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Maximum Power Reference Tracking Algorithm for
Power Curtailment of Photovoltaic Systems
Victor Paduani, Lidong Song, Bei Xu, Dr. Ning Lu
Department of Electrical and Computer Engineering
North Carolina State University, Raleigh, NC
Abstract—This paper presents an algorithm for power curtail-
ment of photovoltaic (PV) systems under fast solar irradiance
intermittency. Based on the Perturb and Observe (P&O) tech-
nique, the method contains an adaptive gain that is compensated
in real-time to account for moments of lower power availability.
In addition, an accumulator is added to the calculation of the step
size to reduce the overshoot caused by large irradiance swings.
A testbed of a three-phase single-stage, 500 kVA PV system
is developed on the OPAL-RT eMEGAsim real-time simulator.
Field irradiance data and a regulation signal from PJM (RTO)
are used to compare the performance of the proposed method
with other techniques found in the literature. Results indicate an
operation with smaller overshoot, less dc-link voltage oscillations,
and improved power reference tracking capability.
Index Terms—Maximum power point tracking, PV Power
curtailment, PV system control, PV system modeling, Real-time
simulation.
I. INTRODUCTION
In grids with high penetration of photovoltaic (PV) systems,
there is an increasing need for employing inverters with superb
controllability to provide grid support functions (GSFs). IEEE
Standard 1547-2018 [1] imposes new operational requirements
on smart inverters to accelerate technological developments for
enabling PV systems at all levels to provide high quality grid
services. Main GSFs of interest include power curtailment,
voltage and frequency droop control, fast frequency response,
and reactive power regulation [2].
Although maximum power point tracking (MPPT) algo-
rithms for improving irradiance tracking are well researched in
the literature, power curtailment algorithms remained under-
examined until the last decade. In [3], Wandhare and Agarwal
introduce a power curtailment algorithm for single-staged PV
systems, in which the dc-link voltage is controlled by the
inverter in fixed voltage steps based on the perturb and observe
(P&O) algorithm. Meanwhile, other power limiting strategies
based on proportional-integral controllers are introduced by
Cao et al. in [4].
In [5], Sangwongwanich et al. compare the P&O- algorithm
with the PI-based approach for constant power generation
(CPG) applications and they show that the CPG–P&O method
achieved the highest robustness at the cost of worse dynamic
response. In [6], Tafti et al. introduced an adaptive algorithm
This research is supported by the U.S. Department of Energy’s Office of
Energy Efficiency and Renewable Energy (EERE) under the Solar Energy
Technologies Office Award Number DE-EE0008770.
in which the step size during transients is proportional to
the difference between the output power and the power refer-
ence. Results confirmed less overshoot and faster convergence
during irradiance changes. However, as the error increases
drastically during sudden irradiance drops caused by passing
clouds, the calculated variable voltage step size from the
adaptive algorithm may become too big and cause large
oscillations in the dc-link voltage, adding unwanted ripple to
the output power and extra stress to circuit components.
Furthermore, despite the successful classification between
transient and steady-state conditions, the issue of large power
overshoots remains as noted in [7]. In normal operation, when
solar irradiance increases, the PV output power also increases.
Thus, as the system approaches its setpoint, the step size
becomes smaller and smaller because of the diminishing power
tracking error. This leads to a slower system response until the
error becomes large again. Hence, overshoots are inevitable.
Consequently, there is a trade-off between the size of the
adaptive gain and the power overshoot.
To address the aforementioned issues, in this paper we
propose two independent solutions that do not require external
irradiance or temperature sensors. First, we introduce a com-
pensation factor that is used to adjust the adaptive gain used
to calculate the voltage step size in real-time. Second, we use
an accumulator, which serves as a memory for tracking the
most recent changes in PV output power, so the algorithm
can maintain a large voltage step size during fast irradiance
changes when the PV is operating closer to its power reference.
The rest of the paper is organized as follows: Section II
introduces the grid-scale PV system model and the proposed
algorithms; Section III describes the simulation environment;
Section IV presents the simulation results; Section V con-
cludes the paper.
II. METHODOLOGY
In this section, we will present models and control methods
used in grid-scale PV systems and introduce the improvements
proposed to enhance the power curtailment algorithm.
A. PV System Modeling
Figure 1 shows the circuit diagram of a single-stage, cen-
tralized inverter found in a grid-scale PV farm. The PV array
is modeled with a 5-parameter model from [8], whereas the
arXiv:2011.09555v1 [eess.SY] 18 Nov 2020
Fig. 1. Circuit and control system block diagrams of a grid-scale PV system.
inverter is modeled with an averaged model of a two-level
voltage source converter (VSC) developed in [9].
