0-7803-8465-2/04/$20.00 ©2004 IEEE.

Abstract—A mathematical model of grid-connected

photovoltaic energy sources suitable for stability studies is presented. The power electronic conditioning unit is modelled from basic power transfer relations. Using this model, it is demonstrated that there exist two solutions for a given power output, one of which is unstable. By doing eigenvalue and eigenvector analysis, dynamic orbits are presented that help visualize any potential problem that may occur under disturbances. Simulations are carried out showing instances where the voltage at the photovoltaic panel collapses, in particular when operating close to the maximum power point.

Index Terms— Eigenvalues, photovoltaic cells, power system dynamic stability, solar energy.

I. INTRODUCTION

esidential photovoltaic installations are set to increase significantly in the United States, Germany, and Japan, among others. Photovoltaics are particularly attractive as

a renewable source for distributed urban power generation due to their relative small size and noiseless operation. To gain wide scale penetration, residential modules in the range 100 to 250W with dc-ac conversion are an attractive option [1]-[3]. However, there are various concerns associated with photovoltaic modules, such as the impact of their interconnection to the grid and the transparent operation of inverters. Some studies have been carried out on this, for example [4], but in general there is little information on the topic. It is of major concern for utilities in deregulated environments that these devices operate safely and meet specific standards such as those recommended by the IEEE and IEC [15], [16]. Fortunately, many of the readily available commercial inverters comply with these regulations [5], the disadvantage being their generic nature. In particular, photovoltaic panels suffer from nonlinear behaviour which a generic inverter does not take into account. Therefore, we present a study on the nonlinear characteristics of photovoltaic modules at constant power aimed at future development of robust control algorithms for power conditioning units. Results in dynamic stability of grid-connected photovoltaic modules are presented in an effort to

This work was supported in part by ENECSYS, Cambridge, United

Kingdom. C. Rodriguez (e-mail: cr310@cam.ac.uk) and G. Amaratunga (e-mail:

ga@eng.cam.ac.uk) are with the Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK.

enable low power distributed PV generators to become widely pervasive and reliable.

First, the power electronic conversion unit is introduced. The topology presented is very robust and is currently used in research experiments at the University of Cambridge. Additionally, it features electrical isolation, which is a requirement in grid-connected applications in Great Britain. This topology is then approximated with a mathematical model in order to apply stability analysis. Although the model completely eliminates high frequency components resulting from the switching devices, it characterizes accurately the dynamic behaviour of the entire system. We then continue with the stability analysis of a photovoltaic array under constant power. Finally, some simulation results related to the stability of solutions are presented.

II. POWER CONDITIONING UNIT

Fig.1 shows a typical topology for a grid-connected photovoltaic array with isolation. The first stage of conversion comprises of a dc-ac-dc converter operated at 50 kHz. In addition to isolation, the transformer steps up the voltage of the array (typically around 36 volts) to an acceptable level for grid interconnection (around 350 volts). The inductance and capacitance in the dc link filters any ripple resulting from the rectification stage. In fact, the inductance may be neglected if one takes into account the leakage inductance of the transformer. Finally the inverter at the output operates under sinusoidal pulse-width modulation (SPWM). Control of real and reactive power is achieved through the magnitude and angle of the modulating signal.

III. MATHEMATICAL MODEL OF POWER CONDITIONING UNIT

The power conditioning unit shown in Fig.1 comprises several switching elements under various control schemes. It would be excessively complex to characterize the system fully in order to assess its stability considering the nonlinear nature of switching functions. Instead, a mathematical approximation is realized based on the principles of energy conversion, and considering ideal switching devices only.

First we consider an ideal transformer whose well-known

relations are: N2v1=N1v2, N1i1=N2i2. Consequently the dc voltage across the PV capacitor is stepped-up to a value of Nvpv after the diode bridge. Conversely, the current across the dc link is N times smaller to the one leaving the PV capacitor.

