STORAGE ENERGY OPTIMIZED DYNAMIC VOLTAGE RESTORER Mahinda Vilathgamuwa*, A.A.D. Ranjith Perera* and S.S. Choi* *School of Electrical and Electronic Engineering Nanyang Technological University Nanyang Avenue, Singapore 639798. e-mail:
[email protected],
[email protected],
[email protected] Abstract Dynamic Voltage Restorer (DVR) is a custom power device used for voltage compensation of sensitive loads against voltage disturbances in power distribution lines. This paper illustrates a correction technique, which draws a minimum amount of energy from the DVR during the process of compensation of a voltage sag or swell. Using the proposed method it can be shown that a particular disturbance can be corrected with less amount of storage energy compared to that of existing in-phase boosting method where the injected voltage is in phase with pre-sag voltage. This paper also discusses a method to detect supply voltage parameters based on Kalman filtering. Finally a multi-loop feedback control method is presented to improve the dynamic behavior of the DVR. 1. INTRODUCTION The proliferation of voltage sensitive equipment in industrial sector has made industrial processes more vulnerable to supply voltage deviations. Such voltage deviations in the form of voltage sag, swell or temporary outage cause severe process disruptions resulting in millions of dollars of loss of revenue. Therefore power supply authorities as well as customers have been desperately looking for a cost effective solution currently to ride through momentary power supply disturbances. As such, the proposition of a novel custom power device called dynamic voltage restorer (DVR) for compensating voltage disturbances in distribution systems has generated a great deal of interest recently [2][3]. Apart from the DVR, some researchers have proposed several other devices to mitigate momentary disturbances. Among those, static voltage booster [1] and unified voltage controller [2] have been noteworthy. The DVR usually consists of an injection transformer, which is connected in series with the distribution line, a voltage sourced PWM inverter bridge which is connected to the secondary of the injection transformer and an energy storage device connected at the dc-link of the inverter bridge. A typical schematic of the DVR is shown in Fig. (1). The inverter bridge output is filtered before being fed to the injection transformer in order to nullify switching frequency harmonics. The series injected voltage with a variable amplitude, phase and frequency of the DVR is synthesized by modulating
pulse widths of the inverter bridge switches. The injection of an appropriate series voltage component in the face of a voltage disturbance requires a certain amount of real and reactive power supply by the DVR. The real and reactive power supplied by the DVR however depends on the type of voltage disturbance experienced as well as the direction of the DVR injected voltage component with reference to pre-sag voltage. The idea of advancing the injected voltage in order to minimize the real power supplied by the DVR has generated a great deal of research interest recently. In this paper, the necessary conditions for such advance-angle control have been formulated for a given supply voltage disturbance. The fidelity of the DVR output voltage depends on the accuracy and dynamic behavior of the pulse width modulated (PWM) synthesis scheme and the control system adopted. Traditionally, closed-loop control PWM schemes have been used for various types of ac power conditioning systems. The general requirement of such control scheme is to obtain an ac waveform with low total harmonic distortion and good dynamic characteristics against supply and load disturbances. Although conventional sinusoidal PWM schemes and programmed optimal PWM schemes have performed reasonably well for linear loads, the voltage waveforms tend to get distorted for nonlinear loads. To overcome this behavior, a deadbeat control scheme has been proposed where the switch-on times of the inverter switches are calculated such that the filter capacitor voltage would reach the reference voltage at the next sampling time
