Modeling of SRF Current Controllers
The single phase equivalent circuit of the three phase grid-connected VSI of Figure 2.32 is presented in Figure 2.34.
FIGURE 2.33 Grid-connected VSI control in natural reference frame with hysteresis controller.
FIGURE 2.34 Single phase equivalent circuit of inverter feeding the grid.
The differential equations of the system of Figure 2.34 for the shown current/ power flow direction is expressed as,
Equation 2.24 can be rewritten as,
So,
where Av_{ahc} = u_{ahc}-e_{ahc}. The abc to dq transformation of the line currents in matrix form is given as,
where o>_{grid} = 2/r/and/is the grid frequency. Using Equation 2.27 and its differential, equation 2.26 can be transformed into dq reference frame as,
Substituting AV_{d} = u_{d} - e_{d}, and ДV_{q} = u_{q} - e_{q}, the c/-axis and q-axis inverter output voltages to be established to deliver the id- and q-axis currents, i_{d} and i_{q}, to the grid against the voltages, e_{d} and e_{q} are
By taking Laplace transform of equations 2.30 and 2.31, the complex transfer function G(s) is derived as,
FIGURE 2.35 Block diagram of the current controller.
FIGURE 2.36 Current control of VSI in SRF.
With the grid voltage modeled as load disturbance, e, and the inverter pole voltage modeled as a control voltage, u, the required output current vector, /, can be obtained as shown in Figure 2.35. The power electronic circuit which supplies the voltage, u, is a VSI that produces the desired voltages with no time delay and with low' harmonic distortion.
Thus the reference required for the inverter control is established using equations 2.30 and 2.31. Based on these equations, the general block diagram of current control of grid-connected VSI in SRF can be formulated as in Figure 2.36, where //and i* are cl and q axes reference currents, respectively. These references are obtained from the outer loop which will be similar to that in Figure 2.33. A PLL gives the necessary phase information of the grid voltage for the abc-dq transformation blocks.
If the voltage drops due to the line impedance is dynamically compensated for all power values using a PI controller, the equations for u_{d} and u_{q} can be given as,
Thus, the inner current control loop will establish the inverter currents as commanded by the reference currents from the output power loops.
Decoupling of Active-Reactive Power Controllers
The active power P and the reactive power Q delivered by grid-connected VSI in
SRF or dq quantities are given as,
It can be seen from (2.35) and (2.36) that P and Q are dependent on both d- and «/-axis currents delivered by VSI, thus making the independent control of P and Q impossible. The loss of independency in the active-reactive power control caused by such cross coupling is undesirable in any grid-tied inverter. If u_{q} is made zero by aligning the grid voltage space-vector to the d-axis, then the resulting decoupled power equations are,
Now, P and Q can be controlled independently by i_{d} and i_{q}, respectively in two independent current control loops. This condition is achieved when voltage at the synchronizer’s coupling point is taken as the reference for dq frame transformations. In spite of decoupling the power equations, the inverter voltage equations of 2.30 and 2.31 exhibit a cross coupling between the two due to the presence of the complex inductance drops represented by the factors, ±wLi_{dq}. Here, the «/-axis control voltage, u_{d}, depends not only on i_{d} but also on i_{q} and vice versa; in other words, if i_{d} is varied for whatsoever reason, it affects u_{d} as well as u_{q} and vice versa. It is equivalent to two first order systems interacting with each other and resulting in cross coupling. Complete independence in P and Q control is still not achieved owing to this cross coupling.
The mitigation of cross coupling is possible by cancelling the complex inductance drop by moving the pole of the plant G(s) from -(f + yen) to -(f) in SRF and by adding a real zero by the compensator. This is achieved by selecting a control voltage, u_{dq}, as,
By substituting u_{dq} (from equation 2.39) in equation 2.30 and 2.31, a feed forward decoupling is introduced and the decoupled system control equation is obtained as,
FIGURE 2.37 Current control with an inner decoupling loop.
It is evident from equation 2.40 that there is no cross coupling as there is no complex valued coefficients. Equation 2.39 is modeled as an inner feedback loop and a current regulator having output as u_{dq} is designed as an outer loop for the decoupled system.
The block diagram of the decoupled control system with the new control voltage of equation 2.39 is given in Figure 2.37, where i_{dq} is the reference vector expressed as,
The transfer function of the decoupled system between u_{dq} and i_{dq} is expressed as,
This first order complex valued system with no interacting terms can be regulated using PI controller with transfer function as F(s) = k_{p} + к/s. The control is not susceptible to the changes in the impedance, wL, between the inverter and the point of common coupling (PCC), because of the feed forward decoupling term introduced in the control loops. This feed forward decoupling term removes the cross coupling between the P and Q control loops resulting in completely independent P and Q controls.