The terminal voltage of the two-level VSC before its output
filter, ~vt, can be calculated as
~vt=vdc
2~m (1)
where ~m is the modulation signal and vdc is the dc bus voltage.
Then, using KCL, the secondary voltage of the transformer,
(~vs), can be expressed as
~vs=~vt−rL
~
i−Lf
d
~
i
dt (2)
where iis the output current of the converter; rLand Lfis
the filter resistance and inductance, respectively.
As shown in the control system block diagram in Fig. 1, the
PV farm is operated in the grid-following mode, and the PV
inverter controller will regulate the output current so that the
power supplied from the dc-link to the grid follows the real
and reactive power reference signals, Pref and Qref. Clarke-
Park transformation is used to convert the 3-phase voltage
measurements to dq0 coordinates whereas a phase-locked loop
(PLL) is used to extract the phase of the grid.
In dq0 coordinates, a phasor ~x can be defined as xd+jxq,
and its derivative can be represented as
d~x
dt =dxd
dt +jdxq
dt +jω(xd+jxq)(3)
where ωis the nominal angular frequency of the system.
By separating the real and imaginary parts, (3) can be
rewritten as
Ldid
dt =md
vdc
2+ωLfiq−rLid−vsd (4)
Ldiq
dt =mq
vdc
2−ωLfid−rLiq−vsq (5)
where idand iq,vsd and vsq,mdand mq, are the direct
and quadrature components of i,vs, and m, respectively. To
decouple the currents and remove vsd and vsq from the control
system, mdand mq, can be calculated as
md=2
vdc vsd −ωLfiq+rLid+ud(6)
mq=2
vdc vsq +ωLfid+rLiq+uq(7)
where udand uqare the outputs of the current controllers.
Notice that the filter resistance is also included as a feed-
forward term to improve the control performance. The three-
phase modulation signal, ~m, is then generated by mdand mq
using the inverse Clarke-Park transformation.
B. Power Curtailment Algorithm
The principle of operation of the P&O technique is well
established in the literature. Based on its strategy, a power cur-
tailment algorithm can be created. According to the flowchart
in Fig. 2, when the output power of the panels, Ppv, is below a
given reference, Pref, the system operates as the P&O method;
when Ppv is above Pref, the voltage reference, v∗
dc, is adjusted
to curtail Ppv. Whether the current operation point is on the
right side or the left side of the maximum power point (MPP)
can be determined by increasing or decreasing v∗
dc, respec-
tively, with a step size of Vstep , whenever Ppv is above Pref .
In a single-stage PV system, because of inherent limitations
of inverters [10], the dc-link voltage, Vdc, should always be
kept above a minimum voltage threshold, so operation on the
right side of the MPP is necessary.
The algorithm’s convergence speed is primarily determined
by Vstep at each iteration. Ideally, Vstep should be small in
steady-state operation to reduce power oscillations around
Pref , and large during transients for a rapid response to
changes in Pref or irradiance. As introduced in [6], a fast
convergence during transient operation can be achieved by
making Vstep a function of the error from the power reference
Perr, so we have
Vstep,n =α×Vmin
step + (1 −α)×Ktr|Perr,n |
Perr,n =Pn−Pref,n
(8)
Fig. 2. Flowchart of a basic P&O power curtailment algorithm.
where αis used to switch between the power error consider-
ation in the transient operation and the fixed minimum step
change in the steady-state operation, and Ktr is a constant
gain for considering Perr.
C. Adaptive Gain Adjustment
There is a major limitation when using (8) to calculate
Vstep. When irradiance suddenly drops to a level where it is
impossible for the PV system to converge to Pref,Vstep will
be too large, causing the DC link voltage to change rapidly. To
attenuate this issue, in [6], a mode classification method for
separating steady-state and transient modes is used to switch
Vstep to a smaller value when irradiance is low. However,
the tuning method for the operation mode classification is not
clearly defined, and if it is not well designed, the issue can be
further aggravated. Besides, irradiance variations happen very
often amid the region of lower irradiance, which causes the
system to switch to transient mode frequently.
Therefore, in this paper, we introduce a compensation factor
for adjusting Ktr in real-time to provide a robust solution to
this issue. First, we calculate the moving average of the power
output of the system at time interval nover a window of length
N, as
Ppv,n =1
N
N−1
X
i=0
Ppv,n-i(9)
Then, by comparing Ppv,n with Pref ,Ktr is calculated as
Ktr =Max"CminKbase , KbasePpv,n
Pref 2#(10)
where Cmin defines the lower boundary related to the mini-
mum Ktr acceptable, and Kbase is the original gain.