Dynamic stability of grid-connected photovoltaic systems

C. Rodriguez, Student Member, IEEE, G. A. J. Amaratunga, Member, IEEE

R

The second approximation is to consider the output inverter as a true sinusoidal source. Although this is a simplification, the power transfer is not considerably affected. The magnitude of the voltage source is proportional to the voltage in the dc link and is related through the amplitude modulation index, K, of the PWM scheme. In order for this approximation to be complete and to satisfy instantaneous power balance, the current drawn from the dc link has also a sinusoidal component proportional to the output current.

Fig.1. Power conditioning unit for a grid-connected PV array.

The circuit equivalent with the simplifications aforementioned is shown in Fig.2. We have included the equivalent model of a photovoltaic array where IL is the light generated current, ION is the dark diode characteristics of the photocells, and Rsh and Rs are the shunt and series resistances of the array respectively. Kirchhoff’s current law (KCL) in the PV terminals yields,

exp 1

/0

pvL s pv s pv pv

pv s pv pv pvpv

sh

diI I v R i L

dt

v R i L di dti

R

α − + + −

+ +− − =

(1) where α=q/nskT, k=1.3807x10-23 JK-1is Boltzmann’s constant, q=1.6022x10-19 C is the electronic charge, T = 298K is the temperature, and ns is the number of series cells in the array. Since Lpv and 1/Rsh are very small, it is possible to

neglect the term ( )/ /pv sh pvL R di dt and we can solve for

dipv/dt,

( )

( )

/1ln 1

1

L pv pv s pv shpv

pv s

pv s pvpv

I i v R i Rdi

dt L I

v R iL

α

− − + = +

− + (2)

KCL at the PV capacitor yields,

1pvpv dc

pv

dvi Ni

dt C = − (3)

Similarly, KCL and Kirchhoff’s voltage law (KVL) at the dc link results in,

1dc

pv dc dc dcdc

diNv R i v

dt L = − − (4)

( )1cosdc

dc outdc

dvi Ki t

dt Cω δ= − + (5)

Finally, KVL at the output of the inverter yields,

( )cosdc out out gout

out

Kv t R i vdi

dt L

ω δ+ − −= (6)

where 0 1; / 2 / 2K π δ π≤ ≤ − ≤ ≤ are control variables.

Fig.2. Mathematical circuit equivalent of the power conditioning unit.

Equations (2)-(6) constitute a system of nonlinear, time-varying differential equations defining the dynamics of grid connected photovoltaic modules, and has the form,

( ), tx = f x& (7)

where nR∈x is the vector of states, t R+∈ are time values

greater than zero, and n nR R R+∈ × →f is a vector function.

Hence, an equilibrium of (7) must satisfy, ( ), 0t =f x (8)

Strictly speaking, some states will exhibit a periodic

solution. These, nevertheless, can be transported into a new time-invariant coordinate system, thus satisfying (8). The details of this procedure are beyond the scope of the present document and can be found in [14].

IV. EXISTENCE, UNIQUENESS, AND DYNAMIC STABILITY OF

SOLUTIONS

Grid-connected photovoltaic modules are characterized by the constant power injection into the mains. However, for a given power injection set point P* there exist two fixed points satisfying equation (8) as is explained below.

For the sake of simplicity we focus first on the dynamics of the PV array with constant power load and neglect any resistive component. The simplified circuit is shown in Fig. 3,

where the state vector is defined by, [ ]Ti v=x .

Kirchhoff’s laws at both nodes yield,

exp 1 0L s

diI I v L i

dtα

− + − − =

(9)

0dv p

i Cdt v

− − = (10)

Therefore the dynamics of the system are given by,

1 1ln 1

1

L

s

I idiv

dt L I

dv pi

dt C v

α −

= + −

= − (11)

Fig. 3. Simplified PV array with output elements.

Fig. 4 shows the I-V characteristic curve for a 170 Wp panel and three curves at different load levels.

Maximum power is achieved when the power curve, I=P/V is tangent to the PV array’s I-V characteristics. Several algorithms have been developed in order to operate at this peak power point [6]-[10].

However, with changing atmospheric conditions the I-V curve may shift thus creating the possibility of two solutions for a given set point P*. It is therefore essential to assess the stability of either point.

Fig. 4. I-V characteristics of a PV array and power loading.

The fixed points of the system are found by solving the

system of coupled nonlinear equations,

10 ln 1

0

L

s

I iv

I

pi

v

α −

= + −

= −

(12)

This requires a numerical solution, for example, through Newton-Raphson.

