1
Power System Small-Signal Oscillation Stability
as Affected by Large-scale PV Penetration
W. Du, H. F. Wang, and R. Dunn
aspect, one of the most important issues is the impact of large
Abstract — This paper investigates the impact of a large penetration of PV generation on power system stability, which
photovoltaic (PV) penetration on power system small signal must be examined carefully [1]. Dynamic of PV generation is
oscillation stability. A comprehensive model of a single-machine considerably different with that of conventional generation
infinite-bus power system integrated with a PV power generation involving rotating machines. Hence a thorough study on the
power plant is established. Numerical computation of damping case of interaction of PV generation with conventional power
torque contribution from the PV power plant is carried out,
generation and transmission systems is an urgent task to be
which is confirmed by the results of calculation of system
oscillation model and non-linear simulation. Those results
pursued. [8] and [9] have reported the results of case studies
indicate that power system oscillation stability can be affected about the effect of PV generation on power system dynamics,
either positively or negatively. There exists an operational limit of which have provided good foundation for further
the PV power plant as far as system oscillation stability is investigations.
concerned. Beyond the operational limit, the PV generation
supplies negative damping torque, thus damaging system The objective of this paper is to examine the effect of
oscillation stability. Hence for the safe penetration of PV operation of PV generation jointly with that of conventional
generation into power systems, the operational limit of oscillation generator and transmission system on power system small
stability of the PV power plant must be considered.
signal oscillation stability. The paper beings with the
Index Terms — PV generation, power system oscillations,
establishment of a comprehensive model of a single-machine
power system small signal oscillation stability, DTA. infinite-bus power system integrated with a PV power
generation plant. Then an example single-machine infinite-bus
I. INTRODUCTION power system integrated with a PV power plant is presented.
Results of numerical computation and non-linear simulation at
T he fast development of photovoltaic (PV) technology in
recent two decades strongly indicates that in medium to
long term, PV generation will become one of most
different operating conditions are given when the load
conditions of conventional power generation and levels of
mixture of conventional and FC generation change. Results
attractive renewable energy sources in large-scale from the example power system indicate that power system
applications. Evidence of such indication is that the growth of oscillation stability can be affected either positively or
demand on solar energy has consistently been by over 20% negatively. There exists an operational limit of the PV power
per annum due to the decreasing cost and price. The European plant as far as system oscillation stability is concerned.
Union is well on the track to fulfilling its own target of 3GW Beyond the operational limit, the PV generation supplies
PV generation by 2010; While capacity of PV generation in negative damping torque, thus damaging system oscillation
the UK is forecasted to be 30GW by 2050 [1][2]. Grid stability. Hence for the safe penetration of PV generation into
connection is the operational practice of large-scale PV power systems, the operational limit of oscillation stability of
generation for its best utilization. This has attracted the most the PV power plant must be considered.
R& D interests in the area [3]. So far, majority of work on the
grid connection of PV generation has focused on the R&D II. MATHEMATICAL MODEL
issues of efficient and effective PV generation itself, such as
generation control and design of interface power electronics
[4]-[7]. Figure 1 shows the configuration of a single-machine infinite-
bus power system, where a large-scale PV power generation
Large scale grid-connected PV generation will significantly plant is connected at busbar s. Typical voltage-current
affect power transmission and generation systems. In this characteristic of PV generation is non-linear, given by the
following expression [1].
The authors would like to acknowledge the support of the EPSRC UK-
China joint research consortium (EP/F061242/1), Supergen 1 – flexible ⎛ N p I sc I r ⎞
− I pv
network consortium, Supergen 3 – energy storage consortium, UK, and the N s nkT ⎜ 100 ⎟
Fund of Best Post-Graduate Students of Southeast University, China. V pv = ln ⎜ + 1⎟ (1)
W. Du is with the Southeast University, Nanjing China, and working at q ⎜ N p I0 ⎟
⎜ ⎟
the Queen’s University of Belfast, Belfast, UK at present (e-mail: ⎝ ⎠
ddwenjuan@googlemail.com).