The pseudo code for adaptively adjusting Ktr is shown in
Algorithm 1. When Ppv < Pref , the steady-state operation
point is the MPP corresponding to the solar irradiance, which
is varying constantly. As illustrated in Fig. 3, to determine
whether or not an MPP is reached, a counter, Ct, is used to
count the number of continuous crossings between Ppv and
Ppv,n. If Ppv,ncrosses Ppv,n,Ctis incremented; otherwise
Ctis reset to zero. Once the counter threshold, Ct,max, is
reached, Ktr will be adjusted. In this paper, we set Ct,max = 3
to balance the adjustment speed of Ktr and the MPP detection
accuracy. Once adjusted, Ktr will maintain its new value.
If Ct=Ct,max again, another adjustment will be made. If
irradiance increases quickly causing a large increase in Ppv,n
compared with Pref −τ1or entering the control deadband so
that |Ppv,n −Ppv,n|> τ2,Ktr will be reset to Kbase, as shown
in operation 6 of Algorithm 1 and Fig. 3.
Algorithm 1 Adaptive Ktr adjustment
1: Calculate Ppv,n .(9)
2: if Ppv,n −Ppv,nPpv,n-1−Ppv,n>0then
3: Ct= 0
4: else
5: Ct+= 1
6: if Ppv,n > Pref −τ1or |Ppv,n −Ppv,n|> τ2then
7: Ktr,n =Kbase
8: else if Ct≥Ct,max then
9: Update Ktr .(10)
10: else
11: Ktr,n =Ktr,n-1
Fig. 3. Example of the adaptive Ktr adjustment.
D. Accumulator for Overshoot Suppression
Overshoot is still a critical issue found in P&O-based power
curtailment algorithms (see Fig. 16c in [5]). The common
cause of an overshoot is the rapid increase of solar irradiance
after a passing cloud. If a controller cannot curtail Ppv fast
enough, an overshoot over Pref will occur. Recall when we
adjust Vstep based on Perr in (8), Vstep becomes smaller as
Ppv,n moves closer to Pref. Thus, with a reducing Vstep and
an increasing solar irradiance, Ppv will overshoot Pref and the
overshoot can be long-lasting unless Vstep can be increased
accordingly. To suppress the overshoots, we introduce an
accumulator, γ, into the Vstep calculation. At the beginning
of each iteration, the moving average of the absolute error
variation, ∆Perr,n, is calculated by
∆Perr,n =|Perr,n|−|Perr,n−1|
∆Perr,n =1
M
M−1
X
i=0
∆Perr,n−i
(11)
Then, if ∆Perr,n >0and Ppv,n > Pref , marking the
detection of an overshoot, we will add γnto the Vstep,n
calculated from (8). If there is no overshoot detected, but
variations in Ppv,n have been positive in the previous two
iterations, then γnis increased by (12). Otherwise, γnis
multiplied by a resetting rate, λr.
γn=γn−1+Kacc ×Kbase × |Perr,n|(12)
The overshoot suppression is presented in Algorithm 2.
Algorithm 2 Accumulator γn
Calculate ∆Perr,n .(11)
2: if ∆Perr,n >0and Ppv,n > Pref then
Vstep =Vstep +γn
4: else if Ppv,n > Ppv,n-1> Ppv,n-2then
Increase γn.(12)
6: else
γn=λrγn
III. SIMULATION RE SU LTS
To test the proposed adaptive gain adjustment and accu-
mulator algorithms, we set up a testbed of a 500 kVA PV
system on an OPAL-RT real-time simulation platform. One-
second irradiance data collected from a 1.04 MW solar farm
by EPRI [11] is used. To demonstrate the power following
capability in a realistic application, we apply a unidirectional
200 kW PJM regulation signal to the estimated available power
of the system to generate the power reference command. The
estimated power is obtained by applying a low-pass filter to the
irradiance. The simulation timestep is 100 µs. To account for
measurement noise due to sensor imperfections, a Gaussian
noise was added to vpv,ipv, and vdc , creating a signal-to-
noise-ratio of 71 dB. The rest of the simulation parameters
are listed in Table I.
TABLE I
SIMULATION PARAMETERS
Power 612 kW
Module CS6P-250P
Size (parallel×series) 153 ×16
PV Array
Vmpp, Impp 481.6 V,1270 A
Power, Frequency 500 kVA,60 Hz
Lf, rL100 µH,3 mΩ
Cdc 5000 µF
PI (vdc) Kp= 1, Ki= 250
Inverter
PI (id, iq) Kp= 0.7, Ki= 50
Power, Frequency 500 kVA,60 Hz
VLL (rms) 200 V /22.86 kV
Xleak, rloss (pu) 0.06, 0.0024
Lm, rm(pu) 200, 200
Transformer
Core type Three-limb
Sampling frequency 5 Hz
Kbase, Kacc , Cmin 0.00006, 0.3, 0.2
Vmin
step /Vmax
step 0.3 / 12 V
τ2, τ17.5 / 10 kW
MPRT
N, M, Ct,max,λr4, 3, 3, 0.5
The performance of three control methods are compared.