We now determine the stability of both solutions by letting the state vector, x, vary slightly by an interval ∆x, so that the change in the sensitivity is given by, ∆ = ∆x J x& (13) where J is the Jacobian matrix,

( )

2

1 1 1

1

L sL I i I L

p

C Cv

α − − − + =

J (14)

whose eigenvalues are found through ( )det 0Iλ − =J .

( ) ( )2

11 22 11 22 12 21

1,2

4

2

J J J J J Jλ

+ ± − += (15)

Let us assume that the radical is positive. Then the system will be stable if,

( ) ( )2

11 22 11 22 12 214 0J J J J J J+ + − + < (16)

Since the radical is assumed to be positive, then

( )11 22 0J J+ < for the system to be stable. Therefore,

( ) ( )2

11 22 11 22 12 214J J J J J J+ > − + (17)

which is equivalent to,

( ) ( )2 2

11 22 11 22 12 214J J J J J J+ > − + (18)

11 22 12 21J J J J> (19)

Substituting the values in (14), knowing beforehand that

0; Lv i I≥ ≤ , and following some manipulation results in,

[ ]2

L s

pv

I i Iα>

− + (20)

Fig. 5 shows the stability of the solutions for a load of p =

130W.

Fig. 5. Stability of solutions using criterion (20).

We finally discuss the positive-definiteness of the

discriminant in (15). Following an algebraic procedure it can be proved that the argument will be non-negative under the following conditions,

2

2

1

22

L s

p Li I I if v

CL C p

C L vα

> + − >

−

(21)

2

2

1

22

L s

p Li I I if v

CL C p

C L vα

< + − <

− (22)

In practice, the inductance is due only to the intrinsic

parameters of a cable and is in the order of µH, whereas the

capacitance is introduced in the circuit for the purpose of filtering and is in the order of mF. Therefore the ratio L/C is in the order of 0.001 and the roots of the characteristic polynomial of (14) are real. As a result, stability of solutions is assessed by relation (20) as depicted in Fig. 5.

A. Phase Portrait Analysis

Assuming we have real valued eigenvalues and eigenvectors, the dynamical orbits can be studied through the phase portrait. A stable node is that with two negative eigenvalues. In this case, the eigenvectors point into the node. A saddle node has one negative and one positive eigenvalues and the eigenvectors point into and out of the node respectively. If both eigenvalues are positive then it is an unstable node and the eigenvectors point out of it [11]. According to the Hartman-Grobman theorem [11], the direction of the eigenvectors for nonlinear systems is valid in a relatively close neighbourhood to the fixed points, while they may curve in an outer region. Fig. 6 shows the phase portrait in the state space of the PV array where a saddle node (considered unstable) and a stable node are present. From the figure we can observe that any state with a voltage higher to the one for the saddle node solution will eventually converge to the stable solution. In the opposite case, the voltage will eventually collapse. In fact, since the inductance is very low, the dynamics of the states are confined to the I-V characteristics of the PV array, that is, any initial condition will swerve towards the I-V curve in a vertical direction. To make the analysis complete, the saddle node will be a stable solution if and only if the initial state is along its stable eigenvector.

Fig. 6. Phase portrait of photovoltaic array solutions.

The stable basin or region of attraction for this case can be defined as follows,

( ) ( ) 2 *lim 0stable

tB R t

→∞= ∈ − =x x x x

(23) where *

stablex is the stable solution. Consequently,

( ) 2 *2 2 _ unstableB R x x= ∈ >x x (24)

where *2_ unstablex is the voltage of the saddle node.

It is important to observe that the regions of attraction

depicted in Fig. 6 will change when all the states and elements in the power conditioning unit are taken into account. It is not possible to carry out graphical analysis in this case, and we must rely upon numerical methods.

