GB2545235A - Current-limiting droop controller for power converters - Google Patents

Abstract

A controller for a grid-tied inverter provides ancillary services to the grid while limiting the current under both normal and abnormal grid conditions. It involves the measurement of the inverter output voltage, the grid voltage and the inverter current in order to calculate the real, reactive power and RMS output voltage, but does not require the knowledge of the system parameters. The controller generates an output consisting of a feed-forwarded voltage term, a controllable voltage source and a virtual resistance term multiplied with the inverter current. The virtual resistance is controlled to stay within a given range having a positive lower bound and the controllable voltage source is generated according to the droop function. As a result, the inverter current remains below a given value at all times, even during transients, independently from grid variations or faults (short circuit or voltage sag). The invention introduces a controller with a continuous-time structure and does not require any switches, additional limiters or monitoring devices to achieve the current-limiting property.

Description

DESCRIPTION

The present invention is concerned with control devices for grid-tied inverters to operate with the droop function to support the power grid while having an inherent property to limit the current independently from grid variations (voltage, frequency) and faults.

During the last decades, there has been a continuous increase in the number of distributed generation (DG) units connected to the power network. Although this fact has provided substantial economical benefits for both utility companies and customers, it has also led to more stringent demands regarding the interconnection of DGs with the grid because they directly affect the stability of the power network. Therefore, the operation and control of grid-tied inverters, that link the DG units with the utility grid, are crucial and should be maintained inside some given limits under both normal and abnormal conditions of the grid.

Droop control represents the most widely used technique to provide ancillary services to the grid for frequency and voltage regulation via adjusting the real and reactive power injected by the inverter-controlled sources. Several control methods have been developed to improve the droop control performance, such as the addition of an extra phase shift term, adapting the control parameters using a grid-impedance estimator, or changing the output impedance of the inverter. Among these methods, the recently proposed robust droop controller not only can achieve the desired droop functions but can also guarantee tight output voltage regulation near the rated value independently from parameter variations or external disturbances. However, the increased complexity of the closed-loop system, resulting from the nonlinear expressions of the real and the reactive power, leads to difficulties in rigorously proving the stability. Moreover, there is no guarantee that the current will remain inside a given range. Limiting the inverter current is crucial for the stable and reliable operation of the gird-tied inverter during normal grid operation, but more importantly during grid faults.

During a grid-fault, the grid voltage drops and the inverter equipped with a droop controller often tries to regulate the output voltage close to the rated value, which leads to high inverter currents. Hence, fault current-limiting controllers are essential for protection purposes. This can be achieved by either triggering suitably designed protection circuits or by using several low-voltage ride-through controllers, which will keep injecting power to the grid with a limited current. In terms of control design, most of the current protection methods are based on a switching control action between the droop controller or power regulation during normal grid operation and the current-limiting scheme after the fault has occurred. Virtual impedance methods have been also proposed in order to guarantee a given limit of the inverter current. However, most of the existing methods are based on algorithmic control schemes and lack from a rigorous stability proof in order to mathematically prove that the current will always remain below a given maximum value, even during transients. In terms of stability, introducing a current saturation unit in the control structure can be devastating for the inverter and lead to undesired oscillations, mainly caused by integration wind-up. Additionally, the inverter should be able to return to its initial condition after the fault is cleared and avoid latch-up issues.

To this end, in this invention, a nonlinear control unit for single-phase inverters connected to the grid via an LCL filter is disclosed to guarantee a current-limiting property under both normal grid operation and grid faults. The controller is independent from the system parameters, has the same structure under both normal and faulty grid conditions and can change between different operating modes including the droop function and accurate real and reactive power regulation to some reference values whenever required. Since the controller is based on the structure of the robust droop controller, tight output voltage regulation is also guaranteed. Therefore, the desired performance can be achieved without additional saturation units, switches or monitoring devices. The continuous-time structure of the controller allows a rigorous proof of the current-limiting property at all times independently from grid variations. An analytic framework of selecting the controller parameters and overcoming practical implementation issues is also disclosed.

The combination of the above leads to an easily implementable control strategy for grid-tied inverters compromising • a feed-forwarded voltage term • a controllable voltage source • a dynamic virtual resistance.