In method 1, Vstep = 0.3 V in steady-state and Vstep = 4 V
in transient operation. In method 2, Vstep = 0.3 V in steady-
state, and (8) proposed by [6] is used to compute Vstep during
transients. Method 3 is the proposed algorithm.
A. Overshoot Suppression
Figure 4 demonstrates the efficacy of employing an ac-
cumulator mechanism for calculating Vstep during large ir-
radiance variations. Because the initial irradiance increase
makes Ppv approach Pref,Vstep calculated by (8) (method
2) will continuously decrease even when the irradiance is still
rapidly increasing. This can cause a large overshoot. In method
3, γvalue accumulates when irradiance increases. Once an
overshoot is detected, the accumulator is activated and its value
is added to Vstep, providing a faster response to suppress the
overshoot, as shown in the first plot in Fig. 4. Because the
resetting rate of the accumulator is greater than zero, it can
be activated multiple times throughout one overshoot event, as
shown in the second plot of Fig. 4.
Fig. 4. Overshoot response comparison.
B. Smoothing DC-link Voltage Oscillations
Figure 5 displays the operation of each method during
periods of low irradiance. By adjusting Ktr in real-time,
method 3 can reduce the oscillations in vdc during low-
irradiance conditions when Pref > Pmpp, making the system
more stable and causing less wear-and-tear. In Table II, we use
the cumulative voltage oscillations (P|vdc(t)−vdc(t)|) in three
consecutive days to compare the performance of each method
at different irradiance levels. Note that lower vdc corresponds
to lower irradiance cases. It can be seen clearly that method
3 outperforms methods 1 and 2 in low irradiance cases.
TABLE II
DC-LI NK VOLTAGE OSCILLATIONS
Method Vdc <450 V 450 ≤Vdc <500 V Vdc ≥500 V
1 121k 69k 417k
2 149k 63k 171k
3 77k 50k 171k
C. Power Reference Tracking
The performance of the three methods when tracking Pref
for providing regulation signals is shown in Fig. 6. Method 1
presents large oscillations in transient responses because of its
Fig. 5. Comparison of DC-link voltage oscillations: (a) method 1, (b) method
2, (c) method 3.
fixed transient Vstep. Method 2 can mitigate the oscillations,
but it is susceptible to overshoots caused by large irradiance
changes. Method 3, equipped with adaptive Ktr adjusted
by accumulator value γ, can suppress overshoots during the
irradiance recovery process.
Fig. 6. Pref tracking comparison: (a) method 1, (b) method 2, (c) method 3.
Tracking errors with a 0.1s resolution are calculated for
three days of operation. When Pref ≤Pmpp, the error is
calculated as Perr =|Ppv −Pref |; when Pref > Pmpp, the
error is calculated as Perr =|Ppv −Pmpp|. The cumulative
percentage of tracking error, Esum is calculated by
Esum =R|Perr |
R|Ppv|(13)
Figure 7 displays violin plots overlaid by scatter plots of Perr
for the three methods. The dashed red lines indicate the region
of concentrated data points from scatter plot of method 3. It
can be seen that the error range of method 3 is much smaller
compared to methods 1 and 2. Furthermore, in this analysis,
the adaptive gains in methods 2 and 3 are kept the same for a
fair comparison. Yet, the real-time adjustment of the adaptive
gain introduced in method 3 permits its system to be designed
with a higher adaptive gain base value, which can be leveraged
to reduce the tracking error even further.
ACKNOWLEDGMENT
The authors thank PJ Rhem with ElectriCities, Paul Darden,
Steven Hamlett and Daniel Gillen with Wilson Energy for their
inputs, suggestions and technical support.
Fig. 7. Tracking error comparison.
IV. CONCLUSION
In this paper, we presented an algorithm for adjusting the dc-
link voltage of a PV system to achieve a better power reference
tracking capability. The algorithm added two new mechanisms
for adjusting the dc-link voltage. First, by comparing the PV
power output with its average output, it can identify when
the MPP is below the power reference. Then, the gain to
adjust voltage step size can be reduced in real-time so that
it is only large when needed. Second, an overshoot detection
mechanism is used to trigger an accumulator for suppressing
overshoots. Simulation results demonstrate a dc-link with
reduced oscillations in lower-irradiance operation conditions,
and an improved overshoot response when compared to exist-
ing P&O-based methods in the literature.
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