Tuning of the PI Controllers for the Crid-Connected VSI
The constants, K_{p} and K_{h} decide the transfer function as well as the location of poles and zeros of the PI compensators of equations 2.33 and 2.34. Various tuning methods are available in the literature ranging from trial and error, Ziegler-Nichols, Tyreus Luyben, soft computing based tuning, loop shaping, etc. The desirable PI controller gains are obtained through the loop shaping method to suit the grid-connected converter system requirements and presented in this section as an example for PI tuning. Loop shaping method demands a desirable rise time to be assigned for the closed loop system based on its constituent components. For example, a rise time of 1 ms can be considered acceptable while working with Transistor/IGBT inverters, and can be further reduced when working with MOSFET devices. G'(s) of equation 2.42 being a complex first order system, the resulting closed loop transfer function is intended to be obtained as first order system with closed loop bandwidth of « as,
The standard relationship between the rise time t_{r} and a for a first order system is,
The closed loop transfer function is obtained using the compensator transfer function, F(s), and the plant transfer function, Gs), of equation 2.42. The closed loop transfer function for a negative feedback system is obtained as,
So, if F(s)G'(s) is selected as,
Then,
/ л
Equation 2.47 is found to be equal to equation 2.43, meaning that the desired closed loop response is achieved. Further, equation 2.46 yields,
Equation 2.48 represents the transfer function of the PI controller and so the K_{p} and Кi values can be obtained as,
Thus, the controller parameters are expressed as the parameters of the plant transfer function, L and R, and within the required closed loop bandwidth. Such tuning of PI controller avoids the trial and error method and gives better stability for the entire bandwidth, especially for systems with PWM switching converters.
Performance Evaluation of Grid-Tied VSI with SRF Current Controllers
The grid-tied VSI with SRF current controllers is simulated with the specifications of Table 2.1 and the system performance like dynamic response, steady state response, frequency tracking, harmonic content, active power delivery, and reactive power consumption has been studied in this section.
- • Steady state performance: Figures 2.38 and 2.39 shows the current delivered to the grid and the voltage at PCC for a power reference of 275 W till 0.6 s and then subjected to a step change to 1450 W. That the rise time of the step change response is less than 2.5 ms and shows the ability of the controller to track step changes.
- • Harmonic analysis of the injected current: Figure 2.40 shows the FFT analysis of the grid current as a frequency spectrum. TFID is only 0.17%. All the components are less than 3% which satisfies all grid codes. The lowest order harmonic appears at the switching frequency of 4 kHz and the subsequent harmonics are at its multiples.
Figure 2.41 shows the active and reactive powers delivered with step changes in the references and also the current delivered. Since power is fed at the unity power factor into the grid, the reactive power component is nearly zero for Q_{rcf} = 0. During the period from t = 0.6-1.2 s, the P_{Kf} is given a step change from 275 to 1450 W and it is observed in Figure 2.41 that the value of power injected increases instantaneously. But the reactive power is unaffected by the change in active power. This shows that active and reactive components of power could be independently controlled in the case of SRF-PI current control.
TABLE 2.1
Grid-Tied VSI Specifications
Rated power |
2 kW |
Output line voltage |
400 V |
Output current |
1.44 A |
DC link voltage |
680 V |
Converter |
3 leg IGBT inverter |
Output filter |
L= 11 mH, r, = 0.23 £2 |
Grid frequency |
50 Hz |
K_{p} and K, |
24 and 505 |
Inverter switching frequency |
4 kHz |
FIGURE 2.38 Steady state waveforms of current injected to the grid and voltage at PCC.
FIGURE 2.39 Steady state current zoomed.
FIGURE 2.41 Active power, reactive power, and line currents.
FIGURE 2.42 Line current with step change in grid frequency.
Response to step change in grid frequency : The simulated grid-connected system is tested for step change in the grid frequency from 50 to 52 Hz introduced from 0.4 to 0.6 s; Figure 2.42 shows the corresponding variation in grid current. The frequency change in current waveform is visible in the figure. The current change is instantaneous and without losing stability. The magnitude of injected current remains the same in spite of frequency change. This shows that the controller is immune to frequency changes. Such step change in frequency is not practical, yet this test has been done to demonstrate the precision of tracking.
• Response to step change in grid voltage: The system is then tested for variation in grid voltage as this is a common scenario expected in microgrids.
FIGURE 2.43 Inverter voltage and current response to step change in grid voltage.