Specific embodiments of the present invention will now be described, by way of example only, with reference to the accompanying drawings, in which:

Figure 1 depicts the structure of an single-phase inverter connected to the grid.

Figure 2 depicts the equivalent circuit diagram of the disclosed controller.

Figure 3 depicts the two ellipses on which the trajectories of the control states stay.

Figure 4 depicts the invented nonlinear controller for grid-tied inverters.

Figure 5 shows the experimental results of the inverter operation under normal grid, where Figure 5(a) shows the time response of the real and reactive power, the RMS values of the inverter current and the capacitor voltage, and the grid frequency, Figure 5(b) shows the transient response at t = 15 s (current-limiting property) and Figure 5(c) shows the steady-state response after t = 30 s.

Figure 6 shows experimental results of the controller states, where Figure 6(a) shows the time response of the controller states w and wq, Figure 6(b) shows the time response of the controller states δ and 6q, Figure 6(c) shows the controller states trajectory on w — wq plane and Figure 6(d) shows the controller states trajectory on δ — δη plane.

Figure 7 shows the experimental results of the inverter operation under grid voltage sag (110 V —> 90 V), where Figure 7(a) shows the time response of the real and reactive power, the RMS values of the inverter current and the capacitor voltage, and the grid frequency, Figure 7(b) shows the transient response when the fault occurs and Figure 7(c) shows the transient response when the fault is cleared.

Figure 8 shows the experimental results of the controller states during the grid voltage sag (110 V —s> 90 V), where Figure 8(a) shows the time response of the controller states w and wq, Figure 8(b) shows the time response of the controller states δ and 6q, Figure 8(c) shows the controller states trajectory on w — wq plane and Figure 8(d) shows the controller states trajectory on δ — δη plane.

Figure 9 shows the experimental results of the inverter operation under grid voltage sag (110 V —> 55 V), where Figure 9(a) shows the time response of the real and reactive power, the RMS values of the inverter current and the capacitor voltage, and the grid frequency, Figure 9(b) shows the transient response when the fault occurs and Figure 9(c) shows the transient response when the fault is cleared.

Figure 10 shows the experimental results of the controller states during the grid voltage sag (110 V —> 55 V), where Figure 10(a) shows the time response of the controller states w and wq and Figure 10(b) shows the time response of the controller states δ and 6q, Figure 10(c) shows the transient response when the fault occurs and Figure 10(d) shows the transient response when the fault is cleared.

The rest of this description is organised as follows. The dynamic model of a grid-tied inverter is presented and the control design problem is formulated. In Section II, the nonlinear current-limiting droop controller is developed and analysed. The controller implementation is analytically described by defining all the parameters and some practical implementation issues are also underlined. In Section III, the closed-loop system performance under grid faults is investigated. In Section IV, experimental results are presented for both normal and faulty grid conditions. I. Dynamic model of grid-tied inverter

Figure 1 describes a single-phase inverter connected to the grid through an LCL filter consisting of the inductances L and Lg with small parasitic resistances in series r and rg, respectively, and a capacitor C with a large parasitic resistance Rc in parallel. The grid voltage is denoted as vg = V2Vg sin jjgL where Vg is the RMS grid voltage and Dg is the grid angular frequency, while the inverter voltage and current are v and i, respectively, with the output capacitor voltage being vc and the grid current being ig. The inverter is assumed to be controlled using a pulse-width-modulated (PWM) generator with high frequency and therefore the voltage v can be assumed the same as the average voltage over a switching period. Here, the grid is considered stiff, although the grid voltage and frequency can vary slightly from their rated values.

The dynamic model of the system is given as

(1) Γ π τ which is obviously linear with state vector x = i vc ig and control input the inverter voltage v, while vg represents an uncontrolled external input.

