1. Introduction to direct current grids and power sharing issues

2. Proposed system topology and features

2.2. Synchronous generator and DC-DC converer output stage control principles

Control of self excited generator involves regulating frequency and voltage by adjusting active and reactive power. In the generating sets arrangement, a combustion engine provides active power, while excitation windings current ensures reactive power. A rotational speed governor and automatic voltage regulator provide data for the two independent control loops. Simple, straightforward scalar control procedures like this omit some phenomena, such as the coupling effect between synchronous generator electrical axes [16]. Among the many suitable methods for this purpose is field orientation vector control. Since AC machines use vector control, their performance during transient operation modes is quite similar to that of direct current machines. Space-phasor theory provides the mathematical foundation for the dynamic model and vector control of AC machines [17]. A rotor flux oriented synchronous machine model can be used to simulate DC generator operation, but a field oriented model is used for controlling the generator. In a sense, this concept resembles a DC machine in which the two variables are controlled independently to produce electromechanical torque. The DC machine equipped with commutator produces top torque in comparison with almost any AC machine due to its mechanical alignment of the commutator thus the perpendicular orientation of armature current and a stator flux. Nowadays such case is not considered because of certain problems arising from the brushes and commutator and it can be stated that only brushless machines are in use in maritime industry today.

In the synchronous generator the stator currents and magnetic fluxes depicted in Figure 3 are given by:

$$i_s=i_{sd}+j{i}_{sq}$$

(1)

$$\begin{gathered} {\Psi }_{sd}=-\int \left(v_{sd}+R_s{i}_{sd}+{\omega }_e{\Psi }_{sq}\right)dt \\ {\Psi }_{sq}=-\int \left(v_{sq}+R_s{i}_{sq}-{\omega }_e{\Psi }_{sd}\right)dt \end{gathered}$$

(2)

where: Ψ_sd, and Ψ_sq are the stator fluxes in d-q plane and ω_e is the stator magnetic flux rotational speed.

Figure 3. Simplified scheme of the 2 poles SESG with rotor and stator fluxes in d-q axis depicted.

Figure 3. Simplified scheme of the 2 poles SESG with rotor and stator fluxes in d-q axis depicted.

Quadrature in the generating mode determines active power and intermediate circuit voltage and, due to reversed energy flow, is negative. The flux component is also negative due to its demagnetizing properties and corresponds to the reactive power generated. An active current component creates torque by influencing the exciting current. Field oriented exciting currents have a longitudinal component due to decoupled control, which contributes to their magnetizing properties.

For power flow to be controlled, the stator current must be oriented in the direction of the magnetizing flux which is implemented in a control algorithm using Clarke and Park (α-β and d-q) plane transformations. A constant torque angle control strategy maintains the current vector on the q axis aligned with the current vector on the d axis at all times. Considering both i_sd and i_sq currents in a synchronous generator, the torque equation is as follows:

$$T_e=\frac{3}{2}p\left[{\Psi }_m{i}_{sq}+\left(L_{sd}-L_{sq}\right)i_{sq}{i}_{sd}\right]$$

(3)

where: Ψ_m is field winding flux, p is the number of pole pairs and L_sd and L_sq are values of the stator self inductances. As a result of some simplifications, the torque equation becomes:

$$T_e=\frac{3}{2}p{\Psi }_m{i}_{sq}$$

(4)

Because machine torque and active current are linearly related, equation (4) shows that the generator can be controlled by means of stator current and flux mutual regulation. In the d-q coordinates, the i_sq independent control loop assures that the DC intermediate circuit voltage U_dc is always a non-zero value even when the reactive current i_sd equals to zero. As it can be seen in Figure 3 to control PWM inverter the signals of d-q plane voltages are applied. To achieve such there is a need to translate current signals into the voltages. From a theory this conversion is done with following dependencies coming from (2):

$$v_{sd}=R_s{i}_{sd}+L_{sd}\frac{d{i}_{sd}}{dt}-{\omega }_r{L}_{sq}{i}_{sq}$$

(5)

$$v_{sq}=R_s{i}_{sq}+L_{sq}\frac{d{i}_{sq}}{dt}+{\omega }_r{L}_{sd}{i}_{sd}+{\omega }_r{\Psi }_m$$

(6)

where v_sd and v_sq are the d-axis and q-axis stator terminal voltages and R_s describes the resistance of the stator phase windings.