A swell in grid voltage from 220 to 240 V is introduced from 0.6 to 0.7 s as depicted in Figure 2.43. The line currents delivered show no variation in spite of the variations in grid voltage. A corresponding increase in the inverter voltage is observed, in order to inject the same current at the higher grid voltage.
Modeling of PR Controllers
PR controllers can be used in grid-connected applications with control variables converted to stationary reference frame. The computation sequence of PR controller is not complex because there is no transformation from the stationary frame to synchronous frame. The active and reactive powers can be expressed in «/3 reference frame as P = vj_{a} + Vpip and Q = -v_{p}i_{a} + v_{a}i_{p}. So, power control can be achieved by controlling the a/) current components. The three phase currents obtained as feedback signals are first converted to two phase «/? components and then compared with afi reference values. The resultant error signal is sinusoidal and not DC as in the case of SRF-PI controllers. A PR controller has the capability to track AC sinusoidal quantities with zero steady state error and without any phase delay. It can be inferred from the transfer function of PR controller in equation 2.50 that it becomes a simple PI controller when со = 0. Thus, PR controller can be viewed as a generalized 2^{nd} order AC integrator tuned to the grid frequency, со. A PR controller can therefore handle AC quantities directly without any DC transformation. But, a PR controller is very sensitive to the grid frequency fluctuations, as it introduces infinite gain only at the tuned grid frequency. If the grid frequency drifts outside the tuned band of the controller, the system may fail to track the reference. It is possible to maintain the reference tracking with wide tuned bands, but at the cost of increased steady state error. However, PR controllers have a great advantage in comparison with other current regulators for grid-connected applications that harmonic compensation can be done without affecting the fundamental reference tracking control.
The transfer function of the PR controller is,
(2.50)
where K,, is the proportional gain, K, is integral gain and
In equation 2.51, (,v + jw) is the positive sequence integrator and (,v - jw) is the negative sequence integrator. The block diagram of PR controller is depicted in Figure 2.44, in which the input, u(s), will be an error signal and the output, y(s), will be an actuation signal.
The gain of the PR controller at resonant frequency can be varied by varying K_{h} therefore A", has to be very high for better dynamic response. The value of K_{P }adjusts the bandwidth of the controller while simultaneously deciding the stability of the plant. A proper selection of can adjust the gain outside the tuned frequency and thus a considerable gain can be maintained even with minor grid frequency deviations. Figures 2.45 and 2.46 show the frequency responses of PR controller for various combinations of K_{P} and K, values. Usually, К, is much higher than 100 K_{P}.
The ideal PR controller gives an infinite gain at resonant frequency. The gain is brought down to usable range by introducing a damping factor <5, which also increases the bandwidth of the controller. The frequency response of the controller with the damping factor is presented in Figure 2.47, wherein the gain at resonance is found to be reduced from its undamped value. Thus, the damped PR controller is designed to operate on partially loaded (below the rated power) condition due to a huge loss in the damping resistor. The transfer function of the damped PR controller can be defined as,
FIGURE 2.44 PR controller.
FIGURE 2.45 Frequency response of PR controller for different values of K, with K_{P} = 1.
FIGURE 2.46 Frequency response of PR controller for different values of K_{P} with K, = 100.
Another important characteristic feature of PR controller is the possibility of inclusion of selective harmonic compensation in the same control structure. This is achieved by cascading several generalized integrators tuned to resonate at the harmonic frequencies required to be eliminated. Since the PR controller acts on a very narrow band around its resonant frequency u>, harmonic compensation can be implemented without any adverse effect on the behavior of the current controller. The transfer function of a typical harmonic compensator could be designed to compensate for the 3rd, 5th, and 7th harmonics, as these are the most prominent ones in the current spectrum. The controller transfer function for compensation of any harmonic order “h” is,
FIGURE 2.47 Frequency responses of damped and undamped PR controller.
FIGURE 2.48 Frequency response of PR controller with harmonic compensation.
The frequency response of the harmonic compensated PR controller is presented in Figure 2.48, wherein it is tuned to eliminate three harmonic frequencies.
The complete control implementation of the PR controller for the grid-tied VSI with outer loops of power control and harmonic compensation is depicted in Figure 2.49.
Digital Implementation of PR Controller
The real time implementation of PR controller in a typical digital platform necessitates discretization of the transfer function. It is possible through bilinear transformation to transform the quantities into z domain by substituting ,v = in
equation 2.52 and the discrete transfer function of the PR controller will be,
FIGURE 2.49 Grid-connected VSI control in stationary reference frame with PR controller, where n_{0}, n_{p} n_{2}, d_{2} are numerator and denominator coefficients. Therefore,
where u(k) and y(k) are the sampled input and the output signals of the discrete PR controller, respectively.