With an increasing number of photovoltaic (PV) inverters in the distribution system, their impact is no longer negligible, especially in the aspect of dynamic interaction. Accordingly, a comparison is done among PV inverters of different reactive power control modes, to determine their impact on the system voltage profile, power loss and small-signal stability. Generalized Nyquist Criteria (GNC) based on impedances in DQ frames is used for stability assessment, which is validated by time domain simulation results and also system eigenvalues calculation results from MATLAB. From these, guidelines are formulated to manage PV inverter reactive power control strategies. Reactive power control mode of volt-var Q=f(V) is preferred to other reactive power modes to avoid voltage profile problem and reduce power loss, but will induce small-signal instability and cause PV terminal voltage oscillations. There’s tradeoff between static influence and dynamic impact in choosing the local reactive power control strategies.

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Abstract
With an increasing number of photovoltaic (PV)
inverters in the distribution system, their impact is no
longer negligible, especially in the aspect of dynamic
interaction. Accordingly, a comparison is done among
PV inverters of different reactive power control modes,
to determine their impact on the system voltage profile,
power loss and small-signal stability. Generalized
Nyquist Criteria (GNC) based on impedances in DQ
frames is used for stability assessment, which is validated
by time domain simulation results and also system
eigenvalues calculation results from MATLAB. From
these, guidelines are formulated to manage PV inverter
reactive power control strategies. Reactive power
control mode of volt-var Q=f(V) is preferred to other
reactive power modes to avoid voltage profile problem
and reduce power loss, but will induce small-signal
instability and cause PV terminal voltage oscillations.
There’s tradeoff between static influence and dynamic
impact in choosing the local reactive power control
strategies.
1. Introduction
DUE to environment problems caused by fossil fuels,
installation of photovoltaic (PV) systems is increasing
rapidly worldwide. The impact on voltage profile is the
most commonly recognized problem caused by high PV
inverter penetration in distribution systems. The impact
on voltage profile is bigger as the PV capacity increases,
which has a limit in the amount of PV power to be
installed with respect to the overvoltage problem [1-7].
However, this scenario has been improved as according
to the revised IEEE 1547 standard, distribution resources
can actively participate in the voltage regulation [8].
Different local reactive power control strategies have
been designed and compared for PV generators to
regulate system voltage [9,10]. The comparison of
different Q control modes should not only consider the
regulation of voltage profile, but also take in effect of
the system power loss. Contrary to PV impact on voltage
profile, system power loss forms a U shape trajectory as
PV generator capacity increases [11]. The static analysis
in this paper demonstrates that system power loss is also
a U shape trajectory as reactive power injected by PV
varies.
IEEE 1547 newest version requires PV inverters to have
reactive power control, thus increasing inverter control
complexity. As negative incremental resistance caused
by constant power behavior of power converters may
bring stability problem, and stability of PV integration
to distributed system attracts more and more attention
[12-17]. Different from real-time simulation or
character root method, which require full models of
all components in distributed system, the Generalized
Nyquist stability Criterion (GNC) for stability analysis
of three-phase AC power system only uses measured
D-Q Frame impedances [18]. Compared to the positive
sequence impedance method used in [19], D-Q frame
impedance matrix method is more accurate for stability
assessment of system with PV inverter that has non-
symmetrical control in DQ frame including DC voltage
loop and phase locked loop (PLL) [20]. As there is
neither frequency control nor generator inertia dynamics
in the distribution system studied, the stability discussed
in this paper is mainly voltage stability. And only small-
signal stability can be investigated by GNC method
which is based on a certain steady state operation point
instead of stability during the transients including short-
term voltage stability or long-term voltage stability.
The goal of this paper is to analyze the small-signal
stability impact of utility scale PV farm composed by
multiple PV inverters under different reactive power
control modes to medium voltage distributed system.The
small-signal assessment approach is GNC based on grid
Small-signal tability mpact of tility PV
with eactive ower ontrol on the edium
oltage istributed stems
Y TANG*, R BURGOS
Center for Power Electronics Systems (CPES), Virginia Tech
KEYWORDS
Distribution system, photovoltaic (PV), power loss, reactive power control, small-signal stability, voltage profile.
*yetang@vt.edu

