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This is an author produced version of a paper published in :

IEEE Transactions on Energy Conversion

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http://cronfa.swan.ac.uk/Record/cronfa29262

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Paper:

Egwebe, A., Fazeli, M., Igic, P. & Holland, P. (2016). Implementation and Stability Study of Dynamic Droop in

Islanded Microgrids. IEEE Transactions on Energy Conversion, 31(3), 821-832.

http://dx.doi.org/10.1109/TEC.2016.2540922

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1

Abstract—This paper presents a dynamic droop load

sharing scheme based on the available generation capacity

of the distributed generation (DG) units. Since

conventional droop schemes share loads proportional to

units’ ratings, they suffer from the inability to maintain an

efficient operating point when their input renewable power

varies; without imposing their new operating point on

other connected DGs in the microgrid. This problem is

mainly due to the insensitivity of the droop scheme to the

varying nature of the renewable resources used including

wind and solar photovoltaic. A control method is proposed

for PV systems; however, it is applicable for all types of

droop-controlled renewable DG. A stability analysis of the

proposed scheme on DG units is also presented to identify

theoretical and practical limits. The proposed scheme

identifies the DC operating zone of the inverter-based

source as irradiance level changes and conditions the

droop parameters appropriately for an efficient load

sharing based on available generation while the rating of

each unit is also taken into account. The proposed scheme

provides energy saving; since energy demand from a local

auxiliary generator is reduced. The proposed method is

validated using MATLAB/SIMULINK simulations.

Index Terms—Droop control, Distributed Generation, Stability

analysis, Photovoltaic, Microgrids

I. INTRODUCTION

istributed generation (DG) is a term commonly used to

describe small renewable power generators and storage

facilities that are located as close as possible to users. DG

continues to gain numerous applications in modern day power

system engineering - it flourishes from the fact that

advancement in distributed energy technologies enhances the

further implementation of distributed generation systems

(DGS) on the Grid/microgrid networks [1]-[3]. Rapidly

Submission Date: “This work was supported by the Electronics System

Design Centre, College of Engineering, Swansea University, Wales”.

A. M. Egwebe is a PhD student with the Electrical/Electronic Engineering

Department, Swansea University, Wales, (e-mail: 570477@swansea.ac.uk).

M. Fazeli is a Lecturer with the Electrical/Electronic Engineering Department,

Swansea University, Wales, (e-mail: M.Fazeli@swansea.ac.uk). P. Igic is a Reader with the Electrical/Electronic Engineering Department,

Swansea University, Wales, (e-mail: P.Igic@swansea.ac.uk).

P. Holland is an Associate Professor with the Electrical/Electronic Engineering Department, Swansea University, Wales, (e-mail:

P.Holland@swansea.ac.uk).

growing technologies and enhanced penetration of DGs aids

their potential to provide local energy as well as more

advanced ancillary services to the National Grid (NG)

including operating reserves, spinning reserve, frequency and

voltage regulation [4].

One of the main challenges facing the use of renewable

energy sources globally is how they can be efficiently

interfaced with the existing NG. The NG was designed for a

centralized distribution model characterized by large fossil-

fuelled thermal power plants at the centre and a one way

directional flow of electrical energy from high to low voltage

at the point of use. Renewable generation requires a

distributed generation model where the grid can accept

generation from any point in the transmission or distribution

network without causing technical issues for the equipment

used to control it or cause supply issues due to the large

amount of intermittent generation encountered with wind and

solar [5]. An intermediate solution to the problem highlighted

above is the concept of the microgrid - an interactive customer

friendly cluster of distributed energy resources, loads and

energy storage. It can operate in grid-connected mode or

islanded mode (without NG) to improve power quality and

network reliability. Modern approaches to microgrids promote

autonomous control in a peer-to-peer and plug-and-play

operation model for each DG on the microgrid [3], [6]-[7].

The concept of peer-to-peer ensures there is no master

controller or central storage unit that is critical for the

operation of the microgrid; hence the microgrid will continue

to operate with loss of any DG [1], [6]. Plug-and-play implies

that the DG can seamlessly operate when placed or mounted at

any point in the microgrid network without re-engineering the

control scheme [8].

In Grid-connected mode, control measures are relatively

easy to be implemented since voltage and frequency are

regulated by the utility grid for loads within the microgrid;

whereas in islanded-mode voltage and frequency must be

actively controlled for the continuous and stable performance

of the network [7], [9]. In order to balance generated energy

with demand in a microgrid, renewable energy generation are

often supplemented with dispatchable resources such as

energy storage system and local auxiliary generation (AG) [7]

;absence of such resources can result in the possibility of

stressing the inverter-based sources leading to excessive

voltage rise due to over-modulation, yielding poor power

sharing and circulating current among the inverters that can

harm the switching components and result in overloading and

excessive total harmonic distortion (THD) on the AC-side [6],

[10]-[12].

