0]/2([J.sub.Z] + [[lambda].sub.66])[v.sup.2.sub.0] (83)

are the dynamic similarity criteria; [X.sub.R] is the distance from the center of the coordinate system to the rudder; [P.sub.ej] and [K.sub.Pj] are the useful thrust of the propeller and its share in the total flow, respectively; L is the length of the vessel; m is the mass of the vessel; [rho] is the specific density of water; [[lambda].sub.11] [??] [[lambda].sub.22] are the connected masses of water along the X and Y axes; [[lambda].sub.66] is the connected moment of inertia of water; [[mu].sub.rx] is the drag coefficient of the rudder; [[mu].sub.K] is coefficient of lateral power of the rudder; [psi] is the angle of the course; [S.sub.C] is the reduced area of the rudder; [c.sub.4] is the coefficient of the longitudinal positional force of the water resistance; [C.sup.[beta].sub.Y], [c.sub.2], [c.sub.3] are the coefficients of the body strength; [m.sub.1], [m.sub.2], [m.sub.3], [m.sub.4] are the coefficients of the positional moment of resistance; FD is the reduced area of the submerged part of the ship's diametrical plane; [M.sub.PZ] and [M.sub.DZ] are the positional and damping moments of resistance; [C.sub.m0.sup.[omega]] is the coefficient of the damping moment of resistance; [[beta].sub.dr] is the drift angle; [J.sub.Z] is the moment of inertia of the vessel when it rotates about the Z axis.

Angle of attack of the helm:

[[beta].sub.RP] = [K.sub.R] [[beta].sub.R] - [[chi].sub.C] (arctg [[beta].sub.dr] - [epsilon] [[bar.[OMEGA]].sub.Z]/[bar.v]), (84)

where [[beta].sub.R] is the angle of rotation of the helm; [[chi].sub.C] is the reduced coefficient of impact of the body and propellers on the rudder; [epsilon] is the value determined by the ratio [l.sub.R]/L ([l.sub.R] is the distance between the rudder and the midriangle frame). The relations (6i) - (84) were obtained in [5].

The developed mathematical model allows to comprehensively analyze transients of propulsion complexes of electric boats on maneuvers.

To analyze the maneuvering modes of operation, a package of applied programs has been developed. The base program is a program that allows calculating the laws of time variation of relative regime indicators, when the electric power performs a variety of maneuvers. At the same time during the analysis of maneuvering regimes:

* dimensionless parameters of the component parts of the complex are calculated;

* for the maneuver under study, the control parameters are entered in accordance with the positions of the handles of the control posts and the shifting of the rudder;

* initial conditions are given;

* according to the selected maneuvers, the laws governing each power circuit are formed;

* current values of the main mode parameters of each power circuit are calculated in the course of the maneuver;

* the hydrodynamic forces and torques acting on the vessel are determined; the current values of the ship's motion parameters in the coordinate system associated with it are calculated, and then in the unrelated coordinate system.

The solutions of the system of equations (i) - (84) presented above is a solution of the Cauchy problem. The Runge-Kutta-Merson method was used as the solution method.

The final results of the calculations are presented in numerical form and in the form of ready-made graphs of the time variation of the mode indicators:

a) for each power circuit:

--angular speed of rotation--[[omega].sub.D], torque--[M.sub.D] and power of the heat engine--[P.sub.D];

--output voltage--[U.sub.G] and current--[I.sub.G] of the generator;

--excitation current of the generator--[I.sub.fG];

--relative control voltage of the converter--[gamma];

--voltage--[U.sub.M] and current--[I.sub.M] of the propeller motor;

--torque--[M.sub.M] and angular rotation speed--[[omega].sub.M] of the propulsion motor;

b) on the parameters of the vessel's movement:

--speed of movement--v;

--components of the speed v along the longitudinal axis X - [v.sub.X] and along the transverse axis Y - [v.sub.Y];

--angular velocity of rotation about the axis Z - [[OMEGA].sub.Z];

--angle of drift - [[beta].sub.dr] and heading angle--[psi] of the vessel.

If necessary, any other parameters, obtained during the calculation, can be recorded.

To confirm the adequacy of the developed mathematical model and the calculation method based on it, we compare the results of mathematical modeling with the data published in [20, 21] obtained in the course of field experiments by other researchers.

We use the oscillograms of acceleration and reversal of the propulsion power plant of the nuclear-powered ship <<Arktika>> presented in these publications.

Recalculated (for convenience of comparative analysis) into relative values, these oscillograms are shown, respectively, in Fig. 2, 3 by solid lines. Here, the dashed lines show the current values of the mode parameters obtained with the aid of the developed calculation method.

The main parameters of the propulsion complex necessary for calculating the dimensionless parameters and the dynamic similarity criteria are presented in Table 1.

The generalized dimensionless parameters and the dynamic similarity criteria calculated from the above relationships are given in Table 2.

a) at acceleration 0 [less than or equal to] T [less than or equal to] [T.sub.1]:

[mathematical expression not reproducible], (85)

[bar.[U.sub.Mset]] = 1; (86)

b) at reversing from the forward to the rear [T.sub.1] [less than or equal to] T [less than or equal to] [T.sub.2]:

[mathematical expression not reproducible], (87)

[mathematical expression not reproducible], (88)

for [[omega].sub.M] > 0;

[mathematical expression not reproducible], (89)

[mathematical expression not reproducible] (90)

for [[omega].sub.M] [less than or equal to] 0;

c) at reversing from the rear to forward [T.sub.2] [less than or equal to] T [less than or equal to] [T.sub.2]

[mathematical expression not reproducible], (91)

[mathematical expression not reproducible] (92)

for [[omega].sub.M] < 0;

[mathematical expression not reproducible], (93)

[mathematical expression not reproducible] (94)

for [[omega].sub.M] [greater than or equal to] 0;

where [U.sub.Gset], [U.sub.Mset] are the control signals in the excitation systems of the SG and IPM, respectively; [K.sub.1], [K.sub.2], [K.sub.3] are the time constants.