As an example consider the following operating conditions. The light generated current of a PV array with 72 cells in series, at 1000 W/m2 of irradiance is IL = 4.7 A, the saturation current of the diode is Is = 9e-11 A. The constant power load is drawing p = 130 W. The series inductance and shunt capacitance are L = 1µH, and C = 10 mF respectively. Through Newton-Raphson we find the two solutions are i1=4.6997; v1=27.6612 and i2=2.9691; v2=43.7837. The eigenvalues corresponding to the first solution and their respective eigenvectors are: e11=-6.54x109; e12=16.97, and v11=[-1 0]T; v12=[0 -1]T. The eigenvalues corresponding to the second solution are: e21=-1.06x106; e22=-86.83 and their respective eigenvectors: v21=[-1.0 0.0]T; v22=[0.6834 -0.73]T. These are shown in Fig. 6 in the phase plane with their respective directions.

B. Stability Analysis of Entire Converter

We now extend the analysis to include the whole converter. The Jacobian matrix of f in (7) is defined as,

1 1

1

1

n

n n

n

f f

x x

f f

x x

∂ ∂ ∂ ∂ ∂ = =

∂ ∂ ∂ ∂ ∂

fJ

x

L

M (25)

where the state vector x is denoted by,

T

pv pv dc dc outi v i v i = x (26) Consequently the non-zero partial derivatives of the Jacobian matrix are,

( )1

1

1 11

1 / /s

pv L s s sh pv pv sh sh

s

pv

Rdf

dx L I I R R i v R R

R

L

α

= − + + − + −

−

(27)

( )

1

2

1 1

1 / /

1

pv sh L s s sh pv pv sh

pv

df

dx L R I I R R i v R

L

α

= − + − + −

− (28)

2 2

1 3

1;

pv pv

f f N

x C x C

∂ ∂= = −∂ ∂

(29)

3 3 3

2 3 4

1; ;dc

dc dc dc

f f R fN

x L x L x L

∂ ∂ ∂= = − = −

∂ ∂ ∂ (30)

( )4 4

3 5

cos1;

dc dc

K tf f

x C x C

ω δ+∂ ∂= = −

∂ ∂ (31)

( )5 5

4 5

cos; out

out out

K tf f R

x L x L

ω δ+∂ ∂= = −

∂ ∂ (32)

Notice that the states directly affecting the Jacobian matrix are only the PV array voltage and current. The system parameters are the following: Rs = 0.0819 Ω, Rsh = 72kΩ, Lpv = 1µH, Cpv = 10mH, Rdc=0.1Ω, Ldc=1mH, Cdc=400µF, Rout=0.1Ω, Lout=10mH, transformer ratio N = 15. The system’s fixed points (state 5 is actually a periodic solution) and control parameters in steady state are listed in TABLE I.

TABLE I. FIXED POINTS OF PHOTOVOLTAIC CONVERTER

Since the Jacobian is time-varying, the eigenvalues can be

obtained for an entire period T = 20ms resulting in values with half the period. However, the first eigenvalue is constant with a real part equal to e11 = -1.146x106 for fixed point 1 and e21 = 6.38x109 for fixed point 2. Therefore, only fixed point 1 is stable.

V. SIMULATION EXAMPLES

All simulations were carried out with the software EMTP. The converter includes all stages with ideal switching devices. The set-points K and δ are determined through phasor analysis and the state values. In addition, a PID controller is included to track the set-point.

Case 1) The power injected to the grid is 130 W. At time instant t = 2s, the grid voltage magnitude decreases by 300 volts for 200 ms. Fig. 7 and Fig. 8 show the voltage and the current across the photovoltaic array.

Fig. 7. PV array voltage with a fault duration of 200 ms.

Fig. 8. PV array current with a fault duration of 200 ms.

The photovoltaic array is operating at its stable point when the disturbance occurs. The voltage drops sharply causing an inrush current to flow. When the fault is cleared after 200 ms the original state cannot be re-established, causing the voltage to droop and eventually collapse. Notice however, that since the inverter is still connected to the mains, the voltage at the dc-link is maintained at approximately 325V. Therefore the voltage at the PV terminals does not go to zero but rather stays at 22V due to the transformer turns ratio as shown in the Figure. At this point the photovoltaic array is injecting real power into the grid (100W only) but the reactive elements are absorbing a great amount of reactive power.