In (6) substituting term ω_r Ψ_m which describes back EMF with E and applying a Laplace transform we get:

$$v_{sq}-E=R_s{i}_{sq}+L_{sq}\frac{d{i}_{sq}}{dt}+{\omega }_r{L}_{sd}{i}_{sd}$$

(7)

In the FOC control method applied in this case the magnetizing current i_sd is kept at 0 (by decoupling term) so the part including active current in (7) can be canceled thus the transfer function of the machine is given by:

$$\frac{i_{sq}}{v_{sq}-E}=\frac{1}{{{sL}_{sq}+R}_s}$$

(8)

Introducing electrical time constant τ_sq$=\frac{L_{sq}}{{ R}_s}$ in (8) yields

$$\frac{i_{sq}}{v_{sq}-E}=\frac{1}{R_s(s{\tau }_{sq}+1)}$$

(9)

In the presented solution a sensorless system was utilized to obtain rotational speed and stator field angle values needed for Clarke and Park transformations. This control method utilizes flux angle calculation block which takes information from observer which in fact is a block of the DSP software running real-time parallel with machine. The current values used in the flux calculation block are filtered out due to noise coming from current sensors, so it provides smooth operation of the control system. The speed calculation equations are fed with measured current values and known voltages applied to the windings. This allows calculation of flux in reference α-β plane which in turn leads to reference frame angle calculation. This is needed to provide information for forward and inverse Park transformation crucial in FOC control and can be useful in rotational speed estimation.

There are three main control loops in the proposed structure: the DC voltage outer loop and the active along with magnetizing current fast loops (in Figure 4 denoted as i_sq and i_sd currents) which is consistent with the classical FOC scheme. Besides providing long-term voltage regulation for DC intermediate circuits, the field oriented control technique also enables the distribution of power between parallel DC sources thanks to introducing active current limitations. What distinguishes the presented method from classical solutions is presence of the fourth PI control loop which produces signal denoted as U_corr. In the islanding mode of operation, the DC link voltage is main controlled quantity, so the regulators parameters are tuned for dynamics of such. Regarding the active current limitation correcting voltage signal provides power distribution instead of limiting i_sq current in the main loop which should be set to the nominal value of the machine and inverter value. The correction loop is enabled in the case of parallel operation and can be limited not to exceed safe operation area of the inverter’s DC intermediate circuit voltage. This additional control loop eliminates nonlinear regulators moreover enables precise and fast voltage control when power distribution mode is required. As it may be deduced additional corrector control loop should be used in parallel operation mode limiting active current contribution to the consumer. Another important issue is parameter variations occurring during system operation. Proposed parallel connected system inherently is not ideally stiff which means that both sources cannot keep intermediate voltages on the required level in the operational range. In the case of simple DC voltage control, fluctuations of the intermediate voltage generated by the DAB contributing to incorrect power distribution. It follows from the methodology of a system without an additional regulator that, in order to be able to limit the current, the DC bus control voltage must be increased by adding a certain value to the reference level. This value should depend on the current state of the bus voltage so that the load current distribution can be precisely regulated. Because in the parallel operation the voltage measurements are common for DC bus and intermediate circuits connected these values cannot be precisely determined without active currents control.

In the Figure 5 there is presented situation of unequal voltage intermediate circuits in two cases. In the Figure 5 a) the control over DC voltage relies solely on the main loop PI control while in the Figure 5 b) the compensating controller was used. In the time of bus voltages equalization the system without active compensator does not take any action when the current drops below limit thus the power sharing operation is not maintained. Such unfavorable situation leads to conclusion that the reference voltage setpoint should be changed either on the master or the slave controlled voltage source every time power sharing is needed. In such condition the only way to determine shared load ratio between paralleled sources is by active currents control so the additional compensator loop is necessary. Over a limited range of parameters, proposed method uses a PI controller configuration and a first-order plus dead time model. Further, all approaches involve tuning cascade control systems in two steps: first, the secondary controller is tuned based on the dynamic model of the inner process; second, the primary controller is tuned based on the dynamic model of the outer process, including the secondary loop. Consequently, if the secondary controller is retuned for some reason, an unfavorable identification step can result in an additional retune process for the primary controller.