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119
as 5 MW. The circuit diagram of the distribution system
is shown by Fig.1, and impedance and load parameters
for the system are given in [21].
Fig. 1. Diagram of case systems. 56 bus system.
The commonly used photovoltaic generator local
reactive power control modes include mode 1 unity
power factor, mode 2 fixed reactive power, mode 3
fixed power factor, mode 4 reactive power as a function
of active power (Watt-var mode) and mode 5 reactive
power as a function of bus voltage (Volt-var mode) [8].
Mode 4 and mode 5 are illustrated by Fig.2.
Fig. 2. (a) Watt-var mode curve (b) Volt-var Q= f(V) mode droop curve
All five modes are applied on the PV farm connected to
bus 45. Fig.3 is the voltage profile of PV farm terminal
bus 45 when the PV active power increases from 0 to full
power in correspond to irradiance variation. In mode 2,
reactive power reference is set to be 1.5 MVar inductive
to compensate the voltage boost caused by active power
injection shown by mode 1 curve, but this setting has
the problem of under-voltage when irradiance is low,
showing that this mode is lack of flexibility for voltage
regulation as a local control strategy. In mode 3, power
factor is set to be 0.91 lagging. In mode 4, P1
= 3.5MW
and the slope is -1 in the curve in Fig.2(a). In mode 5,
and PV terminal D-Q frame impedances. Of all reactive
power local control modes in IEEE 1547, the volt-var
mode is the most preferred mode considering the static
impact on the system voltage profile and grid power loss.
Then a comparison is done among PV inverter terminal
D-Q frame impedances when PV is under different
reactive power control modes, which shows that volt-
var mode control have a significant impact on PV
terminal D-Q frame impedances and cause unstable PV
integration case in weak grids. When instability occurs,
PV terminal sees oscillation voltage magnitude. The
instability phenomena discovered by GNC application
is proved by both the time domain simulation of the grid
with the PV farm and the eigenvalue acquisition from the
state-space model of the PV connection. Finally, several
alternate solutions are provided to mitigate instability
issues of volt-var control schemes.
The paper is organized as follows: Section II introduces
the test-bed system of a radial 56 bus system with high
PV penetration. Different reactive power control modes
are listed and compared in the aspect of impact on
system voltage and power loss. Section III describes the
dynamic model of utility PV farm including switching
model and the average model of PV generator and
its controller, based on which PV generator terminal
impedance is derived in DQ frame. Section IV shows the
GNC application result using the DQ frame impedances
of the grid and the PV generators under different Q
control modes. Section V validates small-signal stability
assessment result of GNC by time domain waveform of
PV output currents and the eigenvalues of state space
model as well. Section VI is the conclusion about the
comparison of different Q control modes and impact of
PV penetration on distribution system voltage and small-
signal stability, and guidelines about PV allocation and
control for utility to avoid voltage profile problem and
ensure system stability.
2. Static Comparison of
Different Q Control
The test-bed distribution system in this paper is one of
Southern California Edison (SCE)’s distribution feeders
with very high penetration of Photovoltaics. It is a 12 kV
and very lightly loaded rural distribution feeder, with a
radial topology and a PV integration that can be as high

3. ฀ ฀ ฀ ฀ ฀ ฀ ฀
120
bus 45 in 56 bus system. System power loss is plotted
in Fig.4, Fig.4 reveals that for the typical load condition
of 56 bus system, system power loss forms a U shape
curve both for active power injection and reactive power
injection. The typical loading condition for the testbed
is that all the loads are 40% constant power and 60%
constant impedance load with a lagging power factor.And
as the voltage magnitudes of the buses change very little
in percentage, so the active currents and reactive currents
can be estimated to be proportional to the active power and
reactive power flowing through the lines. In addition, the
power loss of the lines are quadratic functions in terms of
the active currents and reactive currents, so system power
loss forms a U shape curve both for active power injection
and reactive power injection. The dip of the U shape is the
point that active currents and reactive currents on the lines
caused by load are starting to change flowing direction
because of PV injection. The fact that certain amount
of active power and reactive power should be injected
to achieve minimum power loss helps to understand the
comparison of impact on system power loss of different
reactive power control modes in Fig.5. Power loss under
unity power factor is set as base value of 100% in each
set of bar curves. Mode 2 and 3 cause higher power loss
than other options because PV inverter in mode 2 and 3
is consuming reactive power all the way. In contrast, PV
under mode 4 and mode 5 is only inductive when needed
(the active power injected exceeds P1
or PV terminal
voltage is over the threshold of V3
). In addition, when
irradiance is zero, mode 5 injects reactive power because
voltage bus 45 is below V2
=1.00p.u., which reduces
system power loss compared to other modes.
Fig. 4. System loss in terms of P and Q injected to PV connection bus
Fig. 3. Voltage of under PV injection bus
V1
=0.975p.u., V2
=1.00p.u., V3
=1.025p.u., V4
=1.05p.u. and
Qmax
=2.5MVar. Mode 3 to mode 5 are all able to regulate
the PV bus voltage within the limit ofASIC standard [22].
The difference of different Q control modes in voltage
profile of Fig.3 can be interpreted more by employing
the sensitivity analysis results in [23] that active power
injection increases bus voltage by a ratio of resistor sum
on the line and reactive power consumption reduces bus
voltagebyaratioofreactancesumontheline.Unitypower
factor curve is a rising straight line, and constant Q is also
a straight line with the same ratio but parallelly lowered
because of a constant voltage reduction by a constant Q
consumption. The watt-var mode align with unity power
factor mode before the active power injected reaches the
inflection point of P1
in the watt-var curve and after that
point the reactive power consumption starts to compensate
the voltage rising caused by active power, so that the bus
voltage can be controlled within upper limit. As for the
fixed PF mode, with a lagging PF, the increasing ratio
of voltage over active power is lower compared to unity
power factor as a combination of rising ratio of active
power and falling ratio of reactive power consumption.
And different from all these Q modes, volt-var mode
regulates the AC terminal voltages directly so that the
PV produces reactive power if AC voltage magnitude is
lower than V2
=1.00p.u. and consumes reactive power if
AC voltage magnitude is higher than V3
=1.025p.u. and
the voltage profile can be controlled within lower limit
and upper limit.
To analyze the impact on system power loss, different
amount of active power and reactive power is injected to

