Modeling of Active-Reactive Power Controllers with Droop Control
Active Power Controller
As seen in Figure 2.24, the power balance controller receives DC link voltage as input and compares with its reference quantity for instantaneous power balance control. The DC capacitor voltage depends on the energy balance between the power received by the VSI and that delivered by it; only when these two are equal, the DC link capacitor voltage will remain constant as depicted in Figure 2.28. It thus ensures that the entire active power from the preceding stage is delivered to the grid.
The MPPT algorithms of RE generators will be used to provide the power reference to the power balance controller for the sake of maximum power tracking; the actual power delivered can be computed from the inverter output quantities in synchronous reference frame from equations 2.35 and 2.36. The active and reactive power references in microgrids will be suggested by a central control station intended to maintain the generation-demand balance within the local area.
Consider a grid-tied microgrid that injects power P, through a VSI at the instant fj. Suppose the main grid fails at this instant and the VSI has to migrate to islanded mode. Let the main grid be at its rated frequency of
FIGURE 2.50 P-f droop characteristics of VSI.
Also, the VSI will deliver the power required by the local load at the rated frequency of o>_{raled} from the instant t_{2} onwards. The droop characteristics thus help to decide the limit of local demand that can be met by the islanded microgrid. It also tells that (P,-P_{2}) had been the power imported from the grid just before the islanding.
If the droop rate is constant and uniform, then the inverter will ramp up at uniform rate towards its rated power, P_{raled}; then the droop coefficient, m, is defined as,
Equation 2.56 assumes that P_{2} equals P_{rated} of the inverter.
When multiple inverters are present on the microgrid at the instant of islanding, the smallest VSI (in terms of rated power) is preferred to move into grid-forming mode first. Once islanding is detected and grid is isolated, then the first VSI forms the grid and other VSIs get tied to it.
Therefore, whenever there is a deviation in frequency on the microgrid, then the power support by any VSI can be computed from equation 2.56 and it will serve as the reference in the respective active power control loop. This loop uses either a conventional PI controller or an advanced PR controller and forces the actual power to follow the reference power or force the DC link voltage to follow its reference as elaborated in section 2.4.3 and further in the later sections.
Reactive Power Controller
The reactive power controller seen in Figure 2.24 receives the reference quantities according to one or more of the following requirements as described earlier and presented again here: (i) The reactive power support to be provided by the converter, (ii) the power factor at which the current is to be delivered, and (iii) voltage control required on the AC grid. However, the reactive power reference should be set at zero if the inverter current is required to be delivered at unity power factor; this is often advocated in grid-feeding mode.
The controller can receive the reactive power reference (2-reference) in any operating mode from the microgrid central controller, depending on the reactive power
FIGURE 2.51 Q-V droop characteristics of VSI.
support to be provided by the VSI. The Q-delivered is the other input to the reactive power controller, which is calculated from the inverter output quantities in synchronous reference frame from equation 2.36. Often, a Q-V droop characteristic similar to that in active power control is adopted to accomplish AC voltage regulation on microgrid through reactive power control. It is designed to deliver a reference value to the inner current controller, be it inductive or capacitive reactive current. It can be noted from Figure 2.51 that a reactive power injection in an appropriate direction as AV_{la}g or AV_{lead} can cause a change in voltage, ДК in a required direction. The limit on the reactive power injection, £>_{max}, of Figure 2.51 is determined by the kVA rating of the VSI, which decides the maximum voltage sag or swell that can be corrected by the VSI. The Q-V droop equation is,
where m_{Q} is the reactive power droop slope, Q_{mj} is the reactive power needed for voltage restoration, V is the required AC bus voltage and V_{rated} is the rated AC bus voltage. Therefore, whenever a voltage deviation is observed on the AC bus of the microgrid, an appropriate value of Q_{mi} will be calculated by the reactive power controller through Equation 2.57 and will be applied as ^-reference to this controller.
Emerging Non-unear Controllers
Non-linear controllers are robust which exhibit good dynamic response in reference tracking. Non-linear controllers developed in the past include model predictive controller, H-infinity controller, sliding mode controller, neural network-based controller, and fuzzy logic controllers. Model predictive controllers (MPC) have an edge over the others thanks to their multi-objective control through a single cost function optimization to achieve multiple targets. Modeling of MPC for grid-tied VSI is presented in detail in the following sections along with brief descriptions of control features and functionalities of other non-linear controllers.
MPC for Grid-Following Inverters
New age converter controllers are expected to accomplish additional capabilities such as seamless bi-directional power transfer, multiple control targets, adaptability to system non-linearities, faster dynamic response, etc., besides the prime objective of reference tracking. The MPC excels in simultaneous achievement of multiple control objectives of a heterogeneous nature.