Implementation and stability study of Dynamic

Droop in islanded MicroGrids Augustine M. Egwebe, Student Member, IEEE, Megdad Fazeli, Member, IEEE, Petar Igic, Senior

Member, IEEE and Paul Holland, Member, IEEE

D

2

In islanded network, DGs as well as fossil-fuel sources must

be able to operate autonomously in balancing generation and

demand, voltage and frequency regulation, cost minimization,

while ensuring system stability [13]-[16]. The droop-sharing

scheme is usually employed in a microgrid for voltage and

frequency control between the available DGs, this scheme

adopts an autonomous load sharing approach, where each

connected DG uses their local parameters for accurate load

sharing. Therefore droop method provides higher flexibility

and reliability when compared to master-slave techniques [2],

[17]-[18].

It is noted that in a microgrid, most line can be mainly

resistive. In these cases, two main methods are proposed in

literature: (1) it is shown in [18]-[19] that in resistive lines,

droops are active power-voltage and reactive power-frequency

slopes. (2) References [19]-[21] proposed a method called

“Virtual Impedance” to reduce the coupling between real and

reactive power flow in low-voltage distribution network due to

non-trivial feeder impedances. Both approaches are applicable

to the method proposed in this paper. However, this paper

considers the classic active power-frequency droop in order to

emphasise that the method is not limited to resistive

microgrids. Moreover; inductive microgrid is studied in many

literatures such as [10], [12], [16], [22]-[24].

Different aspects of droop control have been investigated in

different references such as [1], [6], [13], [14], [20], [25] to

curb the drawbacks with droop-based control i.e. instability

issues due to sudden load perturbation, poor transient

response, inaccurate load sharing, steady state error of voltage

and frequency [26]. Reference [13] proposed the angular

droop method to perform load sharing with minimum

frequency variation; real power droop coefficient can thus be

chosen depending on the maximum/minimum value of load

demand and load sharing ratio [13]. Numerous researches

have also been carried to improved droop sharing scheme

while maintaining a relatively steady frequency and voltage

[13], [25]-[28]. For example [29] highlights the droop control

and average power control via two independent control

variables on each DG for accurate sharing of active and

reactive power- hence eliminating the sensitivity of the droop

method to measurement error and wire mismatches. Reference

[28] proposed a three stage mutually interactive droop scheme

based on DC link voltage variation, for power regulation

between the DG and the AC bus. The scheme operates by the

introduction of a power offset to the calculated power from the

conventional droop scheme, in order to modify voltage

reference to the inverter. The scheme was tested to show

power balance based on variation in load without

consideration for scenario when demand exceeds generation.

Modern approach towards improving the flexibility and

reliability of the microgrid favours a hybrid DG networks

(compromising of renewable sources, energy storage systems

(ESS) and fossil-fuelled AG), with adaptive hierarchy

controller schemes employed for power conditioning and

management [3], [17], [30]. However, the systems studied in

these literatures do not include an auxiliary generator whereas

in any practical microgrid a fossil-fuelled AG is necessary to

supply the critical loads in case of shortage of energy. It is

noted that the role of the AG is not similar to that of a master

unit (in a master-slave paradigm) since unlike in a master-

slave control, the operation of other units are not dependent on

the AG. The ESS often provides power deficit compensation

when all connected DG are fully utilized, with droop

techniques employed for accurate load sharing. For example,

in reference [30] the frequency droop was regulated using

power reference from the ESS. The approach presented in [30]

relies mainly on proper understanding of the state of Charge

(SOC) of the battery in defining operating conditions for

charging the battery and for load sharing. However, the

current paper considers networks without battery storage (in

order to comply with current UK regulations on distributed PV

systems). Moreover, unlike in previous literatures, the control

of an auxiliary generator (AG) is presented and the droop gain

is properly tuned to minimize the energy required from the

AG. Reference [16] proposed a complex multi-stage

optimization scheme to minimize fuel consumption in

microgrids. Beside the fact the proposed method in [16] is

very complex, it does not consider the intermittent nature of

renewable-bases DG units which is a common issue with all of

these studies. This drawback is illustrated in Fig.1:

1f

P2P1P1'

P2'

2

f*

Fig. 1: Steady-state characteristic of traditional droop

The conventional static active power-frequency (P-f) and

reactive power-voltage (Q-V) droop sharing scheme (shown in

Fig. 1) sets a fixed frequency/voltage droop gain irrespective

of the available energy from the renewable source. The

frequency thus only changes due to load change; it is

constrained within the allowable frequency droop gain [14]. In

such cases, a drop in the available power of one of the DG

from P1 to P’1 (e.g. due to a reduction in solar irradiation)

will

shift its frequency (f) to a new operating point (f*). Since the

other DG must comply with the new operating frequency (f*),

its power reduces from P2 to P’2 (even though it might have

the capacity to generate more power).

In order to mitigate this problem references [7] and [22],

[31] proposed a droop scheme for wind turbine generation to

control local demand. Besides the fact that the method was

proposed only for wind generation, the stability analysis of the

scheme was not presented while the authors admitted that the

method may affect the system stability.