[5]. However this scheme also suffers from its sensitivity to parameter variations, dependence on load parameters for control and complex structure. These drawbacks can be avoided if a multiloop feedback controller is used where filter capacitor current is fed back to achieve a sinusoidal capacitor current while an outer voltage loop is used to regulate the output voltage. Therefore in this paper, the multi-loop controller is proposed for the DVR to achieve good dynamic performance. For energy optimized operation of the DVR, it is imperative to determine the supply voltage parameters on-line as the DVR injected voltage advanced angle depends on these parameters. The Discrete Fourier Transform (DFT) method and the Kalman filtering approach have been proposed to predict the supply voltage parameters [7]. As most disturbances in power systems belong to nonstationary category, the Kalman filtering approach gives more accurate prediction compared to that of DFT methods [8]. In this project, the Kalman filter approach is adopted to detect supply voltage magnitude and phase angle on-line to obtain an accurate and a fast DVR compensation. This paper is organized as follows. The general principles of the DVR operation along with the DVR energy optimization is presented in section 2. The detection of supply voltage parameters using Kalman filter approach is discussed in section 3. In section 4, the authors present the multi-loop controller design for the DVR. Finally, the effectiveness of proposed DVR control structure is evaluated and simulation results are presented in section 5. 2. PRINCIPLES OF THE DVR OPERATION The DVR is connected in series with power distribution line as shown in Fig. (1). The DVR is able to control the voltage across a sensitive load by injecting an appropriate voltage phasor through an injection transformer. As a result, any voltage disturbance appears in up-stream can be compensated through the DVR and the disturbance is unseen to the load. Vdvr
VS
V1
Supply
I
V2 Load
Fig. 1: Typical schematic of a power system compensated by the DVR The in-phase boosting technique where the correction voltage is in-phase with the supply voltage has been used widely for correcting voltage disturbances [3]. Under in-phase boosting, the DVR is required to inject a certain amount of active power during the period of compensation. Therefore, the stored energy becomes the limiting factor in the disturbance compensation process especially for sags which last longer. Consider a phasor diagram of a typical sag situation as shown in Fig. (2) where only an unbalance in a certain phase is shown for clarity.
V2 α φ
V1 I
Fig. 2: Phasor Diagram of power distribution system during a sag Where V1, V2, Vdvr are the post-sag supply voltage magnitude, compensated load voltage magnitude and the DVR injected voltage magnitude respectively. Moreover I, φ, δ, α represent load current, load power factor angle, supply voltage phase angle deviation and load voltage advance angle respectively. If Pin and Pout are the input power from the source and load power respectively, then Pin = ∑ V1 j I j Cos( φ − α + δ j ) (1) ∀j
Pout = ∑ V 2 j I j Cos( φ )
DVR
(2)
∀j
Where j = A, B , C Assuming a balanced load (Ij = I) and a balanced output voltage (V2j = V2), Pout = 3V 2 I Cos( φ ) Pdvr = Pout − Pin
∀j
PWM Inverter
δ
Pdvr = 3V2 ICos( φ ) − ∑ V1 j ICos( φ − α + δ j )
Filter Circuit
Strorage Unit
Vd
(3)
It has been shown that the supply of energy by the DVR for voltage restoration can be kept minimum, by advancing all three phases with a certain advance angle α [6]. This control method is usually known as α -control. The complete theoretical analysis for the derivation of necessary phase advance angle for generalized sag situations is given in [6] where any magnitude or phase change of the supply voltage has been accommodated. For the sake of completeness, the expressions for the optimum phase advance angle αopt are summarized as follows: If the supply voltage parameters satisfy the condition, ∑ V Cos δ j ∀j 1 j
2
+ ∑ V1 j Sin δ j ∀j
( )
2
( )
1
2
≥ 3V2 Cos( φ )
then
(8) X 2 + Y 2 ≥ 3V 2 Cos( φ ) i.e. when the sag parameters satisfy the equation (8), the DVR does not supply any energy to the load during the compensation by means of α -control. In reality, the DVR absorbs power from uncollapsed phases under such situations to minimize the need of supply of active energy from the stored energy source. Fig. (3) shows the variation of the DVR supplied power against advance angle for two cases of voltage Per unit DVR Power vs Advance Angle 6.00 5.00 Per Unit DVR Power
Where Pdvr is the DVR supplied power during the sag.