In order for the inverter to support the grid in voltage and frequency regulation, droop control is adopted into the control design. By defining the control input as v = \flE sin Θ with Θ = ω being the angular frequency of the inverter, the conventional droop control takes the form: ω = ω* - n(P - Pset) (2) E = E* -m(Q - Qset), (3) where ω* and E* are the rated angular frequency and voltage, respectively, Pset, Qset correspond to the reference values of the real and the reactive power and n, m are the droop coefficients. The measured real and reactive power P and Q are usually obtained at the capacitor node as the average values of the instantaneous power expressions over where vcq is the the capacitor voltage delayed by | rad. It is then obvious that the power expressions are nonlinear due to the multiplication of the system states, resulting in a nonlinear closed-loop system that is difficult to analyse in terms of stability. Using the robust droop controller disclosed in GB2483879, tight voltage regulation can be accomplished. Since the robust droop controller introduces an additional term to make the output impedance resistive, the droop functions result in ω = ω* + mn(Q - Qset) (5) E = Ke(E* — Vc) — n(P — Pset), (6) where Ke is a positive constant gain and Vc is the RMS value of the capacitor voltage, which is a nonlinear expression as well. II. Nonlinear current-limiting droop controller A. Controller structure

The disclosed generic controller for grid-tied inverters is given in the following form v = vQ T AU sin(cu0t + 5)- rai, (7) where vQ, AV, ω0, δ and r0 should be suitably defined to achieve different controller operating modes.

The equivalent circuit of the controller is given in Figure 2. In particular, the proposed generic controller consists of three main parts: 1. a voltage term v0 2. a controllable voltage source AV sin(iu0t + δ) 3. a dynamic virtual resistance rQ.

In order to inherit the droop control functions of (5)-(6) and a crucial current-limiting property, the controller terms are defined as explained below. B. Controller design

For the design of a current-limiting droop controller, the controller components are defined with 1. v0 = vc 2. AV = y/2(l - wq)Vg 3. UJ0 = UJg 4. rQ = (1 - wq)w so the controller takes the form v = vc + (1 — wq)(s/2Vg sm(u)gt + δ) — wi). (8)

Variables w, wq and δ represent the controller states with dynamics:

with cw, eg, wm, Awm, Αδηι, kw and k§ being positive constants. The initial conditions of w, wq and δ, δ(1 are defined as w0 = wm, wq0 = 1 and 50 = 0, δφ = 1, respectively. Note that both Vg and ω9 can be obtained by using a traditional PLL.

By considering, for the controller dynamics (9)-(10), the Lyapunov function candidate

(13) its time derivative is

(14)

According to the initial conditions w0 and wq0, it results that both w and wq start and travel at all times on the ellipse

as depicted in Figure 3(a). This guarantees that w G [wmin, wmax] = [wm — Awm, wm + Awm], Vi > 0. By choosing wm > Awm > 0, the ellipse is defined on the right half plane and w > 0. V/ > 0. This indicates that when Ke(E* — Vc) = n(P — P3Ct), as the robust droop controller results in the steady state, then 0 = 0 and then w and wq will converge to some constant values we and wqe, respectively, corresponding to the desired equilibrium point.

It should be also noted that the angular velocity φ, given from (15), can be zero on the horizontal axis as well, i.e., when wq = 0. This is desirable in order to avoid a possible limit cycle behavior of the controller dynamics on the w — wq plane. Particularly, if the controller states pass the desired equilibrium point during transients and tend to reach the horizontal axis, then wq —> 0 and as a result Φ —> 0 independently from the term Ke(E* — Vc) — n(P — Pset)· Thus, the controller states slow down until the angular velocity changes sign and forces them to return to the desired equilibrium. Consequently, w and wq cannot travel around the whole ellipse Wo and therefore it holds true that wq G [0,1], Vt > 0.

In the same framework, the controller states δ and 5q given from (11)-(12) will operate exclusively on the ellipse

with angular velocity

(16) with similar properties as described above, as shown in Figure 3(b). C. Current-limiting property

By substituting the controller (8)-(12) to the original plant (1), the inverter current dynamics become

(17) which can be handled as a time-varying system with w £ icmax] > 0 and wq £ [0,1] for all t > 0.