The current paper proposes a dynamic droop scheme for

Photovoltaic (PV) systems. The proposed scheme uses the PV

array’s current vs voltage characteristics in defining an

operating range for the inverter-based source to ensure an

efficient load sharing interaction with other DG(s); as the DC

link voltages varies due to varying irradiance of solar energy.

The practical and theoretical stability of the proposed method

is also investigated for one DG unit. This study will be based

on PV sources but it is necessary for all renewable energy

sources i.e. wind, wave, etc.

3

To the best of our knowledge, there is no prior work that

precisely focuses on the intermittent nature of the renewable

source(s) in configuring the droop scheme i.e. droop gain

sensitive to source variation, in order to minimise the energy

required from an auxiliary generator. This is especially useful

in cases where ESS is not allowed, hence energy support is

often provided from utility side or a centralized auxiliary

generator within the microgrid.

II. MICROGRID NETWORK UNDER STUDY

Paux*

L1

P1 Q1

VDC1

PL QL LoadVt δt

V1 δ1

VDC1

AG

PEC

+

-

D1P1, Q1

ω,V P1

Q1

ω

V

L2

P2 Q2

VDC2V2 δ2

+

-

D2P2, Q2

ω,V P2

Q2

ω

V

VDC2

+

DG2

DG1

Paux Qaux

PV Array

Lline1

Lline2

PV Array

Fig. 2: MicroGrid Network Under study

Fig. 2 shows a microgrid network in islanded connection

with 3- phase inverter-based DGs (DG1, DG2). The power

electronic controller (PEC) is used to control the flow of

energy from local auxiliary generator (AG) using local

information from the DGs. This study aims to analyze the load

sharing interaction between these DG sources in the islanded

microgrid network.

The three phase inverter-based sources above are PWM

controlled with PQ, voltage and current controllers. The

traditional PQ controller uses the droop scheme (f vs P and V

vs Q) to autonomously respond to changes in connected loads.

In the absence of maximum power tracking, the PV operating

point is usually determined by the AC-side load demand;

hence the DC link voltage (VDC) will be perturbed

continuously from the minimum operating voltage (VDC-min) to

the PV array’s open circuit voltage (VOC) as the irradiance

level or load varies.

The proposed dynamic droop scheme uses the variation in

irradiance (i.e. VDC) in conditioning the conventional droop

scheme for an efficient load sharing while constraining VDC

within VDC-min to VOC (since inverter output voltage, Vinv ≈

0.5DVDC for a three phase system, where D is the modulation

index [6]). This will involve a linear approximation of the PV

maximum power point characteristic curve and the subsequent

droop gain tracking of irradiance variation within the DG’s

operating zone.

The current study does not consider reactive power sharing;

hence reactive power compensation control is not considered

in the AG control scheme [32].

A. Inverter Operating Zone in DG Application

The mathematical model of a PV array is described in [33]

with P-V characteristic shown in Fig. 3. It was also shown in

[10], [34] that a three-phase inverter (with sinusoidal PWM)

can averagely be modelled using d-q frame transformation

techniques:

DCdqdq VDV

2

1 (1)

Where Vdq is the d-q frame park transform of the AC bus

voltage, Ddq is the modulating index (in d-q frame) and VDC is

the DC link voltage. When there is a reduction in solar

irradiance level (hence decreasing VDC), D must increase to

maintain (1). At D =1; a constant Vdq depends solely on VDC.

Further reduction in VDC due to irradiance perturbation will

reduce Vdq. Generally VDC perturbs in response to irradiance

level and demanded load. Hence, in order to accurately control

AC bus voltage (Vdq), minimum DC voltage VDC-min must

ensure (1) while D = 1; e.g. for a nominal RMS 240V DG(s)

system explained in section II, VDC ≥ 678.8-V (i.e. operating

point limit with modulating index, D = 1). Thus, the PV array

must be designed such that the DC voltage of the maximum

power at a small irradiation (say 0.05 pu) = VDC-min= 678.8 V

(see Fig. 3).

Fig. 3: PV curve for varying irradiance level. (Curve B): Maximum point

curve; (PLoad): Constant Load demand curve

B. Conventional Static Droop Load Sharing Scheme

Using droop control, two or more parallel wired DGs can be

controlled to deliver the required real and reactive power. In

published droop schemes, two systems independent

parameters are controlled to achieve load sharing with

minimal communication between the sources [6], [13], [17],

[30]. The real and reactive powers injected from the DG to

the microgrid are sensed and averaged; the resulting signals

are used to adjust the frequency and voltage amplitude of the

DG [6], [14], [20].

The averaged real and reactive power (P and Q) of the DG(s)

(deduced in [6]) shown in Fig. 2 is given below:

4

X

VV

X

VVP tt sin (2)

)(cos2

tt VV

X

V

X

VVVQ

(3)

Equation (2) and (3) shows P varies with the phase angle

difference (ϕ) between the inverter output voltage (V) and the

common AC bus voltage (Vt) while Q varies with the

amplitude difference (V - Vt). Where X is the output reactance

of the inverter.