4.00 3.00
Case II Case I
2.00 1.00 0.00
α opt = φ + β − Arc cos
0 50 100 150 200 250 300 350 Advance angle in Degrees unbalances. 2 2 ∑V1 j Cosδ j + ∑V1 j Sinδ j Fig. 3: Energy supplied by the DVR for two different sags. Case I: Eq. (8) is not satisfied. Case (4) II: Eq. (8) is satisfied. -1.00
(
α opt = φ + β
else
3V2 Cos( φ )
) (
)
Y Where β = Arc tan , X = ∑ V1 j Cos( δ j ) and X ∀j Y = ∑ V1 j Sin( δ j ) ∀j
th
V1j – post-sag supply voltage of the j phase δj – post-sag angle deviation of phase j of the supply j = A,B,C The magnitude of the DVR injection voltage and the real power supplied by the DVR can be calculated for αopt as follows; j V dvr = V 22 + V12j − 2V 2 V1 j Cos( α opt − δ j )
(5)
opt Pdvr = 3V2 ICos(φ) − ∑ V1 j ICos(φ − α opt + δ j ) (6) ∀j
Also, the minimum energy needs to be stored in the DVR storage unit can be formulated as: opt opt Edvr = Pdvr ∗ Tsag
(7)
Where Tsag is the maximum anticipated sag duration. The necessary condition for correcting a sag without supplying energy from DVR is,
3. DETECTION OF DISTURBANCE PARAMETERS BY KALMAN FILTERING It can be seen that the equation (4) requires the knowledge of the voltage disturbance parameters (magnitude (V1j) and phase change (δj)) to determine the optimum α. As it may not be possible to estimate these parameters as soon as a disturbance has occurred, the optimized energy operation of the DVR needs to be delayed. Therefore it is proposed that until the supply parameters are made available by the estimator, the voltage disturbance has to be corrected by operating the DVR in the in-phase boosting mode. The supply voltage parameter estimation has been carried out using Discrete Fourier Transform (DFT) method as well as Kalman filter algorithm [7] earlier. However, the DFT and FFT algorithms which have been applied to many important areas of waveform analysis of power systems, suffer from certain limitations, such as the requirement of the input signal to be stationary and number of samples per cycle be an integer number. These limitations can be overcome by Kalman filtering approach. The Kalman filtering technique enables optimal estimation
and tracking of the supply voltage parameters irrespective of the nature of the input signal. Consider a sinusoidal supply voltage signal with an angular frequency ω, V (t ) = Vm ⋅ Sin(ωt + δ ) V (t ) = VmCosδ ⋅ Sin(ωt ) + Vm Sinδ ⋅ Cos(ωt ) If we select x1 =
Vm
Cosδ and x2 =
Vm
Cosδ
2 2 then the state equations can be written in the following form, w x1 1 0 x1 + 1 x = 0 1 ⋅ x 2 k −1 w2 k − 1 2 k xk = φk −1 ⋅ xk −1 + wk −1
(9)
where w1 and w2 allow the state variables to random walk with time. The measurement matrix is given by, x zk = 2 Sin(ωt k ) 2Cos(ωt k ) ⋅ 1 + vk (10) x2 k
[
and resulting error covariance is given by, Pk ( + ) = [I − K k H k ]Pk ( − ) (14) The equations 11 and 14 describe the discontinuous state estimate and the covariance matrix behavior across a measurement while the following equations describe these behaviors between measurements, xˆk (−) = φk −1 xˆk −1 (+) (15) Pk ( − ) = φk −1 Pk −1( + )φkT−1 + Qk −1
(16)
Therefore it is possible to estimate the supply voltage parameters on-line by sequentially applying recursive equations 11, 13, 14, 15 and 16. The Fig. (4) shows the effectiveness of the Kalman filter algorithm and is compared with the DFT algorithm developed. In this example, a voltage sag of 50% occurs at t=0.05 sec with an angular deviation of –0.5 radians. It can be seen that the Kalman filter estimates the parameters far quicker than the DFT algorithm and requires less memory space for the computation.