By choosing the controller parameter wmin as

(18) where Imax is the maximum allowed RMS current, then there is

for an initial RMS value of the current satisfying /(0) < Imax, since (1 — wq) > 0 and rlmax > o. In other words, the current will never exceed Iniax, which guarantees the current-limiting property of the disclosed controller. D. Selection of controller parameters

Since the term (1 — wq)w represents a dynamic virtual resistance at the output of the inverter and wmin corresponds to a maximum current Imax, similarly, the initial value wq = wm will correspond to an initial current Im as

Note that initially the inverter is not connected to the grid but a small amount of current still flows through the LC filter. In particular, since the RMS capacitor voltage is almost at Vq to have a smooth connection (vc pa v9), then the inverter current before the connection will be

Im = W*CVq.

As a result

(19)

According to the ellipse W0, the parameter Awm is given as

(20)

It should be underlined that since Awm > 0, then

Imax > UJ*CV9 « u*CE* for a normal grid, which makes sense since one cannot set Imax to a lower value than the current needed for the capacitor of the filter.

Regarding parameter Αδηι, this corresponds to the maximum absolute value of δ. According to (17), the controller state δ describes the phase shifting applied to the inverter voltage. By neglecting the small phase shifting applied by the filter inductor L, the value of δ will correspond to the reactive power of the inverter, i.e. 5 = 0 and δ = — | will approximately correspond to O = 0 and Q = Sn, respectively, where ,S'„ is the rated power of the inverter. Therefore, Αδ,η is chosen as | in order to allow the control of the reactive power in the range Q E [—Sn, .5',,]. In practice, Αδηι can be chosen smaller to cope with the small inductance L.

The controller gains kw and /¾ should be any arbitrary positive number since they increase the robustness of the wq and 6q dynamics, respectively, with respect to calculation errors during an actual implementation.

Finally, the controller parameters cw and c§ affect the dynamic performance of the controller since they are found inside the angular velocities φ and φ in (15) and (16), respectively. By considering as ts the settling time of the inverter and assuming a worst case scenario, the minimum value of parameter cw and c§ can be chosen as

In practice, cw and c§ can be further increased to improve the dynamic performance and achieve an acceptable time response. Therefore, the settling time ts can be chosen smaller than the original value until a satisfactory response is achieved. E. Practical implementation

Since during the grid-connected operation, Lg and rg are relatively small, one can ignore the small phase shifting and voltage drop across the inductor which gives vg ps vc. This means that the controller structure (8) can become v = vg + (1 — Wg)(vg cos δ + vgq sin δ — wi) (23) and result in the same performance. This can also help with the initial connection to the grid since according to the initial condition of the controller state wg0 = 1, before the connection with the grid it yields v = vg and a smooth connection can be achieved if the small voltage phase shifting and voltage drop across the L inductor is also ignored. After connecting with the grid, the controller can be enabled at any time; thus no pre-synchronisation unit is required.

Based on the controller structure (23), the measured signals vg and i are directly used in the control input v and therefore they represent feed-forward terms which can introduce a small delay due to the measurement and communication circuits. To overcome this small delay, a phase-lead low-pass filter

can be used for the measurements of vg and i. The final controller implementation structure is shown in Figure 4. III. Operation under grid faults

When the inverter is connected to a stiff grid, there is Vg = E* and ω9 = ω*. Since the selection of wmin is given from (18) where Vg = E* (considering a stiff grid), the current-limiting property is guaranteed for any Vg < E* and for any frequency ug. This includes the inverter operation under grid faults as it is better explained below: A. Case 1: Short circuit

Assuming that a grid fault occurs with a short circuit at the grid voltage, then Vg = 0 and the closed-loop system (17) becomes

(24) which means that the current i will exponentially converge to zero since w e [//·„„„. wmax\ > 0 and 1 — wq > 0. Opposed to the traditional droop control methods that will try to regulate the output voltage which will force the current to increase to high values and violate the maximum limit, the controller forces the inverter current to exponentially converge to zero, satisfying the current-limiting property and protecting the inverter. When the fault is cleared, the closed-loop system becomes again as the one in (17) which forces the current i to increase and converge again to the desired value. Furthermore, during the fault wq —> 0 which results from (9) that w —> 0 and the integration automatically slows down. As a result, the disclosed controller can overcome windup and latch-up problems without additional switches or monitoring devices. B. Case 2: Voltage sag

Assume now that instead of a short circuit, a voltage sag occurs to the grid with a percentage p x 100%, i.e. the grid voltage Vg becomes (1 — p)Vg. Then the closed-loop system is given as

(25) which according to the same current-limiting property yields I < {l - p)Iniax. (26)

Therefore the RMS voltage of the inverter current I still remains less than I max. Once again, when the fault is cleared, the closed-loop system is expected to return to its original values after a small transient.