H F Wang is with the Queen’s University of Belfast, Belfast, UK and R.
Dunn is with the University of Bath, Bath, UK.
2
that the characteristic of PV generation of Eq.(1) is given such
that the curve Ppv − max can be obtained. Therefore, the
function of MPPT can be modeled as,
d c = dc 0 + K pv ( s )( Ppv − Ppv max ) (2)
no matter what scheme it is implemented in practice, where
K pv ( s ) is the transfer function of the MPPT controller.
Dynamic of the DC/DC inverter can be modeled simply to be
1
Ipv = [V pv − (1 − d c )Vdc ] (3)
Ldc
DC/AC inverter usually employs the PWM (pulse width
modulation) control to regulate the exchange of active and
Figure 1 A power system integrated with a PV generation plant
reactive power between the PV generation and the rest of the
power system. This can be achieved by controlling the
where T is the junction temperature, I r the irradiance, N s modulation ratio m and phase φ of the PWM algorithm
and N p number of cells in series and parallel respectively, n through the following AC and DC voltage control function
ideality factor, k Boltzmann’s constant, q charge of the respectively
electron, I sc short-circuit current and I 0 saturation current. m = m0 + K ac ( s )(Vs − Vsref ) (4)
Ipv
φ = φ0 + K dc ( s )(Vdc − Vdcref ) (5)
where K ac ( s ) and K dc ( s ) is the transfer function of the ac
Ppv-max and dc voltage controller respectively.
Ir
Ppv
Vpv
Figure 2 Non-linear V-I characteristic of PV generation
In Figure 1, the PV power plant is connected to the power
system through a two-stage topology of inverter system, which
is usually employed by gird-connected PV generation [2][3].
In addition to the function of enhancing output DC voltage,
DC/DC inverter is mainly for implementing the maximum
power point tracking (MPPT) control. As illustrated by Figure Figure 3 Phasor diagram of power system of Figure 1
2, change of PV output current and voltage change are
affected by variations of various factors, such as the AC voltage at terminal of the DC/AC inverter, VC , can be
irradiance, I r . In order to make full use of PV cells (due to expressed in the d-q coordinate of the generator (Figure 3) to
their limited life span and high initial cost), MPPT is be [13]
employed in order to extract as much power from them as Vc = mkVdc (cosψ + j sinψ ) = mkVdc ∠ψ (6)
possible by controlling the cycle duty of the DC/DC inverter, where k is the converter ratio dependent of inverter structure,
dc . Design and implementation of MPPT control has been Vdc is the DC voltage across the capacitor Cdc . Active power
one of the most important issues in developing PV generation. received by the DC/AC inverter from the power system is
Many methods have been proposed, such as the curve-fitting Vdc I dc1 = isd vcd + isq vcq = isd mkVdc cosψ + isq mkVdc sinψ
technique, perturb-and observe method, etc., considering the
need of practical operation of PV generation [4]-[7]. Basically, where subscript d and q denotes the d and q component of the
the objective of MPPT is to ensure the operation of PV variable respectively. Hence
generation to be on the curve, Ppv − max , by controlling dc . I dc1 = isd mk cosψ + isq mk sinψ (7)
Because
Because in establishing the mathematical model of PV
generation integrated with the power system of Figure 1, the Ppv = I pvV pv = I dc 2Vdc = (1 − dc )Vdc I pv (8)
purpose is to investigate the impact of PV generation on the Hence I dc 2 = (1 − dc ) I pv such that dynamic of the DC/AC
power system by theoretical analysis, numerical computation
inverter is expressed to be
and simulation (not experiment). Hence it must be assumed
3
1 PV power plant of Figure 1 are given in Appendix I. The
Vdc = ( I dc1 + I dc 2 ) effective technique of damping torque analysis (DTA) [10]-
Cdc
(9) [12] was used to study the oscillation stability of the power
1 system as affected by the PV power plant. Appendix I presents
= [isd mk cosψ + isq mk sinψ + (1 − d c ) I pv ]
Cdc the details of deriving the linearized model of the power
system integrated with the PV power plant of Figure 1, given
General mathematical model of the synchronous generator can by (A-2) and (A-8)-(A-11). Computational results of damping
be written as X g = F ( X g , I ts ) , where X g is the state torque contribution from the PV power plant, confirmed by
the computational results of system oscillation mode, from the
variable vector associated with generator dynamics and I ts (or complete linearized model of Figure 4 and 5 are given in
itsd and itsq ) is the interface variable between the generator Table 1 and 2. In Table 1, the total active power supplied by
the generator and PV power plant to the load at the infinite
and rest of the system. In this paper, the following generator busbar is fixed at 1.0 p.u., but level of mixture of conventional
model is used, which is sufficient for the study of power and PV power generation varies. In Table 2, the PV power
system small signal oscillation stability [10]. generation is fixed to be 0.3 p.u., but active power supply from
δ = ωo (ω − 1) the generator changes.