2.3. Synchronous generator dc voltage cascaded control

One method allowing precise control is proposed, modified control closed loop enhanced in comparison to basic current regulator form with additional corrector controller. According to the Figure 4 cascaded control loop part can be presented as follows.

where: U_dc bus is actual voltage value of DC bus, U^*_errdc is the voltage error differential signal, U_corr voltage correcting signal, i_sqlim denotes active current limit, i_sqcorr is the corrector current, G1, G2, G3 are PI controllers, H1 is the generator plant and H2 is the external circuits block.

The control loop consists of SESG plant H1 block which can be presented as follows:

$$H1=\frac{1}{R_s\left(\frac{L_{sq}}{R_s}\cdot s+1\right)}$$

(10)

The block H2 describes external circuits which in presented circuit include capacitances and resistances of common DC bus bar.

$$H2=\frac{R_{ext}}{R_{ext}\cdot C_{ext}\cdot s+1}$$

(11)

where R_ext and C_ext are the external circuits resistances and capacitances.

The PI controllers are responsible for control signals of DC voltage outer loop, internally calculated active current and the corrector current. The auxiliary measured variable is an active current i_sq which feeds the DC bus consumers and is crucial to maintain the proper value of direct voltage. At the same time the value of the i_sq is critical for the sharing load control method. The current limit value comes from energy management system and fulfills the load sharing requirements which means the control of outer loop main measured variable rely solely on active current limiting.

The scheme shown in Figure 6 depicts dependencies between controlled variables, PI controllers and the active current limiter.

In the presented loop, the controlled and corrected variable is the active current i_sq. This current charges intermediate circuit capacitors and parallelly powers up external circuits. According to Figure 6 the equations describing inner control loop of active current and resulting DC voltage (all components are in the s-domain) can be expressed as:

$$i_{sq}=\left(i_{sq}^{\mathrm{*}}-i_{sq}\right)·G3·H1$$

(12)

Thus:

$$i_{sq}=\frac{i_{sq}^{\mathrm{*}}·G3·H1}{1+G3·H1}$$

(13)

The outer, voltage control loop can be presented as:

$$U_{dc}=U_{errdc}^{\mathrm{*}}\cdot G2\cdot \frac{G3\cdot H1}{1+G3\cdot H1}\cdot H2$$

(14)

From Figure 6 the voltage error signal is as follows:

$$U_{errdc}^{\mathrm{*}}={U_{dcbus}+U}_{corr}-U_{dc}$$

(15)

After applying (15) into (14) we get:

$$U_{dc}=\left(U_{dc \mathrm{bus}}+U_{corr}\right)\frac{G2\cdot \frac{G3\cdot H1}{1+G3\cdot H1}\cdot H2}{1+G2\cdot \frac{G3\cdot H1}{1+G3\cdot H1}\cdot H2}$$

(16)

In the proposed solution the error voltage along with G2 controller signal gives the information proportional to required active current. In the slave loop commanded active current is compared to actual value and the result is transformed to machine side inverter voltages. Such a structure is used widely in control applications regarding real-time machines control and works flawless in the single mode of operation. Presented configuration refers to the cascaded control and is functional enhancement of a single loop control. The quality and performance of the cascaded control loop mainly depends on parameters [18] and coefficients of master and auxiliary loops controllers. In theory for such cascaded systems the frequency response methods are used to set the controllers properly [19]. Open loop transfer function of the main loop has higher order dynamics than inner loop in the control algorithm. There are available tuning charts allowing prediction of the master controller settings for minimizing integral of time-weighted absolute error due to load disturbances on the secondary loop in cascade control systems. Because of the use of controllers one needed information deals with stability of the system and can be obtained with means of classic analysis in s-domain. To achieve such, the transfer function of the control loops should be defined. After some simplifications on voltage equations (5,6) the transfer functions of current loops for SESG are as follows:

$${Tf}_q(s)=\frac{1}{s\cdot L_{sq}+R_s}=\frac{1}{R_s\left(s\cdot \frac{L_{sq}}{R_s}+1\right)}$$

(17)

$${Tf}_d(s)=\frac{1}{s\cdot L_{sd}+R_s}=\frac{1}{R_s\left(s\cdot \frac{L_{sd}}{R_s}+1\right)}$$

(18)

where s is the complex variable.

Having the parameters of the real-time system, such as computational delays and circuit time constants, it is possible to put together a control system loop without corrector.

The detailed active current control loop in its generic form is shown in the Figure 7. The active current set point is compared with its measured value and the error signal feeds PI controller. The PI controller in continuous domain transfer function according to [20] can be expressed as:

$$PI=k_{p_{\omega }}\frac{1+{\tau }_{\omega }\cdot s}{{\tau }_{\omega }\cdot s}$$

(19)

In order to obtain unitary feedback the control loop from Figure 7 can be rearranged and presented in the following figure.