The P-ω and Q-V droop are strictly computed to ensure that

accurate load sharing is possible without a significant steady

state frequency and voltage drop across the overall system

[14].

The droop block (defined by ((4) and (5)) is used for

proportional sharing of P & Q; where P varies with system

frequency and Q with system voltage.

rating

prefprefP

mPPm

);( (4)

rating

qrefqrefdQ

VnQQnVV

);( (5)

Where ω, Vd, P and Q are the DG’s angular frequency,

terminal voltage, active and reactive power; ωref and Vref are

the rated angular frequency and voltage of the DG. ∆ω and ∆V

are the allowed frequency and voltage deviation. mp and nq are

the droop coefficients (i.e. the gradient of droop lines in Fig.

4) – which ensures the desired proportional power sharing

based on the DG’s rating (i.e. Prating and Qrating) [13], [23],

[29].

ω*

ω

PratingP

Δω ω = ω* - mpP

V

QratingQ

ΔVV = V

* - nqQ V

*

Fig. 4: Steady-state characteristic of traditional droop

Using (4) and (5), it was shown in [6, 13]:

2

1

1

2

2

1

rating

rating

p

p

P

P

m

m

P

P (6)

A problem arises if the available energy of one DG is not

enough to meet the demanded load. A new frequency

operating point will surface, which forces all other DG on the

network to its new operating point irrespective of the

generating capacity of the other DG(s) as shown in Fig. 1 and

Fig. 3. In other words, a drop in generation of one unit (due to

a reduction in irradiation) causes reductions in all the other

units’ generation (see simulation results in Fig. 8). The

shortage of supply is compensated by the energy stored in the

DC-links’ capacitors (or a local energy storage) which causes

a drop in the DC link voltage. Hence, the DC-link voltage (or

the energy level of the energy storage) can be used to trigger

an auxiliary generator (AG) via a Power Electronic Converter

(PEC) to compensate for the shortage of energy. It is noted

that since the other DGs are forced to reduce their generation,

the energy demanded from the AG will not be optimized. This

is due to the insensitivity of a static droop control to the input

solar energy (Fig. 8).

C. Proposed Dynamic Droop Scheme

This section proposes a dynamic droop control in which the

droop coefficient varies as solar irradiation changes without

the need to measure the irradiation. The method also ensures

that the maximum power from each unit is generated if

required by the load.

Fig. 3 depicts the PV curve of a DG as irradiance level varies.

When available solar power is more than the load power, the

system operates normally within its operating zone (right hand

side of curve B). As solar irradiation (S) drops, the DG will

continue to supply the load, until the available solar power is

not enough to meet the demand (point O); AG is thus

triggered on (when VDC becomes less than a threshold) to

compensate for the shortage in power.

Fig.5: Steady-state characteristic of PV operating zone

The maximum power of a PV system varies according to S. In

order to increase the efficiency of droop controlled PV

systems and ensure proportional load sharing, the maximum

power curve for various level of S can be used to define a

reference for the droop gain. The maximum power points

curve (curve B in Fig. 3) of the PV array can be accurately

approximated by (7) [10] (shown in Fig. 5):

dcVbVaVP DCDCDCDC 23

max (7)

Where a, b, c, and d were deduced using the Matlab “polyfit”

command, VDC is calculated from the Ppv - Vpv characteristics

of the array (as given by the PV manufacturer).

The operating zone is the area specified by VDC-min, PPV-rated

and VOC in Fig.5. As S drops, the operating point (for a given

load) moves towards the PDC-max curve (Fig. 5). At the

intersection of PLoad and PDC-max (point O), for any reduction in

S, the AG is turned on (when VDC becomes less than a

threshold) to compensate for the shortage of energy. In cases

with conventional static droop, drops in S of one unit causes

reductions in power output of all other units (to comply with

the new operating frequency), regardless of their available

generation capacities. As a result, more energy will be

demanded from AG. To solve this problem, the droop gains

5

p = b(VdIgd + VqIgq )

q = b(VqIgd – VdIgq)

δ = ω dt

Vα = V.cosδ

Vβ = V.sinδ

P ω

QV

Vd*

Vq*

Dd

Iabc

Vabcp

q Vd

Vq

Igq

Igd abc

dq

Vdq Ii-dq

+-

ω

P(W)

ω0

ω

P0 P

V

Q(VAR)

V0

V

Q0 Q

P-ω Droop

(mp)

Q-V Droop

(nq)

LPF

Dq

Vd Ii-d

Ii-d* +

--

+Kp

i( )s + ai

s

VqIi-q

Ii-q* +

--

+

Control Block

AC Bus

Transform Block

Kpi( )s + ai

sKp

v( )s + av

s

Kpv( )s + av

s

Plant

Fig. 6: Control block for DG in d-q frame

must vary according to the maximum power point curve of

their associated PV array (i.e. PDC-max):

(8)

Where n is number of PV array = 1, 2,…, N

Although (7) can be used in denominator of (8), it will impose

an unnecessary complexity. After all, it is only needed to have

an inverse relationship between mp and VDC (which in turns

varies with S). So (7) can be approximated as PDC-max≈kVDC+c

where, k and c are constants which are used to get a linear

approximation of (7). Please note that the idea is to make the

droop mechanism sensitive to the available solar power.