If Imax is chosen according to the ratings of the inverter, e.g., given a rated power

Table I

System and controller parameters

Sn, then (26) is equivalent to S<(l-p)Sn, (27) by ignoring the small voltage drop at the filter, which provides a limit for the apparent power of the inverter at all times, even during faults. IV. Experimental validation

Extensive experimental results of the disclosed invention have been obtained for a single-phase grid connected inverter operating with the developed controller. A switching frequency of 15 kHz was used for the inverter operation and the disclosed controller was implemented using the TMS320F28335 DSP with a sampling frequency of 4 kHz. The controller performance was investigated under both a normal and an abnormal operation of the grid with parameters given in Table I. A. Operation under normal grid

The time response of the system is shown in Figure 5(a), where the grid frequency remains constant at 49.97 Hz. The inverter is connected to the grid at t = 6 s and the real and reactive power references are set to Pset = 50 W and Qset = 0 Var, respectively. Both the real power and the reactive power are regulated to their reference values, although the reactive power is slightly positive (less than 5 Var) due to the limitation of the power analyser used for the results, which cannot show negative reactive power and introduces small inaccuracies close to zero. At t = 9 s the real power reference changes to 100 W and at t = 12 s the reactive power references changes to 50 Var. Figure 5(a) clearly demonstrates the ability of the proposed controller to regulate the injected power to a given level. At the time instant t = 15 s, the reference Pset is changed to 250 W, which violates the technical limits of the inverter and will force the inverter current to exceed Imax. However, as shown in Figure 5(a), the real power is regulated at around 180 W because the current has reached the maximum limit. Particularly, the RMS value of the current is limited at 1.73 A, which is less that Imax = 2 A for three main reasons: i) the reactive power is not zero, which reduces the maximum current, ii) the parasitic resistance of the L inductor is not zero and iii) the proposed controller uses the feed-forward voltage term vg from (23) instead of vc, which are slightly different. The transient response of the inverter and grid currents and voltages is shown in Figure 5(b), where the inverter current increases and reaches the maximum allowed value. At t = 18 s, Pset is changed to 150 W. In order to check the droop functions of the proposed controller, at / = 21 s, the P ~ V droop function is enabled and the real power drops in order to bring the capacitor voltage Vc closer to the rated value E*. This is clearly observed in Figure 5(a). Finally, at t = 23 s, the Q ~ — Θ droop function is enabled and the reactive power drops since the system frequency is lower than the rated. This verifies the capability of the proposed controller to operate in different modes. The steady-state response of the system is shown in Figure 5(c). Finally, the responses of the controller states w, wq and δ, 5q as shown in Figures 6(a) and 6(b), respectively, while their trajectory on the desired ellipses Wo and Δ0 is verified in Figures 6(c) and 6(d), respectively. B. Operation under grid faults

To further evaluate the disclosed controller, two different grid-fault scenarios are investigated while the system is operating with the droop functions.

Case 1: 110 V —> 90 V grid voltage sag

In this case, the grid voltage drops rapidly from 110 V to 90 V and the fault is cleared after 9 s. As shown in Figure 7(a), during the fault, the current remains limited below 1.64 A (calculated from (26)). Figure 7(b) indicates the drop of the grid and the capacitor voltage and the smooth increase of the current. When the fault is cleared, the system returns to its initial operation after a short transient (Figures 7(a) and 7(c)). This transient is caused due to the fact that the controller state w, which represents the main part of the virtual resistance, is regulated at wmin for the duration of the fault. Then, when the grid voltage rapidly increases, the current will increase due to the small resistance wmin but it will never violate the limit as rigorously proven in the theory. After the short transient, w increases and converges again to its initial value. This is clearly shown in the time response of the controller states w and wq in Figure 8(a). Since the frequency of the grid remains constant and (27) is not violated according to the given Qset = 50 Var and the droop control, the reactive power is regulated at the same value during the fault (Figure 7(a)), which is achieved from the performance of the controller states δ and 6q (Figure 8(b)). The trajectories on Wq and Δ0 are once again verified in Figures 8(c) and 8(d), respectively.