To get required transfer function the input function 0.5sTs+1 should be approximated and presented in more convenient form which is $\frac{1}{ 1-0.5T_{s}s}$. The way to achieve this is to linearize this function around fixed point described as (x0, y0) = P(0, f(0)) = P(0,1).

The equation of the tangent in the general case has the form $y-y_0=m\left(x-x_0\right)$, while

$m=f\text{'}(x){|}_{x=x_0}={\left.-\frac{1}{{\left(1-x\right)}^2}\cdot \left(-1\right)\right|}_{x=x_0}=1$. In the given situation $y=m\left(x-0\right)+1=x+1$ and finally we have an approximation of $\frac{1}{1-x}\approx x+1$.

After inserting the transfer functions of plant H1 and H2 along with G2 and G3 controllers given by (18) into (14) and including required calculation along with the digitizing delays the stability of the cascaded control loop has been proven.

The transfer functions have been implemented to Matlab software and for the chosen set of parameters the obtained step response is as in the Figure 10.

It should be pointed out that tuning process is crucial in order to get needed step response from designed system and values which work flawless in digital simulation probably would need some more work in digitally controlled laboratory setup.

The control signal goes to the real-time DSP-FPGA control system which introduces digitizing and calculation delays and SESG itself also has delay which mainly depends on the electric constant of investigated machine depicted in the Figure 8 as τ_sq. This value was calculated on basis of the data acquired in the previous projects regarding investigated machine. Based on machine parameters and control system properties the open loop transfer function was created and after applying modulus optimum criterion [21] the proportional gain of the active current controller was determined. All data were included into control loop what allowed plot Bode diagram of the investigated open-loop system.

Figure 8. Configuration of unitary closed-loop active current control.

Figure 8. Configuration of unitary closed-loop active current control.

Figure 9. Configuration of unitary closed-loop active current control.

Figure 9. Configuration of unitary closed-loop active current control.

Figure 10. The step response of voltage controller with corrector.

Figure 10. The step response of voltage controller with corrector.

For given data of machine parameters it can be read from the Bode diagram in the Figure 11 that the phase margin equals to 62.5 deg while gain margin is 13.6 dB and both are stable for closed control loop.

Similar calculations and tests were performed to obtain information of stability in the i_sd current loop and U_dc based on (16) so the stability of the control was proved.

2.4. Synchronous generator and DC-DC converer output stage control principles

In addition to the synchronous generator inverter the complete system includes a bidirectional converter tied to the batteries pack or supercapacitors.

In the Figure 10 there is a complete system of DC-DC dual active bridge converter presented. As it can be noted DAB contains low voltage (LV) and high voltage (HV) side which are galvanically separated with high frequency (HF) transformer. This ensures lack of dependency between closing transistors on both sides of converter. The dual active bridge utilizes two modes of operation. In the first mode converter charges batteries and a combination of switches denoted as SW1 and SW2 along with parallel connected auctioneering diodes allows current flow solely into the batteries from power DC bus. It allows the direct voltage input ranging from 300V up to 700V and with use of one phase inverter changes current into 21 kHz quasi-sinusoidal waveform.

High frequency is needed to avoid the typical noise and then is transformed with use of high-frequency transformer of 1.87 voltage ratio. This unusual ratio value ensures almost trapezoidal/square voltage waveform which in turn creates minimal amount of higher order harmonics what makes system more efficient. The core of power flow control is achieved by input/output square voltage phase shift control and with use of presented dual-active bridge it can work with minimal set of voltage and current measurements.

Figure 12. Circuit of bidirectional DC-DC converter with high frequency transformer and auctioneering switch module.

Figure 12. Circuit of bidirectional DC-DC converter with high frequency transformer and auctioneering switch module.

3. The experimental set up and results

Figure 19. Overview of investigated laboratory setup.

Figure 19. Overview of investigated laboratory setup.

Figure 22. Waveforms of battery voltage drop along with battery and generator currents while different ratio power sharing operation obtained for 1600W and 2400W load. Time base 2s/div.

Figure 22. Waveforms of battery voltage drop along with battery and generator currents while different ratio power sharing operation obtained for 1600W and 2400W load. Time base 2s/div.

4. Conclusions and further work

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