However, since droop mechanism is based on the relative ratio

between units, it is not necessary to use (7) and a linear

approximation of it works satisfactorily. This equation

describes the simplest maximum power point tracking method

explained in literature [35]. Droop gains must still be

proportional to the rating of their associated units. Therefore,

(7) is approximated for each PV unit as:

nnDCnnDC cVkP max (9)

where kn and cn are gains to get a linear approximation of the

PDC-max (i.e. (7)) of the nth

PV array. Therefore, the allowed

frequency variation will be:

NNDCNpN

DCpDCp

cVkm

cVkmcVkm

...

22221111 (10)

Doing so, when S is the same on the units, the load is shared

proportional to their ratings which is the same as conventional

units (see result in Fig. 8). However, any reduction in S of one

unit (e.g. unit 1 in Fig. 7), does not force the other units to

reduce their generation as mp is now sensitive to VDC.

Moreover, as illustrated in Fig. 7, the other units will increase

their generation through reducing mp (provided that enough S

is available) to compensate for the power reduction from the

first unit. This significantly reduces the energy demanded

from AG compared to conventional static droop method (see

Fig. 9 for results).

Fig. 7. Dynamic droop operation

D. Three-Phase System Control Scheme

The diagrammatic description of the control scheme of each

inverter-based DG can be approximated using the direct PI

control approach (i.e. ignoring feed forward path for axis

decoupling) as shown in Fig. 6. Synchronous reference frame

parameters are generated using the Park and Clarke

transforms. The droop block is used for accurate sharing of P

& Q; where P varies with system frequency and Q with

system voltage. Where b = 1.5 for a three phase system. The

low pass filter (LPF) is used to deduce average values of P

and Q respectively. The voltage controller regulates the

terminal voltage and generates reference for the current

controller [9], [17], [27], [36]-[37].

A voltage feed forward approach was employed in the current

controller to combat bus voltage disturbance and for voltage

drop compensation; with the bus voltage used as reference.

Compensation terms are added to erode coupling effects. The

current controller generates the control signal for the PWM.

Hard limit block sets the current limit to protect the system

from over current (i.e. restrict the upper and lower limits of

allowable reference voltage). The d & q components are

controlled independently. Where‘d’ regulates active power (P)

and ‘q’ regulates the reactive power (Q) [37]-[38].

The dynamics of the control scheme depends mainly on the

bandwidth of the PQ controller, since the bandwidth of the

current and voltage controller are much higher than that of the

nDCnp

Pm

max

f

PP1' P1 P2 P2'

6

PQ controller [37]. The instantaneous active and reactive

power is given by (11).

)(2

3

)(2

3

gqdgdq

gqqgdd

IVIVq

IVIVp

(11)

LC filter is interfaced at the output of the inverter to reduce

switching harmonics of the output voltage (THD < 5%); filter

parameters were chosen as discussed in [10].

E. Auxiliary Generator Control

The DC link voltage of DGs is an indicator for regulating the

AG (see Fig. 2). When the DC link voltage of either DG

decreases below a threshold (here 0.85pu), AG is switched ON

to compensate for the energy shortage in order to maintain the

demand power. The AG reference power (Fig. 2) is:

N

nnDCaux VP

1

*3 (12)

The proposed method in this paper studies the active power

sharing not reactive power. Thus the PQ control scheme was

adopted in the AG control [32], [39] for injecting active power

into the network when needed, Where Qref for the PQ

controller is zero and Pref is set by (12). The load reactive

power is shared appropriately using the classical Q-V droop.

III. SIMULATION RESULTS

TABLE I

SYSTEM’S PARAMETER

Variable Value

Prating1 0.64 pu

k1 and c1 (Eq. 9) 76.48 and -50692.01

Prating2 0.36 pu

k2 and c2 (Eq. 9) 43.05 and -28442.60

Δf and ΔV 2% and 5%

PLoad 0.75 pu

DC link capacitor, C0 800 μF

Line to line voltage VL-L 415 V

LCL line parameters

10mH/6 μF/0.2mH

LPF bandwidth (ωf) 45 rad/s

The test model consists of two DGs and one AG feeding a 3-

phase load. Each DG has its own control scheme as shown in

Fig. 2 and the load sharing scheme is simulated for both

conventional static droop gain and the proposed dynamic

droop gain.

Various testing scenario were observed in

MATLAB/SIMULINK to analyze the test network depicted in

Fig. 2. Note that all results are presented in pu based on the

total system rating (not each PV system).