1
ω = [Pm − Pt − D(ω − 1)] Table 1 Computational results of the example power system when total active
M power received at the infinite busbar is fixed at 1.0 p.u. ( Pt 0 + Ppv 0 = 1.0 p.u. )
(10)
1
E q '= ( − Eq + E fd ) Pt 0 Ppv 0 ψ0 ΔTdt ΔTddt ΔTdt −ac Oscillation
Td 0 ' (degree)
mode
E fd '=TE ( s )(Vtref − Vt )
1.0 0.0 54.3 1.80 1.43 0.0006 -0.57 ± j 3.86
where for simplicity of discussion, transfer function of the 0.9 0.1 58.6 1.12 0.79 0.0015 -0.43 ± j 3.97
AVR (automatic voltage regulator) is taken to be a first-order 0.8 0.2 62.9 0.49 0.21 0.0021 -0.31 ± j 4.09
KA 0.7 0.3 67.4 -0.09 -0.32 0.0023 -0.22 ± j 4.49
system TE ( s ) = in this paper and
1 + sTA 0.6 0.4 72.1 -0.62 -0.81 0.0021 -0.15 ± j 4.29
Pt = Eq ' itsq + ( xq − xd ')itsd itsq 0.5 0.5 76.8 -1.11 -1.27 0.0017 -0.09 ± j 4.39
0.4 0.6 81.5 -1.56 -1.68 0.0010 -0.04 ± j 4.47
Eq = Eq '− ( xd − xd ')itsd (11) 0.3 0.7 86.4 -1.97 -2.06 -0.0000 0.01 ± j 4.55
0.2 0.8 91.3 -2.36 -2.42 -0.0012 0.04 ± j 4.62
Vt = vtd 2 + vtq 2 = ( xq itsq )2 + ( Eq '− xd ' itsd ) 2
0.1 0.9 96.1 -2.72 -2.75 -0.0003 0.07 ± j 4.69
From Figure 1 it can have Table 2 Computational results of the example power system when the PV
Vt = jxts I ts + Vs power generation is fixed at 0.3 p.u. ( Ppv 0 = 0.3 p.u. )
Vs = jxs I s + Vc (12) Pt 0 ψ0 ΔTdt ΔTddt ΔTdt −ac Oscillation mode
(degree)
Vs − Vb = jxsb ( I ts − I s )
Those equations above give 0.1 89.6 -2.04 -2.07 -0.0000 -0.01 ± j 8.13
jxs I s + Vc − Vb = jxsb ( Its − I s ) 0.2 85.8 -1.75 -1.81 0.0001 -0.06 ± j 8.09
(13) 0.3 81.9 -1.45 -1.55 0.0003 -0.12 ± j 8.05
Vt = jxts I ts + jxsb ( I ts − I s ) + Vb 0.4 78.2 -1.14 -1.27 0.0006 -0.17 ± j 7.99
In d-q coordinate of the generator, as shown by Figure 3, from 0.5 74. 6 -0.81 -0.99 0.0010 -0.23 ± j 7.93
Eq.(13) it can be obtained that 0.6 70.9 -0.46 -0.67 0.0016 -0.29 ± j 7.85
⎡ xsb − xs − xsb ⎤ ⎡itsq ⎤ ⎡ −Vc cosψ + Vb sin δ ⎤ 0.7 67.5 -0.09 -0.33 0.0023 -0.36 ± j 7.76
⎢x + x + x =
⎣ q ts sb − xsb ⎥⎦ ⎢⎣ isq ⎥⎦ ⎢⎣ Vb sin δ ⎥
⎦ 0.8 64.1 0.33 0.05 0.0031 -0.44 ± j 7.64
(14) 0.9 60.8 0.80 0.49 0.0042 -0.52 ± j 7.51
⎡ x sb − x s − x sb ⎤ ⎡ tsd ⎤
I ⎡Vc sin ψ − Vb cos δ ⎤
=⎢ ⎥ 1.0 57.6 1.34 0.97 0.0056 -0.61 ± j 7.35
⎢ − xsb ⎦⎥ ⎢⎣ I sd ⎥⎦ ⎣ Eq '− Vb cos δ ⎦
⎣ xd '+ xts + xsb
From Table 1 and 2 it can be seen that