Case 2: 110 V —> 55 V (50%) grid voltage sag The responses of the real and reactive power during the 50% voltage sag of the grid voltage are shown in Figure 9(a). During the fault, the real power is regulated to a lower value corresponding to the limit of the inverter current since according to the analysis of subsection III-B, the current I should be limited below Imax x 50% = 1 A, as it is shown from the RMS current response in Figure 9(a) and from the transient in Figure 9(b). As in the previous grid fault scenario, when the fault is cleared, the system returns to its initial operation after a short transient (Figure 9(c)). Some oscillations that appear in the inverter current and capacitor voltage during the fault and during its clearance are caused by the slow response of the PLL and the dynamics of the LCL filter, since for simplicity the controller (23) was used instead of its original form (8). However, this transient lasts for less than a half-cycle, which is acceptable in practice. Although again w —> and wq —> 0 in order to guarantee the current-limiting property (Figure 10(a) and Figure 10(c)), a difference between this and the previous fault scenario is that during the fault and the current limitation (27), the droop control in the reactive power can no longer be accomplished with the given Qset = 50 Var. This means that the controller states δ and δ(1 also decrease and converge to lower values that correspond to a lower reactive power that guarantees (27) for the given current limitation. This is shown in the time response of the controller states in Figure 10(b) and the trajectory on the δ — % plane is shown in Figure 10(d).

Claims (11)

1. A control device for controlling a grid-tied inverter that generates an inverter output voltage comprising • a feed-forwarded voltage term, • a controllable voltage source, and • a virtual resistance multiples with the grid current.

2. A control device as claimed in Claim 1 having • a sensor to measure the inverter output voltage, • a sensor to measure the grid voltage, • a sensor to measure the inverter current, • a power unit to provide real and the reactive power of the inverter, • a PLL unit to obtain the amplitude, frequency and phase of the grid voltage, • a root-means-square (RMS) unit to obtain the RMS value of the output voltage, • a sinusoidal function unit to form the controllable voltage source, • a nonlinear control unit consisting of two double-integrators that take the droop functions for the real and the reactive power as the inputs, • a PWM conversion block to convert the output voltage into PWM pulses to drive the converter switches, • a voltage generation unit to form the output voltage before sending it to the PWM conversion block.

3. A control device as claimed in Claims 1 and 2, in which the virtual resistance stays inside a range having a lower bound larger than zero.

4. A control device as claimed in Claims 1 and 2, in which the output voltage is generated by adding the feed-forwarded voltage, the controllable voltage source and the product of the virtual impedance with the grid current together.

5. A control device as claimed in Claims 1 and 2, in which the power unit provides the real and the reactive power via a series of calculations according to the output voltage and the inverter current.

6. A control device as claimed in Claims 1 and 2, in which two phase-lead low-pass filters are used for the grid voltage and inverter current measurements.

7. A control device as claimed in Claims 1 and 2, in which the nonlinear control unit consists of two double-integrators, each providing two controller states within given bounded ranges.

8. A control device as claimed in Claims 1 and 2, in which the feed-forwarded voltage term is provided by the sensor that measures the grid voltage after passing through one of the two phase-lead low-pass filters.

9. A control device as claimed in Claims 1 and 2, in which the output of the nonlinear control unit consists of the virtual resistance multiplied with the inverter current and the controllable voltage source.

10. A control device as claimed in Claims 1 and 2, in which the controllable voltage source takes the RMS grid voltage multiplied by the output of one doubleintegrator as the RMS value, the grid frequency as the frequency and the output of the other double-integrator as the phase.

11. A control device as claimed in Claims 1, 2 and 10, in which the grid RMS voltage and the grid frequency are obtained by the PLL.