A. Conventional Static Droop Scheme

Conventional static droop load sharing was tested for two

PV DG sources shown in Fig. 2, with droop gain set by (4) &

(5) using values for Prating1 and Prating2 given in Table I. The result in Fig. 8 shows load sharing between the DGs

where solar irradiation of DG2 drops in 4 steps and the load- is

constant at 0.75pu. Up to 20s, the load is appropriately shared

based on their rating since the available solar power (Pa1 and

Pa2) on both systems is the same (i.e. 1pu based on their own

ratings). However, as the available power in DG2 (Fig. 8 (b))

drops due to drops in irradiance level, its frequency changes to

a new operating point resulting in a reduction in power

contribution to the load (Fig. 8 (a)). Therefore, DG1 complies

with this new operating point and reduces its power

contribution (Fig. 8 (a)) although its solar irradiance is

constant (Fig. 8 (b)). As a result, the total generation becomes

less than the load which leads to reduction in VDC. When VDC <

0.85 pu, the AG is turned on to supply the shortage (according

to (12)). It is important to note that over the entire simulation

the total available solar power (Pa1+Pa2) > PLoad i.e. there

should not be any need for AG. Hence it can be seen from the

simulation results that the conventional droop scheme does not

make an optimized used of an AG.

Fig. 8: Simulation results of two DG systems using conventional static droop

(a) active power in pu, 1-PLoad , 2-P1 , 3-P2, 4-Paux , (b) available solar power in pu 1-Pa1, 2-Pa2

7

B. Proposed Dynamic Droop Implementation

The simulation was repeated with droop gains set by (10)

using the available solar power for load sharing. Fig. 9(a)

shows that the power is shared appropriately (i.e. based on

rating) - when the solar irradiances are the same. As the

available power on DG2 begins to reduce, its droop gain

increases which in turn reduces the power contribution of DG2

to the overall load. However, the droop gain of DG1

proportionally reduces to compensate for the power drop in

DG2 (since DG1 has extra capacity to compensate for DG2).

The DGs are hence within their ‘operating zone’, since DG1

compensates for the power reduction in DG2 without the need

for the AG (power contribution from AG = 0 as shown in Fig.

9 (a)). The flexibility of the proposed scheme ensures

complementary energy support between the DG(s)-for

instance at time 30 - 40s, the DG(s) fully supply the demanded

power since the combined total available power (0.86pu) is

more than total load (0.75pu).

The dynamic droop when compared with the conventional

droop scheme saves energy, since DGs compensate for one

another which minimize the energy demanded from AG.

Fig. 9: Simulation results of two DG systems using the available solar power

for dynamic load sharing (a) active power in pu, 1-PLoad , 2-P1 , 3-P2, 4-Paux ,

(b) available solar power in pu 1-Pa1, 2-Pa2.

C. Simulation Results with Real-Time Solar Irradiance

Variation

The proposed scheme was also tested using real-time solar

irradiation profile measured at the College of Engineering

Swansea University, Swansea, U.K. (at 51.6100 northern

latitude and 3.9797 western longitudes) as shown in Fig. 10

(a). Fig.10 (b) shows that the contribution of DG1 is reduced

as DG2 reduces when conventional scheme is employed;

hence more energy support is required from the AG. Compare

to the proposed scheme in Fig.10 (c) where the DGs

compensate for each other and thereby provides energy saving

from AG. Fig. 10 (d) shows the AG energy profile of the

conventional and dynamic droop scheme. It can be seen that

the proposed scheme provides energy saving up to 74%

compared to the conventional droop sharing scheme for the

data set studied.

Fig. 10: Simulation results of two DG systems using the real-time solar data

(a) available solar power in pu 1-Pa1, 2-Pa2 (b) active power using

conventional droop in pu, 1-PLoad , 2-P1 , 3-P2, 4-Paux , (c) active power using proposed dynamic droop in pu, 1-PLoad , 2-P1 , 3-P2, 4-Paux , (d) AG Energy

profile for Conventional and Dynamic droop in pu, 1-Conventional Droop, 2-

Proposed Dynamic droop.

8

ωf

sPower

Eqn. 11

P,Qp,q

-

+ Droop

Eqn. 14

Vd

Vq

Igd

Igq

1s

2

VDCD

ω0 ω0t

V

Vd*

Vq* = 0

TFv , TFi

Vidq*

[mp ,nq]VDC

Droop Power Controller

Trans.

Eqn. 15-17

Fig. 11: Control model of power droop controlled PV system

IV. STABILITY ANALYSIS OF THE PROPOSED

SCHEME

For a linear stable operation the modulation index D must be

kept less than 1. This section investigates the effect of the

variation of p-f droop gain (mp) on the operation of a 3-phase

inverter. It has been shown in various literatures [12], [16],

[37], [40] that the dynamics (low-frequency poles) of the DG

is mainly determined by the bandwidth of the droop power

sharing control model shown in Fig. 11. However, in order to

ensure completeness, this study will be based on the entire

linearized state-model of the DG.

qs

sQ

ps

sP

f

f

f

f

)(

)(

(13)

Eq. (4), (5), (11), and (13) is combined and linearized around

an operating point:

Q

Pn

m

V

V

V

I

I

IIVV

IIVV

Q

P

m

Q

P

q

p

q

d

qg

dg

gdgqdq

gqgdqdf

f

f

p

0

00

0

0000

2

3...