The complete mathematical model of the power system of (1) Damping torque supplied by the PV power plant changes
Figure 1 thus is established, consisted of the model of PV at different levels of mixture of conventional and PV
generation and control of Eq.(1)-(5) and (9), generator of power generation, which can be positive or negative. With
Eq.(10) and (11), and integration of the generator and the PV the fixed load at the infinite busbar, the more power
power plant with the rest of the power system of Eq.(14). contributed by the PV generation, the worse the impact of
PV generation on system oscillation stability, because the
III. AN EXAMPLE POWER SYSTEM more negative damping torque is provided by the PV
Parameters and initial operating conditions of an example power plant.
single-machine infinite-bus power system integrated with a
4
(2) Damping torque supplied by the PV power plant also the DC/DC and DC/AC inverter of the PV power plant
changes from positive to negative at different levels of respectively.
conventional power generation even though the PV power
generation is fixed. In this case, the lighter the load of 1.3
conventional power generation (power supplied by the
generator), the worse the impact of the PV generation on 1.2
power system oscillation stability. 1.1
(3) Damping torque contribution from the PV power plant 1
changes sign between ψ 0 = 62.90 and ψ 0 = 67.40 (Table 0.9
1) or ψ 0 = 64.10 and ψ 0 = 67.50 (Table 2). Hence there
Pt
0.8
must exist a critical operating condition ψ critical when the 0.7
damping torque contribution from the PV power plant 0.6
changes sign. The operation of the PV power plant should 0.5
avoid ψ 0 > ψ critical when it provides negative damping to
0.4
power system oscillation. 0 1 2 3 4 5
t
6 7 8 9 10
5 (a) Pt 0 = 0.9 p.u. and Ppv 0 = 0.1 p.u.
0.35
0.8
0.3
0.6
0.25
0.4
0.2
0.2
0.15
Pt 0
0.1
-0.2
0.05
0
-0.4
0 0.1 0.2 0.3 0.4 0.5 0. 0.7 0.8 0.9
6
-0.6
0 1 2 3 4 5 6 7 8 9 10
4(a) Set of simulated V-I characteristic of PV generation and tracking of the t
maximum power point 5 (b) Pt 0 = 0.1 p.u. and Ppv 0 = 0.9 p.u.
0.135 Figure 5 Simulation at two different levels of mixture of conventional and PV
power generation when the total power received by the load at the infinite
0.13 busbar was fixed at 1.0 p.u.