Case 2) The power injected to the grid is 130 W. At time instant t = 2s, the grid voltage magnitude decreases by 300 volts for 60 ms. Fig. 9 and Fig. 10 show the photovoltaic array voltage and current respectively during the transient.

Fig. 9. PV array voltage with a fault duration of 60 ms.

In this case, the fault is cleared faster and the system doesn’t lose its stability and returns to its original state.

Fig. 10. PV array current with a fault duration of 60 ms.

Extensive numerical analysis has been carried out showing that under many types of disturbances the system states return to its original values. It is important to note, however, that the system is more prone to losing stability when operating close to the maximum power point due to the change in the regions of attraction as shown in Fig. 11. The voltage and current are shown for a power output of p = 150 W and a fault duration of 60 ms. We observe a similar behaviour as in Fig. 7 and Fig. 8 but with a shorter fault duration time. In this instance, however, the system is not capable of re-establishing the original conditions.

Fig. 11. PV array voltage and current for power p =150W and a fault duration of

60 ms.

VI. CONCLUSIONS

We have presented a mathematical model suitable for stability analysis that includes the nonlinear behaviour of grid-connected photovoltaic modules. Utilizing the model we have shown there exist two solutions for a specific power injection into the grid mains, one of which is a saddle node and therefore unstable in practical terms. Moreover, eigenvalue analysis was introduced to define regions of attraction that determine the dynamics of the converter states.

Simulations including the entire power converter were done to support the mathematical analysis. Results showed that the system is more susceptible to instability under high loading levels, i.e. when operating close to its maximum power point. This is mainly due to the small margin available in order to withstand any disturbance.

Further research focuses on the determination of stability margins and operating limits, as well as the development of better control strategies to cope swiftly with disturbances in the grid.

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[2] R.H. Wills, F.E. Hall, S.J. Strong, J.H. Wohlgemuth, “The AC photovoltaic module,” Twenty Fifth IEEE Photovoltaic Specialist Conference, pp. 1231-1234, 1996.

[3] B. Kroposki, R. DeBlasio, “Technologies for the new millennium: photovoltaics as a distributed resource,” IEEE Power Engineering Society Summer Meeting, Vol. 3, pp. 1798-1801, 2000.

[4] Li Wang, Ying-Hao Lin. “Random fluctuations on dynamic stability of a grid-connected photovoltaic array,” IEEE Power Engineering Society Winter Meeting. Vol.3, pp.985-989, 2001.

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VIII. BIOGRAPHIES

Cuauhtemoc Rodriguez Scalise (S’96) was born in Mexico City in 1978. He obtained his B.S. with honours in Electronics and Communications Engineering from ITESM-Monterrey, Mexico, in 2000. He then joined the American-based company Symtx, where he worked as a design engineer. In 2003 he graduated from the M. Eng degree in electrical engineering from McGill University, Montreal, Canada. His research focused on power system stability and load flow analysis. He is currently a PhD candidate at the University of Cambridge, UK, where his main interest falls in the interconnection of photovoltaic sources to the electric

grid.

Gehan Amaratunga (M’ 90) . obtained his B.Sc from Cardiff University and

PhD from Cambridge, both in EE. He has held the 1966 Professorship in Engineering at the University of Cambridge since 1998. He currently heads the Electronics, Power and Energy Conversion Group, one of four major research groups within the Electrical Engineering Division of the Cambridge Engineering Faculty. He has worked for 20 years on integrated and discrete electronic devices for power conversion; and on the science and technology of

carbon based electronics for 15 years. His group was amongst the first to demonstrate integration of logic level electronics for signal processing and high voltage power transistors in a single IC ( chip). His current research on integrated power conversion circuits for connecting solar modules directly to the AC grid draws directly on his research in power ICs . This work on novel solar inverter technology is currently being commercialised through a start –up company, Enecsys Ltd. He also leads an active research effort in novel solar cell technologies. His group has developed a new approach to enhancing the performance of low cost polymer solar cells by combining semiconducting polymers with carbon nanotubes. He has previously held faculty positions at the Universities of Liverpool (Chair in Electrical Engineering), Cambridge, and Southampton. He also held the UK Royal Academy of Engineering Overseas Research Award at Stanford University. He has published over 300 journal and conference papers.