00

00

00

0

.

.

.

(14)

The output of the power droop controller (v(t)) is transformed

to d-q components to yield Vqd* for the voltage controller [13]:

Using Clarke transform, v(t) can be rewritten as:

sin

cos

sin

cos)(

0

0

*

*

0

V

V

tV

tV

V

V

Vtv (15)

Parke Transformation representation equals:

*

*

*

*

cossin

sincos

V

V

V

V

q

d (16)

Vdq* is linearized and simplified using (14) and (16):

V

V

V

V

q

d

10

0*

*

(17)

The DG plant model is described by:

][][][ 21

.

dqtdgdqdgdgdgdg VDxx BBA (18)

Where ∆Ddq from fig.11 is given as:

][2

][2

][2

...

][)(22

][

1

*

11

212

*

dqc

dc

dqcv

dc

dqcv

dc

dgvcc

dc

idq

dc

dq

JV

EV

VV

xV

VV

D

CDCDD

DDD

(19)

‘xdg’ is the state variables of the DG converter model; Adg, Bdg-

1 and Bdg-2 are matrixes from the DG plant; Cv, Dv-1 and Dv-2 are

matrixes from voltage controller; Cc, Dc-1 and Dc-2 are matrixes

from current controller, EՓ-dq and JՓ-dq are integration terms of

the voltage and current controllers respectively, Vt is the

terminal AC bus voltage.

Equation (14), (17), (18) and (19) is combined to yield:

][][...

][][][

43

21

.

tdq

dqdqdgdg

VPQ

EJxx

BB

BBA1

(20)

Hence, all the state variables in (20) can be merged to deduce

the linearized complete state-space small signal dynamic

model of the DG in fig. 6 interfaced to a common reference

frame (as shown in [12], [13], [24]):

].[].[ uxdt

d BAx (21)

Tdqddqgidqdqi EVIJIPx

;;00001

000000

00000000000

0010000000

00001

0001

00

000000000

001

000000

00001000

0000010

000000

00000

00000000

0000000000

0

0

0

dt

T

g

p

ff

gd

g

gq

ivf

ivpv

p

gddf

p

VuL

Vm

CC

I

LI

KCF

KKFk

fhdcb

fedcbam

gIgV

m

B

A

;;;;;i

pvpi

i

pi

i

ii

i

pi

i

pvpi

L

KKe

L

FKd

L

Kc

L

Kb

L

KVKa

9

.;;2

3;

0

ppv

i

piff

i

ivpimVKk

L

KChg

L

KKf

Where Q = Vq = 0 , Li is the filter inductance, Lg is the line

impedance, Cf is the filter capacitance, F is feedforward gain

term for the voltage controller, K-terms are the proportional

and integral gains of the current and voltage controller, Δ

implies a small perturbation around the operating point. The

angular perturbation (δ0) in the reference frame is considered

to be very small in this study, i.e. δ0 ≈ 0.

The open loop poles of the system are the eigenvalues of

matrix A. Hence the stability of the inverter can be analyzed

as the droop gain changes using matrix A.

A. Variation of open loop poles for different active and

reactive powers

Fig. 12: Variation of Open loop poles as output power (p) varies with q = 0

The system has five pairs of complex conjugate open loop

poles. There are four pole pairs far to the left from the

imaginary axis whose dynamics has minimal effect on the

stability (hence not shown in Fig. 12). It can be seen that as

contributed power (p) reduces, two poles moves towards

instability (towards the jω axis). The system poles were also

observed as q, nq and VDC changes. Variation in either of them

does not influence system open loop poles. VDC was set to

VDC-min to account for the worst case scenario when D =1.

Fig. 13: Open loop plot of droop controller gain variation showing the

dominant low-frequency poles with p = 0.05pu (worst case scenario), q = 0,

nq = 6.7 x 10-4 and VDC = VDC-min

B. Variation for different mp

As discussed, the open loop poles moves towards instability as

active power p reduces. So in this section in order to study the

worst case scenario, p=0.05pu while mp varies from 0.11 to

0.13.

Fig. 13 has five pairs of complex conjugate poles. Four pole

pair is far to the left (not shown) whose dynamics has minimal

effect on stability and they move away from the imaginary

axis as mp increases. The other pair, shown in Fig. 13, moves

towards instability as mp increases, and crosses the imaginary

axis at a certain value (mp-limit = 0.122).

Therefore the system’s eigenvalue moves towards instability

for large f-P droop gain. For mp greater than mp-limit, system

becomes unstable as shown in Fig. 13.