0.125
Figure 5 gives the results of simulation (the fault applied was
0.12
the three-phase to-earth short circuit on the transmission line
occurred at 1 second of simulation for 100ms) at two different
0.115
levels of mixture of conventional and PV power generation
0.11
when the total power received by the load at the infinite
busbar was fixed at 1.0 p.u. Figure 6 is the results of
0.105 simulation at two different levels of conventional generation
and the PV power generation was fixed at 0.3 p.u. Those
results of non-linear simulation confirm both the analysis in
0.1
0 1 2 3 4 5 6 7 8 9 10
t
the above section and computational results presented in Table
6(b) Power output from the DC/DC and DC/AC inverter of the PV power 1 and 2.
plant (1) When the total power received by the load is fixed, the
Figure 6 Simulation results when the irradiance changed
heavier load condition the SOFC power plant operates at,
the more it will damage the damping of power system
Non-linear simulation using the non-linear model given in oscillation;
section II of the paper was carried out to verify the (2) When the SOFC power generation is fixed, the lighter
mathematical model and confirm the results of computation the load condition of the conventional power plant is, the
from system linearized model. Figure 4 shows the simulation more negative damping torque is provided by the SOFC
results when the irradiance changed at 1.0 second of the power plant.
simulation. The change was a step increase of 1% every 0.1
second and lasted for 2 seconds. Figure 4(a) shows the set of
V-I curves of PV generation used and tracking of the
maximum power point. Figure 4(b) is the power output from
5
1.3
V. REFERENCES
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1.2
suitable for stability analysis”, IEEE Trans. on Energy Conversion, Vol.19,
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[2] J. M. Carrasco, L. G. Franquelo, etc., “Power electronic systems for the
1
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Industrial Electronics, Vol. 53, No.4, 2006, pp1002-1016
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0.8 grid-connected inverters for photovoltaic modules”, IEEE Trans. on Industry
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Applications, Vol. 41, No.5, 2005, pp1292-1306
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0.1 penetration of photovoltaic generation”, Proc. of IEEE PES General Meeting,
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0.05 [10] Y. N. Yu, Electric Power System Dynamics, Academic Press Inc., 1983
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t III”, IEEE Trans. Power Appar. Syst., Vol. 100, No. 6, 1981, pp3017-3046
[13] “Modeling of power electronics equipment (FACTS) in load flow and
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Figure 6 Simulation when the generator operated at two loading conditions [14] W.G.Heffron and R.A.Phillips, “Effect of a modem amplidyne voltage
and the PV power generation was fixed at 0.3 p.u. regulator on under excited operation of large turbine generator”, AIEE Trans.
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IV. CONCLUSIONS STATCOM and applications”, IEE Proc. Part C, Vol.146, No.5, 1999, pp521-
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Conventional technique of damping torque analysis is used for
the investigation of a single-machine infinite-bus power APPENDIX I
system integrated with a PV power plant. Results of damping Transmission line: xts = 0.2 p.u., xsb = 0.2 p.u., xs = 0.2 p.u. ;
torque computation, confirmed by calculation of system Generator:
oscillation mode and non-linear simulation, of an example xd = 1.3 p.u., xq = 0.47 p.u., x 'd = 0.3 p.u., M = 3s., D = 2 p.u., T 'd 0 = 5s. ;
power system integrated with a PV power plant are presented
AVR: TA = 0.1s. K A = 10 p.u. ;
in the paper. They demonstrate that the effect of PV
penetration on system oscillation stability varies with changes Initial load condition: Vt 0 = 1.0 p.u., Vs 0 = 1.0 p.u., Vb 0 = 1.0 p.u.