]
z k = H k ⋅ xk + vk where vk represents the measurement noise. The covariance matrix Qk of w and the covariance Rk of the noise v are given by,
[
[ ]
]
Q i = k R i = k E wk wiT = k , E vk viT = k ≠ 0 i k 0 i≠k In Kalman filtering, a prior estimate of the system state at time tk , denoted by ˆxk ( − ) , is used to derive an updated estimate, ˆxk ( + ) , based on the measurement vector zk [4]. This estimate is carried out by the following linear recursive equation, ˆxk ( + ) = xˆk ( − ) + K k [zk − H k ˆxk ( − )] (11) ((-) and (+) are used to denote the times immediately before and immediately after a discrete measurement, respectively). The gain K k is optimally evaluated such that a weighted scalar sum of the diagonal elements of error covariance matrix Pk ( + ) is minimized where the error covariance matrix Pk ( + ) is given by,
[
Pk ( + ) = E (ˆxk ( + ) − xk )(ˆxk ( + ) − xk )T
]
(12)
The optimal Kalman gain matrix can be expressed by [4],
[
K k = Pk ( − )H kT H k Pk ( − )H kT + Rk
]
−1
(13)
Fig. 4: The supply voltage parameter estimation using DFT and Kalman filtering (KF) 4. CONTROLLER DESIGN With rapid advances in microelectronics industry, real-time digital control of power conditioners using cost-effective digital signal processor (DSP) based hardware system has become a reality. The DSP hardware architecture enables most of its instructions to be executed in a single instruction cycle and complex control algorithms to be processed at a faster rate. Therefore with the possibility of using a DSP in real-time control, it is proposed to employ a multi-loop control structure for the DVR where the DVR output is regulated such that it tracks a sinusoidal reference. In this control scheme, the filter capacitor current and load-side voltage are sensed as feedback variables and the control algorithm evaluates the necessary switching pulse
widths for the dc-ac inverter in every sampling interval. 4.1
Multi-loop Control Scheme
Fig. (5) shows the schematic of the DVR control system with the incorporation of inner capacitor current loop within the outer voltage loop. The detection of capacitor current enables the controller to predict and correct anticipating variations in the injected voltage thus enhancing the dynamic performance. Moreover, the outermost voltage loop tracks the output voltage stiffly and produces a near sinusoidal output waveform with low total harmonic distortion. V1 Disturbance compensator IC -
V2ref + -
KV+KDs
+
KC
Inverter KI
+ + -
1 Ls
V2
+ -
VC
1 Cs
N
V2
+ + V1
N IL
Fig. 5: Basic Control Diagram of multi-loop PID controller In order to examine the stability of the DVR control system, the following transfer function for the inner current loop can be written by neglecting the filter inductor equivalent series resistance (ESR) and the filter capacitor ESR [9], R s s + L LL K Gc ( S ) = i ⋅ L RL R L + N 2L s+ s3 + L s2 + L LL LC LL LC LL (14) Vdc . where, K i = 2 The open loop transfer function of the outer voltage feedback loop is given by, K c Gc ( S ) 1 Gv ( S ) = (K v + K D s ) ⋅ ⋅ (15) Cs 1 + K c Gc ( S )
(
)
The bode-plot of closed loop transfer function of outer voltage loop is shown in Fig. (6). The outer loop controller gains are selected such that the steady state error between load voltage and its reference waveform is within an acceptable value. With this controller gains the system behaves in stable manner
with a phase margin of 95 deg and a theoretical gain margin of infinity. The bode plot hardly shows any variation with the variation of the load ranging from no-load to highly inductive load. This shows the robustness of the controller in the face of load parameter variations. Fig. 6: Bode diagram of the closed loop transfer function of the control system with Kv = 6.0, Kc = 1.0, KD = 0.0006 5. SIMULATION RESULTS A detailed simulation of the DVR control system was performed using MATLAB/SIMULINK program in order to verify the operation. The parameters of the DVR system are as follows, Supply Voltage Series transformer turns ratio DC link voltage Filter capacitance Filter inductance Load resistance Load inductance
: : : : : :
100V 1: 10 30V : 2mF 0.1mH 34Ω 67mH
Consider an occurrence of a three phase, 40% voltage sag at 0.025 sec which lasts for 5 cycles as shown in Fig. 7(a). The corrected load voltage with the inverter switching at 5kHz is shown in Fig. 7(b). The waveforms are obtained from simulations carried out with a detailed model of the dynamic voltage restorer including the pulse width modulation process. According to the simulation results shown in Fig. 7(b) it can be seen that there is a voltage spike at the end of the sag owing to sudden recovery of the sag. This spike may not appear as sharp as shown in Fig. 7(b) practically because the happening of a sharp sag recovery is a rare exception. The simulations have been carried out for voltage sags up to 90% and an excellent DVR output behavior with a balanced and nominal amplitude sinusoidal waveform has been accomplished. In this control scheme, the phase advancement of the injected voltage as suggested in section 2 is started around 0.035sec once all the supply voltage parameters have been accurately detected by the Kalman filter. During the period 0.025