The results of calculating the mode indicators ([I.sub.G0], [U.sub.M], [[omega].sub.M], [I.sub.M]) carried out according to the developed method in accordance with the relations (18), (33), (34), (37) in the qb64 application package, matlab are shown by dashed lines in Fig. 2, 3.

Comparison of the calculated results with the developed method with the experimental data shows a fairly good convergence. Trends in the change of mode indicators coincide. Certain discrepancies in the initial stages of transient processes are expected, and are explained by the settings and gain factors of the automatic control system, which for each ship have their own values.

Thus, the conducted studies confirm the acceptability of the proposed mathematical model and the developed method for calculating the maneuvering operating modes of electric vessels.

Conclusions:

1. A refined mathematical model and a method for calculating on its basis the operating conditions of propulsion power plants in the composition of propulsion complexes of electric boats are proposed. The method makes it possible to calculate the current values of the basic mode indicators both in the steady-state and in the dynamic modes of electric power plants, and evaluate the performance indicators of electric boats on maneuvers.

2. The adequacy of the developed method of calculation is confirmed by the results of full-scale tests conducted by independent researchers.

3. The use of the developed method of calculation opens up wide possibilities in the study of transient and steady-state operating conditions for propulsion complexes of vessels with electric movement.

doi: 10.20998/2074-272X.2017.6.05

REFERENCES

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[12.] Konoplev K. G. Impul'snoe regulirovanie sinkhronnykh generatorov [Impulse regulation of synchronous generators]. Sevastopol, SevNTU Publ., 2008. (Rus).

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[14.] Kuznetsov N.A., Kuropatkin P.V., Khaikin A.B., Khomyakov N.M. Osnovy proektirovaniia grebnykh elektricheskikh ustanovok [Basic design of rowing electrical installations]. Leningrad, Shipbuilding Publ., 1972. 656 p. (Rus).

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[19.] Voytkunsky Ya.I., Pershitz R.Ya., Titov I.A. Spravochnik po teorii korablia [Handbook of ship theory]. Leningrad, Shipbuilding Publ., 1973. 512 p. (Rus).

[20.] Aizenshtadt E.B., Gilerovich Yu.M., Gorbunov B.A., Serzhantov V.V. Grebnye elektricheskie ustanovki: Spravochnik [Rowing electrical installations. Reference book]. Leningrad, Shipbuilding Publ., 1985. 304 p. (Rus).

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Received 20.09.2017

V.A. Yarovenko (1), Doctor of Technical Science, Professor, P.S. Chernikov (1), Senior Instructor,

(1) Odessa National Maritime University, 34, Mechnikova Str., Odessa, 65007, Ukraine, phone +380 50 5980683, e-mail: yarovenko@3g.ua; chernikov@onmu.odessa.ua

Caption: Fig. 1. Structural diagram of the propulsion complex of the electric ship

Caption: Fig. 2. Comparison of the theoretical calculation of the acceleration of the REI of the nuclear-powered electric ship <<Arktika>> with experimental data

Caption: Fig. 3. Comparison of the theoretical calculation of the alternating reversals of the REI of the nuclear-powered electric ship <<Arktika>> with experimental data
Table 1
The main parameters of the propulsion complex

Length of the vessel by c.w.l.      136 m

  Vessel speed                    21 knots

  Nominal parameters of thermal engines:
    power                         27 600 kW
    rotation speed                3500 RPM

  Nominal parameters of synchronous generators:
    power                          9000 kW
    voltage                         780 V
    rotation speed                3500 RPM
    power factor                    0.88
    winding data, p.u.:
      [x.sub.d]                     0.96
      [x.sub.q]                     1.12
      [x".sub.d]                    0.14
      [x".sub.q]                    0.16

  Nominal parameters of propulsion electric motors:
    power                         17600 kW
    Armature current               9200 A
    rotation speed                 130 RPM

Table 2
Dimensionless parameters and criteria for dynamic similarity of the
nuclear-powered ship <<Arktika>>

[K.sub.m]           [K.sub.d1]   [K.sub.q1]   [K.sub.q2]

1.071                  1.12        0.294          1
[K.sub.G2]          [N.sub.D]    [N.sub.fG]   [N.sub.M]
1.684                 0.641         5.6          6.06
[K.sub.[gamma]G]    [K.sub.P]    [K.sub.Fb]   [K.sub.is]
0.118                  900           2           0.5
[a.sub.11]          [b.sub.21]   [b.sub.11]   [c.sub.21]
1.73                   0.33         0.33        -1.06

[K.sub.m]           [K.sub.G1]

1.071                 -0.029
[K.sub.G2]          [N.sub.x]
1.684                  0.2
[K.sub.[gamma]G]    [a.sub.21]
0.118                  1.73
[a.sub.11]          [c.sub.11]
1.73                  -1.06
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Author:Yarovenko, V.A.; Chernikov, P.S.
Publication:Electrical Engineering & Electromechanics
Article Type:Technical report
Date:Nov 1, 2017
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