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121
Fig.6 Topology of utility PV
With switching ripple ignored, average model of the PV
generator in DQ axis is formed in Fig.7. The average
model of DC side is on the left side and average model
of AC side is on the right side, which is based on primary
side voltage of step up transformer.
Fig.7 Average model of PV generator
For PV controller in Fig.6, AC current regulator is a PI
controller with parameter of kpi
= 0.0012 and kii
=0.36,
making the current control bandwidth to be 220 Hz . DC
voltage regulator is also a PI controller with parameter
of kpv
=-3, kiv
=-30, making the DC voltage control loop
bandwidth to be 22.2 Hz. Vdc_ref
is the DC voltage reference
given by maximum power point tracking (MPPT) block.
The most widely used MPPT principle is perturb and
observe (P&O), which perturbs PV array voltage to find
the right direction of adjusting DC voltage to increase
PV array output power . For utility scale single stage PV
inverter, the frequency of MPPT block to change Vdc_ref
is
smallerthan DC voltage loop bandwidth.The time interval
to perturb PV array voltage is long as environmental
condition changes not very fast in PV farms. So in this
paper, Vdc_ref
is considered to be constant. Phase Lock
Loop (PLL) track grid voltage phases to do transformation
between dq axis and abc axis. PLL bandwidth is set to be
6 Hz with PI block parameter to be kp_pll
=0.1, ki_pll
=1
Fig. 5. System power loss Comparison of different Q modes
In sum, mode 5 volt-var mode is the best option
considering voltage regulation and system power loss.
Mode 1 unity power factor and mode 2 constant Q are
not able to regulate the voltage within range. Mode 3
costs more system power loss than mode 4 and mode 5.
While mode 5 is only inductive when needed and is able
to deal with both overvoltage and under voltage.
3. Dynamic Model of PV
The circuit configuration of a utility PV generator is
shown in Fig.6. The switching model is built in Simulink
with switching frequency set to be 3 kHz. The rated
capacity is 250 kW. The topology contains PV arrays,
a DC capacitor, a single stage DC/AC inverter, a three
phase LCL filter and a step up transformer. The PV
arrays have 29 PV panels in series in each string and
40 strings in parallel. The rated DC voltage is 870 V,
which is also the MPPT point of PV arrays under
standard weather condition (1000 W/m2 and 25°C).
The DC capacitor is chosen to be 8.2 mF to keep DV
voltage ripple under ±5%. Parameters of LCL filter with
resistor in series with cap are L1
= 0.32 mH, L2
=0.32 mH,
C=70.3 µF, Rc
are set to make sure that PV inverter output voltage and
current harmonics are within standard limit. For three
phase step up transformer, the rms values of line to line
voltages of primary and secondary side are 330 V and
12 kV separately. Dual loop control strategy is applied
in which inner current loops regulate output current in
DQ axis separately and outer DC voltage loop regulates
DC voltage. Reactive power can be regulated based on
reactive current loop. The controller may have an outer
reactive power loop according to different reactive
power control strategies.