A drop in VDC due to a decrease in irradiance level causes an

increase in droop gain (mp) to reduce the power contribution

from the DG. On the other hand, a reduction in VDC also

increases the modulation index D to comply with (1). D is

limited to 1 where VDC=VDC-min (see Fig.14). This point defines

a practical stability limit for mp (mp-max) where using (1):

VDC-min ≈ 2Vd; hence using (8) and (9), mp-max for the nth

PV

systems is:

nDCnnp

cVkm

minmax

;

The practical stability limits for the PV systems simulated in

this study are: 4

2max

4

1max 1061.1;1003.1

pp mm

Therefore, it can be concluded that mp-limit>>mp-max which is

illustrated in Fig. 14. Note that mp-max-n is the practical limit

that the droop gain can be reduced to. This happens when the

solar irradiation is very low and Vdc=Vdc-min, D=1.

Fig. 14: Stability limit of the proposed dynamic droop

This study demonstrates that the practical stability limit which

is imposed by modulation index of the inverter is much less

than the theoretical stability limit which is imposed by a very

large droop gain mp. In other words, the dynamic droop

method does not add a further limitation on the operation of an

inverter-based unit.

V. CONCLUSION

Dynamic droop load sharing based on the available solar

power was studied in this paper. The proposed dynamic droop

scheme was validated in MATLAB/SIMULINK. Simulation

results show that the proposed dynamic droop based on the

available solar capacity of DGs can be used for optimum load

sharing.

f

P

mp-max (D=1) mp-limit

10

The presented scheme was validated for multiple PV array

with various irradiance conditions; and it was shown that

power sharing is proportional to the units’ ratings when the

irradiance levels are the same. However, if the solar available

power on one PV array drops, the other inverter-based sources

can generate more power (if the capacity is available) to

compensate for the load demand, without the need for energy

support from local connected auxiliary generators and thereby

providing significant energy saving compared with

conventional static droop control. The scheme was also

validated with real-time (measured) solar irradiation.

A stability analysis was also presented to determine the

theoretical and practical stability limit of the proposed scheme.

It was shown that the dominant low-frequency eigenvalues of

the system are mainly influenced by the parameters of the

active power controller (i.e. the frequency-droop gain, mp). It

was also demonstrated that the proposed dynamic droop will

not make the system unstable as the practical stability limit

(which is defined by modulation index D=1) is much less than

that which is defined by a very large f-p droop gain.

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Augustine M. Egwebe (M’13) received

the B.Eng. degree (with first class Hons.)

in electronic and electrical Engineering

from Swansea University, Wales, in 2012.

He is currently working towards his Ph.D.

degree in Electrical and Electronic

Engineering at Swansea University, with

core focus in renewable energy systems

and control schemes optimization for large scale integration

with smart grids. His current research interest includes

renewable energy power generation systems, distributed

generation, microgrid, energy management systems, power

electronics and power quality.

He is also a student member with the Institute of Electrical

Technology (IET). He obtained the best graduating student

award from the College of Engineering, Swansea University

and the IET best student award in 2012.

Meghdad Fazeli (M’13) received the B.Sc.

degree in electrical engineering from the

Chamran University of Ahwaz, Iran, in

2004, the M.Sc. degree in electrical

engineering and the Ph.D. degree in wind

generator energy storage control schemes

for autonomous grids both from

Nottingham University, U.K., in 2006 and 2010, respectively.

Since January 2011, he has been with the Swansea University,

U.K. He was appointed as a Lecturer in Electrical Power

Engineering in September 2013. His current research is mainly

concentrated on grid integration of photovoltaic systems. His

main research interests include the integration of renewable

energy resources with grids, smartgrids, and distributed

generation.

Petar Igic (SM’15) is Head/Director of the

Electronic System Design Centre and

Director of the EEE Board of Studies at the

College of Engineering, Swansea

University, UK. He also held the esteemed

EPSRC Advanced Fellowship for his study

in the field of High Power IC technology

development. Petar has 20 year experience of research in

power semiconductor devices and technologies, electro-

thermal compact modelling, modelling and characterization of

microelectronic power semiconductor devices and power

electronics and systems. Dr. Igic has been leading Swansea

University’s contribution to the £50M Low Carbon Research

Institute, a pan-Wales university initiative amongst other.

Petar has also done some of the pioneering work in the

development of the compact models for power bipolar

semiconductor devices. He worked on industrial projects or

has been a consultant to several major Japanese, European and

American multinationals, such as TOYOTA, HITACHI,

Vishay SILICONIX, IR, ALSTOM. He has published over

100 scientific papers in journals and international conferences

and technical reports.

Paul M. Holland (M’12–M’14)

received the B.Sc. degree (with Hons.)

in engineering physics from Sheffield

Hallam University, U.K., in 1993, and

the Ph.D. degree in power integrated

circuit technology development at

Swansea University, U.K., in 2007.

He spent the first ten years of his career

working in the U.K. semiconductor industry for GEC Plessey

and ESM Ltd., as a Senior Process and a Device Engineer.

After working as a Researcher at Swansea University from

2002, he was appointed as a Lecturer in 2008 in the College of

Engineering and is now an Associate Professor. His research

interests include the application of CMOS technologies in the

areas of power ICs and Lab-On-A-Chip development which

has been funded by the Engineering and Physical Sciences

Research Council. He has published and presented more than

30 scientific papers in journals and international conferences.