of system operating conditions, because the damping torque PV power plant:
contribution from the PV generation can be positive or
negative. The critical operating condition of the PV power APPENDIX II
For the simplicity of analysis, it is assumed that control functions of PV
plant is when its damping torque contribution changes sign, generation are implemented by proportional control law. Hence linearization
which in fact indicates a stability limit of PV operation as far of Eq.(2), (4) and (5) is
as power system oscillation stability is concerned. The effect Δdc = K pv Δ Ppv , Δm = K ac ΔVs , Δφ = K dc ΔVdc
of PV generation on system oscillation stability is negative (A-1)
From Figure 3 it can be seen that Δψ = − Δφ . Hence it can have
beyond the critical operating condition. When the total load is
fixed, a higher PV penetration should be avoided because this Δψ = K dc ΔVdc (A-2)
increases the chance of its damaging system oscillation
Linearization of Eq.(1) and (8) is
stability. When the PV penetration is fixed, its operation can
ΔV pv = a1Δ I pv ,
also approach to the critical operating condition with the (A-3)
decrease of conventional power generation. Δ Ppv = V pv 0 Δ I pv + I pv0ΔV pv = a2Δ I pv
That gives
Δdc = K pv a2 Δ I pv (A-4)
6
Because Vs = jxs I s + Vc ,Vs = vsd 2 + vsq 2 , it can have
vsd = − xs isq + kmVdc cosψ , vsq = xs isd + kmVdc sinψ
Hence
ΔVs = b1Δδ + b2 ΔEq′ + b3ΔVdc + b4 Δm + b5 Δψ (A-5)
From Eq.(A-1), (A-2) and (A-5) it can have
K ac
Δm = ( B6 Δδ + B7 Δ Eq '+ B8 ΔVdc ) (A-6)
1 − b4 K ac
By using Eq.(A-2) and (A-6), linearization of Eq.(14) can be obtained to be
⎡ Δδ ⎤
⎡ Δitsd ⎤ ⎢ ΔE '⎥ c
⎢ Δi ⎥ ⎢ q ⎥ ⎡ 11 c12 c13 ⎤ ⎡ Δδ ⎤
⎢ tsq ⎥ = C ⎢ ΔV ⎥ = ⎢c c22 c23 ⎥ ⎢ ΔEq '⎥ (A-7)
⎢ Δisd ⎥ ⎢
dc
⎥ ⎢ 21 ⎥⎢ ⎥
⎢ ⎥ ⎢ Δψ ⎥ ⎢⎣c31 c32 c33 ⎥⎦ ⎢⎣ ΔVdc ⎥⎦
⎢⎣ Δisq ⎥⎦ ⎢ Δm ⎥
⎣ ⎦
Linearization of Eq.(3) and (9) is
Δ Ipv = C1Δ I pv + C2 ΔVdc (A-8)
ΔVdc = C3Δδ + C4 ΔEq′ + C5 ΔVdc + C6 ΔI pv + K dm Δm + K dψ Δψ (A-9)
Linearization of Eq.(10) is
Δδ = ωo Δω
1
Δω = ( − Δ Pt − DΔω )
M
1
Δ E q '= ( − Δ Eq + Δ E fd ' )
Td 0 '
Δ E fd '=TE ( s)( − ΔVt )
(A-10)
By using Eq.(A-2), linearization of Eq.(11) can be obtained to be
ΔPt = K1Δδ + K 2 ΔEq′ + ( K pdc + K pψ K vdc )ΔVdc + K pm Δm
ΔEq = K 4 Δδ + K 3ΔEq′ + ( K qdc + K qψ K vdc )ΔVdc + K qm Δm (A-11)
ΔVt = K 5 Δδ + K 6 ΔEq′ + ( K vdc + K vψ K vdc )ΔVdc + K vm Δm
Ldc = 1.5 p.u., Cdc = 1.0 p.u., Vdc 0 = 1.0 p.u., K dc = 0.3, K ac = 0.1, K pv = 5
ACKNOWLEDGEMENTS
The authors would like to acknowledge the support of the
EPSRC UK-China joint research consortium (EP/F061242/1),
Supergen 1 – flexible network consortium, Supergen 3 –
energy storage consortium, UK, and the Fund of Best Post-
Graduate Students of Southeast University, China.