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arrows. In conclusion, Fig.8 includes dynamics of power
stage, PLL and controllers. The loop in Fig.8 is solved to
get PV impedance so that
The four DQ frame impedances in Fig.9 are single PV
inverter impedances under mode (1)-mode (4). In mode
(2-4), PV inverter is inductive in steady state. PV farm
is composed of 12 inverter modules. For each module,
P = 250 kW, Q = –75 kVar (Generator convention,
Output as positive direction). In mode 4, P1
= 175 kW,
ratio of slope is –1 in Fig.2 (a). In all four curves, PV
impedances in DQ frame are diagonal dominant, which
means Zdd
and Zqq
are much bigger than Zdq
and Zqd
in
magnitude. And all four curves are almost the same at
high frequencies over 200 Hz. At lower frequency, blue
curve of unity power factor is different from other curves,
Zdq
and Zqd
are zero in magnitude as Q is zero, while the
other three curves are almost the same. So mode (2-4)
can be represented by mode 2 in dynamics. It can be
observed from Fig.9 that the PV terminal impedances are
diagonal dominant under mode 1 to mode 4, or in other
words the decoupling terms of non-diagonal elements of
Zdq
and Zqq
is much smaller than diagonal elements of Zdd
and Zqq
, so the original system stability can be decoupled
into D axis stability and Q axis stability separately. As
the frequency goes from 10 kHz to 1hz, the bode plots
of Zdd
and Zqq
both reveals the dynamics of LCL filter
above 1kHz and have a rising trend below current
Fig.8 Small signal model of PV inverter controller
Linearization of average model with the controllers
around the steady state yields the small signal model
shown in Fig.8, in which is grid voltage
perturbation in grid DQ axis, T
qd
s
ddd ]
~
,
~
[=
~
is duty ratio
change, is DC voltage change, T
qd
s
iii ]
~
,
~
[=
~
is PV output
current response in grid DQ axis. Power stage dynamics
are shown by red arrows and blocks in Fig.8. c
v~ and
c
i
~
are grid voltage perturbation and PV output current
response in converter controller DQ axis separately. Grid
DQ axis and controller DQ axis don’t align with each
other because of PLL effect, which is reflected by blue
arrows and the corresponding blocks. And inner current,
outer DC voltage, outer reactive power controller and
digital control delay are reflected by black boxes and
Fig.9 PV impedances under mode 1 - mode 4

6. ฀ ฀ ฀ ฀ ฀ ฀ ฀
123
becomes non-diagonal dominant, the interconnection
stability can not be observed directly from the bode
plots of the impedance matrices but need the application
of GNC and obtain of the two eigenvalues of the return
ratio matrix which is the combination of the impedances
of the grid and the PV farm.
4. GNC Application
The Nyquist Criterion can be used to assess DC system
stability by the number of encirclement of Nyquist
diagram of source impedance over load impedance
around (-1,0). GNC is an extended version of Nyquist
Criterion for multivariable systems [18]. Once the grid
impedance Zgrid
and PV impedance ZPV
are obtained, the
return ratio matrix can be calculated as
(1)
The generalized Nyquist stability criterion can be
formulated as [24] : “Let the multivariable system have
no open-loop unobservable or uncontrollable modes
whose corresponding characteristic frequencies lie in
the right-half plane. Then the system will be closed-
loop stable if and only if the net sum of anticlockwise
encirclements of the critical point (–1+j0) by the set of
bandwidth of 220Hz matching the dynamics of ideal
current source and are damped down below DC voltage
bandwidth around 20Hz. And the negative phase in Zqq
of frequency below 6Hz is caused by synchronization
function of PLL [18]. So the biggest stability concern for
mode 1 to mode 4 are high resistive on the lines from PV
farm to the grid, which is not observed in this paper as
the lines impedances are not high enough to trigger this
kind instability.
Fig.10 is the impedance of PV inverter under mode 1,
mode 2 (as it can represent mode 2, mode 3, and mode 4)
and mode 5. In mode 5, the steady state operation point
is the same as other modes, P = 250 kW, Q = –75 kVar
(Generator convention, Output as positive direction).
V1
= 0.0975 p.u., V2
= 1.0 p.u., V3
= 1.025 p.u., V4
=1.05
p.u., Qmax
= 112.5 kVar in Fig.2(b). Compared to mode 1
andmode2,oneobviousdifferenceofmode5isthatsigns
of Zdd
and Zqq
in zero frequency are inverted in mode 5.
The magnitude of Zdd
is decreased, while the magnitude
of Zqd
is increased obviously. This will cause unstable
PV integration case, which can be proved in section IV.
The volt-var control mode builds a huge link between d
axis and q axis by generating iq from vd, that’s why the
decoupling term Zqd becomes much higher compared
to other modes. As the PV terminal impedance matrix
Fig.10 PV impedances under mode 1, mode2 and mode 5