A discrete time model of the system is used in MPC, to predict possible outcomes over a predefined time horizon. Any non-linearity in the system can be thus incorporated while formulating the prediction model. The optimal control is achieved by minimizing a cost function, g, that represents the desired target of the control,
where/,, is the hard constraint or primary target,/,, is the soft constraint or secondary target and Л„ is a weight parameter. In grid-following inverters, the reference current will be the primary target while the secondary target may be one or more of such features as harmonic profile improvement, switching frequency reduction, reactive power control, etc. Conventionally, both of these targets are attained in MPC through optimization of a single cost function. A firm minimization of/„, gives accurate reference tracking, while/,, will be graded relative to/„, by the weight parameter. The values of 1„ vary between 0 and 1 depending upon the priority of the auxiliary control implemented via the corresponding/,,.
The mathematical model of a three-phase grid-feeding inverter with MPC current control is presented here. The possible output currents are predicted for every possible inverter switching state. These predicted currents are passed on to the optimization process with the defined cost function. Optimization of the cost function will suggest the switching state of the inverter that will yield the smallest error between the reference and the actual value of current. Then the selected state will be applied to the inverter in the subsequent iteration.
Modeling of MPC Based Grid-Tied Inverters
The schematic diagram of MPC based grid-tied inverter system is shown in Figure 2.52, with Vdc as the input DC voltage, i*(k) as the reference current of kth sample intended to be delivered by the inverter, and i(k) as the corresponding actual current.
The load model of the grid connected inverter can be derived from the fundamental voltage equations obtained by applying Kirchhoff’s voltage law at the inverter output,
where v_{xN} (x = a, b, c) are the inverter voltages, v_{Cx} are the grid phase to neutral voltages, v,„_{v} is the voltage between load neutral and DC bus ground, / are the phase currents of inverter, R is the filter resistance, and L is the filter inductance.
Equation 2.59 can be represented in space vector form as,
FIGURE 2.52 Three phase grid-tied inverter with MPC.
where v is the space vector of inverter output voltage, vc is the peak value of grid voltage and i is the load current vector. Alternatively, the inverter voltage vector can be expressed as,
where a = е'^{2лП}, and, s_{a}, s_{h} and s_{c} represent the status of the top switches of the inverter legs. Thus, a three-phase inverter with six switches will have eight possible voltage vectors represented as v_{0} to v_{7}, from the inverter states 000-111.
Converting equation 2.60 into its discrete time model by applying Euler’s approximation with a sampling time of T_{s} gives,
Now, the eight possible values of the future current, i^{f}(k+1) will be predicted as,
Equation 2.65 requires the grid voltage value at kth sample. However, this can be estimated by back extrapolation. The estimated peak value of grid voltage can be expressed as,
The back extrapolation estimates the grid voltage for the preceding sample with the assumption that the grid voltage does not vary within the sampling interval. Therefore, the estimated vc(A-l) is applied in place of vc,(k) in Equation 2.65 to complete the current prediction for (k + l)th sample which is then sent for optimization.
Formulation of Cost Function
The cost function can be developed in any reference frame and it does not alter the tracking capability of MPC. The cost function, g, of the grid-tied inverter developed here to track a current reference in stationary reference frame without any secondary targets is expressed as,
where i*g, iP_{tt} and are the stationary reference frame coordinates of the reference current, Г, and the predicted current, respectively.
Eventually, the a and fi current components are predicted at every sampling instance using equations 2.65 and 2.66 and MPC is executed to switch the inverter.
2.4.5.1.3 Performance Evaluation of MPC-Based Grid-Tied Inverter A three phase grid-tied inverter with specifications given in Table 2.2 is simulated and controlled using MPC and its performance is presented in this section. The cost function presented in Equation 2.67 is used to track a rms current of 15.19 A and inject 14.8 kW of power to the grid. The formulated MPC controller is sampled at 25 kHz.
The current delivered by the inverter in Figure 2.53b shows a close compliance with the reference current of Figure 2.53a confirming the tracking ability of the MPC inverter. The current vector trajectory of Figure 2.53e is the plot of the reference and the actual current of the grid-tied inverter system. Inverter current (i) and reference current (/*) exhibit a high degree of agreement with each other except minor deviations. The switching frequency of the inverter will be varying while tracking the reference as it is a current regulated PWM.
TABLE 2.2
Inverter Specifications
Parameters |
Value |
DC source voltage |
700 V |
Inverter VA rating |
15 kVA |
Grid specifications |
ЗФ.400 V(l-l), 50 Hz |
Filter inductance |
0.001 Q, 10 mH |
FIGURE 2.53 (Continued) The MPC based grid-tied inverter performance: (d) inverter leg voltage, and (e) current vector trajectory of I* and / in stationary reference frame.