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124
stability problem under the control mode of volt-war
without reducing the number of PV inverters in parallel.
The solutions include reducing current loop bandwidth
(reducing current controller PI parameters), reducing
reactive power loop bandwidth (reducing current
controller PI parameters) and reducing controller delay
by increasing switching frequency from 3kHz to 6kHz.
Fig. 12 shows the Nyquist diagrams of eigenvalues if
one of the solutions is applied. Compared to the blue
curves of original volt-var droop control, 1
(s) doesn’t
encircles (-1,0) in red, yellow and purple curves, which
shows that the system is stable after the modifications.
Fig.12 GNC application after alternative solutions
5. Time Domain Waveform and
Eigenvalues to Validate GNC
Results
The results from GNC can be validated by time domain
simulation results in Fig.13, which is output current of 3
MW PV farm (12 modules) under volt-var mode in DQ
frame. The oscillation frequency is 256 Hz. Oscillation
magnitude is limited by Q magnitude – Qmax
in droop
mode curve in Fig.2(b) before 0.5 s, which is removed
at 0.5s and the oscillation of current starts to increase,
proving system instability.
The most widely used tool of eigenvalues for small-
signal stability assessment is also applied on the system
with PV generator under volt-var mode. Fig.14 is the
plot of five eigenvalues which are closest to the right
half plan or in the right half plane on the complex plane.
The system has two eigenvalues in the right half plane,
which is a confirmation of the instability discovery from
GNC application from last section.
characteristic loci of L(s) is equal to the total number
of right-half plane poles of Zgrid
and YPV
”.The two
eigenvalues of L are 1
(s) and 2
(s). In all the cases of
this paper, Zgrid
and YPV
don’t have RHP pole, so if the
Nyquist diagrams of 1
(s) and 2
(s) don’t encircle (-1,0),
then the connection of PV inverters to the system is
stable. On the other hand, system is not stable if any of
the eigenvalue loci encircles (-1,0).
PV terminal impedances or admittances are derived in
last section. The grid is the 12 kV distribution grid in
Fig.1, in which PV is connected to bus 45. The PV farm
capacity is 3 MW, which means that 12 PV inverters
are operated in parallel. As this system is very lightly
loaded, grid impedance is determined by the cumulative
resistance R =3.35 W and inductance L=11.7mH of the
direct path from substation bus 1 to bus 45.
(2)
The results of GNC application are shown in Fig.11,
whichincludesNyquistdiagramsofeigenvaluesofreturn
ratio matrix L when PV is at different Q control modes
of mode 1, mode 2 and mode 5. In mode 2 and mode
5, PV generator operation point is P = 3 MW, Q = - 0.9
Mvar. Dynamics of mode 3-4 are represented by mode
2. It can be observed that if the multiple inverters are all
operating at mode 1 or mode 2, either of the eigenvalues
encircles (-1,0), meaning that the connection is stable.
But if PV is under mode 5, 1
(s) encircles (-1,0), and the
connection of PV to the grid becomes unstable.
Fig.11 Chatacteristid loci of PV connection to the grid
Some alternative solutions of modification on the
PV inverter controllers have been found to solve the

8. ฀ ฀ ฀ ฀ ฀ ฀ ฀
125
modes, based on which GNC is used to assess the grid-
PV connection stability. The volt-var control mode
changes PV terminal impedance signs and magnitudes
significantly and may cause unstable connection to the
grid. The stability assessment is proved by time domain
simulationandalsoeigenvaluesacquisitionfromthestate
space model of the whole system with PV generators.
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Fig.14 System eigenvalues in complex plane
5. Conclusion
A known broad statement is that active power injected
by PV inverters increases the system voltage. In what
regards the reactive power compensation capability of
PV inverters, this paper showed that the reactive power
control mode Volt-var droop mode of Q = f (V) is
preferred over other reactive power control modes after
the static analysis of voltage regulation effect and the
impact on grid power loss.
The terminal impedances in DQ frame are derived of
utility-scale PV farm based on small signal model of PV
inverters. A comparison is done among impedances of
PV inverters under 5 different reactive power control

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The authors wish to gratefully acknowledge the support
from Dominion Energy, USA.
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