As the switching frequency and the model prediction depend on the sampling frequency,/,, used in MPC, the tracking performance depends greatly on the sampling frequency. Therefore, the system is tested for sampling frequencies of 50 and 100 kHz, with the same reference currents and the corresponding current vector trajectories are presented in Figure 2.54.
Figure 2.55a-c depict the line current error, i_{aerr}, plotted against samples. The current error per sample is found to get reduced with increase in/. Figure 2.55d shows the frequency spectrum of the delivered current, which is found to have harmonics well within the stipulated limits. The harmonic spectrum with any/ is widely spread up to half of/. Such spread spectrum can be shaped with the addition of a secondary target.
FIGURE 2.54 (a and b) Current vector trajectory of /'* and i in stationary reference frame with /' = 50 and 100 kHz respectively.
FIGURE 2.55 (a-b) Instantaneous current error, i_{acn}; with f_{s} of 25, 50 and 100 kHz
respectively. (Continued)
FIGURE 2.55 (Continued) (c) Instantaneous current error, i_{(lm}; with f_{s} of 25, 50 and 100 kHz respectively, (d) Frequency spectrum of current for 50 kHz.
Tracking accuracy and THDi are the performance indices considered. Tracking accuracy is defined in terms of magnitude, A(mag), and phase angle, A(angle), and expressed in % as,
where V_{rmv} 0 and в are the rms values and phase angles of reference current and inverter delivered current, respectively. The values of tracking accuracy obtained in the test case for both magnitude and phase angle are well above 99.5%. These performance merits clearly show that MPC is a competent contender for the grid connected converter control. It is identified as a prominent alternative to voltage based classical PWM techniques as well as other implicit modulator techniques.
Other Non-linear Controls
- 1. H-Infinity Controller. Hoo control is adopted when robust performance is expected in spite of system parameter variations and large disturbances. An optimization process is formulated from the problem which will be subsequently solved by the controller. The design requirements like disturbance rejection, robustness, tracking performance, etc., are to be formulated as constraints in different control loop transfer functions. The weighting functions are selected so as to tune these loops until the desired performance is reached. This control works well even with unbalanced load, exhibits reduced THD and high tracking accuracy and is easy to implement. Slow dynamic response and requirement of multiple control loops are the disadvantages of Hoo controller.
- 2. Sliding Mode Control: A sliding mode controller (SMC) has inherent robustness against wide range of system parameter deviations, external disturbances even with their strong uncertainties. If any plant response deviates from its normal operating points, a discontinuous control will direct the system’s state trajectories to persist on some desired sliding surface. Because of the discontinuous regulator, a strong control action occurs, which provides an excellent dynamic response for the controller. In basic SMC, a discrete control law is defined for the system under control with the desired performance defined as the desired states of the system. A sliding surface and the switching conditions will be such defined that the system states are made to follow the desired states. A good sliding surface in grid current control can ensure current regulation with better harmonic profile.
A commonly reported limitation of SMC is the chattering problem, which is rectified through optimization of the SMC parameters and by addition of integral terms onto the sliding surface to eliminate tracking errors.
Artificial Intelligence-Based Current Controllers
Artificial intelligence-based controllers have been developed for various applications that employ VSI. Neural network and fuzzy logic controllers are two such tools adapted in these current controllers. ^{[1]}
a model-free approach using the designer’s knowledge base to fine tune the controller actions by simple If-Then rules. A fuzzy logic controller in grid-tied VSI receives the inverter current error and its derivative as inputs and it delivers the desired reference voltage, with the help of the knowledge base, to the PWM generator. Further, fuzzy controllers are also effective in handling the microgrid model dynamics, especially in transient conditions..
- [1] Neural Network Controller: Neural network (NN) controller falls under thecategory of non-linear controllers. Learning ability of the controller makes itan obvious choice for microgrids having uncertain system models with widerange of parameter variation induced by RE source intermittency. Dynamicprogramming can be used to train the NN and the training is offline. It ispossible to implement the real-time control action with the modern highspeed-large memory microcontrollers without significant delay and withlow computing power requirements. The NN controllers show fast operationand good dynamic response especially in MPPT control under fast changing ambient conditions. Though online optimization is a constraint in NNcontrollers, it can be overcome by adopting parallel processing architecture. • Fuzzy logic current controller: Fuzzy logic controller is a replacementfor the traditional PI controllers to work in non-linear dynamic systems.Fuzzy logic methodology handles non-linear dynamics effectively, as it is