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Optimization of a stand-alone hybrid wind/solar generation system for lifespan extension

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Optimization of a stand-alone hybrid wind/solar generation system for lifespan extension
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Mejía, Daniel A
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English
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xv, 68 leaves : ; 28 cm

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Subjects / Keywords:
Hybrid power systems ( lcsh )
Solar energy ( lcsh )
Wind power ( lcsh )
Energy storage ( lcsh )
Energy storage ( fast )
Hybrid power systems ( fast )
Solar energy ( fast )
Wind power ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 66-68).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Daniel C. Mejía.

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|University of Colorado Denver
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Classification:
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Full Text
OPTIMIZATION OF A STAND-ALONE HYBRID WIND/SOLAR
GENERATION SYSTEM FOR LIFESPAN EXTENSION
by
Daniel A. Mejia
B.S.E.E., University of Colorado Denver, 2007
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Masters of Science Electrical Engineering
Electrical Engineering
2011


This thesis for the Masters of Science Electrical Engineering
degree by
Daniel A. Mejia
has been approved
by
Jan Bialasiewiez
Date


Mejia, Daniel A. (M.S., Electrical Engineering)
Optimization of a Stand-Alone Hybrid Wind/Solar Generation System for Lifes-
pan Extension
Thesis directed by Assistant Professor Fernando Mancilla-David and Assistant
Professor Alexander Engan
ABSTRACT
The work presented in this thesis focuses on the optimization of a hybrid
wind/solar system with energy storage for equipment lifespan extension. In
general, electrical equipment is exposed to operational disturbances that result
in temporary or permanent damage to the equipment. This study analyzes
each component of the system, identifies several potential operational distur-
bances, and subsequently proposes a method to minimize the conditions that
create or cause such effects. These conditions are minimized by solving an op-
timization problem that calculates the sources supply set points at which the
systems operational disturbances are minimized. Scenarios with different op-
erating conditions to verify the performance of the proposed formulation along
with illustrative examples in terms of case studies are also presented.


This abstract accurately represents the content of the candidates thesis. We
recommend its publication.
Signed
Signed
Alexander
Engau^'


DEDICATION
This Masters Thesis is dedicated to my Lord and Savior Jesus Christ for His
is all the power, the honor and glory. Amen.


ACKNOWLEDGMENT
I would like to sincerely thank Professor Fernando Mancilla-David for his
support and teaching throughout my entire career. His teaching methodology,
heavily based on conceptual foundation, combined with state-of-the-art technol-
ogy applications has forged my engineering abilities and professional develop-
ment.
I deeply thank Professor Alexander Engau for his continuous support
throughout the entire project. His rich knowledge and talent combined with
his accessibility and delightfulness to teach make him fun and enjoyable to work
with.
I would like to give special thanks to Professor Fernando Valeneiaga from La
Plata National University (UNLP) for his valuable support, answering questions
related to his previous research.
I greatly appreciate Professors Jan Bialasiewicz and Jae Do Park for form-
ing part of my dissertation defense committee.
I would like to also dedicate this work to my brother and best friend Julian
for his support and his unconditional love, support and friendship.
My mother and my father have always been there for me and given me
comfort and support. Their love and support has given me the ability and
courage to accomplish all my goals.


CONTENTS
Figures .............................................................. x
Tables.............................................................. xii
Symbols and Nomenclature........................................... xiii
Chapter
1. Introduction...................................................... 1
1.1 Hybrid Systems Review ............................................ 1
1.2 Optimization in Power Systems..................................... 4
2. System Models Review.............................................. 5
2.1 Wind Subsystem.................................................... 6
2.1.1 Wind Subsystem Operation....................................... 8
2.2 Solar Subsystem ................................................. 12
2.2.1 Solar Subsystem Operation..................................... 13
3. System Operational Contingencies Analysis........................ 17
3.1 Battery Potential Hazards...................................... 17
3.1.1 Batteries Discharging Mode.................................... 18
3.1.2 Batteries Charging Mode ...................................... 18
3.2 Wind Subsystem Potential Hazards............................... 20
3.3 Solar Subsystem Potential Hazards................................ 22
4. System Modes of Operation........................................ 23
4.1 Operation Mode 1 23
vii


4.2 Operation Mode 2 ................................................ 25
4.3 Operation Mode 3 ................................................ 25
4.4 Operation Mode 4 ................................................ 26
4.5 Operation Mode 5 ................................................ 26
4.6 Operation Mode 6 ................................................ 26
4.7 Operation Mode 7 ................................................ 27
4.8 Operation Mode 8 ................................................ 27
5. Problem Formulation................................................. 28
5.1 Total Harmonic Distortion Index................................... 29
5.2 Mathematical Optimization Model................................... 31
6. Problem Solution.................................................... 34
6.1 Solution Approach................................................. 36
6.1.1 Scenario 1....................................................... 37
6.1.2 Scenario 2....................................................... 41
6.1.3 Scenario 3....................................................... 43
7. Discussion.......................................................... 45
8. Conclusion.......................................................... 52
Appendix
A. Simulation Parameters.............................................. 54
B. Computational Code................................................. 56
B.l Scenarios Generator .............................................. 56
B.2 Optimization Solving Function .................................... 57
B.3 Variables Definition Function..................................... 58
B.4 Objective Function................................................ 60
viii


B.5 Non-linear Constraints Function.................................. 61
B. 6 Weighted Objectives Solver Function............................ 62
C. MATLABs fmincon Function........................................ 64
D. System Case Studies.............................................. 65
Bibliography ........................................................ 66
IX


FIGURES
Figure
1.1 Wind/solar hybrid generation system................................... 2
1.2 Energy technologies hybrid combination matrix [1] 3
2.1 Equivalent circuit of a battery....................................... 5
2.2 Wind subsystem block diagram.......................................... 6
2.3 Analytical approximation of Cp A characteristics (blade pitch angle
$ as parameter [2])............................................... 7
2.4 Equivalent circuit and vector diagram for a PMSG [2].................. 9
2.5 Aopt at a blade pitch angle fl 0.................................. 10
2.6 Solar subsystem block diagram........................................ 12
2.7 Solar cell equivalent circuit ....................................... 12
2.8 I-V characteristic curves............................................ 14
2.9 PV Curves maximum values at 25C..................................... 16
3.1 Battery charge characteristic curve.................................. 19
6.1 Pareto frontiers illustration........................................ 35
6.2 Scenario 1 Pareto set................................................ 38
6.3 Te and iow as a function of dw for different wind velocities V ... . 39
6.4 fi{x) as a function of dw............................................ 40
6.5 Scenario 2 Pareto set................................................ 42
7.1 Objective functions variation with respect to II..................... 47
x


7.2 Control signals variation with respect to ii............................. 48
7.3 Wind system design variables variation with respect to ii ............... 49
7.4 Solar system design variables variation with respect to ................. 50
7.5 Battery system design variables variation with respect to ii .... 51
xi


TABLES
Table
4.1 Modes of Operation ............................................ 24
6.1 Scenario 1 Inputs............................................... 37
6.2 Scenario 2 Inputs............................................... 41
6.3 Scenario 3 Inputs............................................... 43
6.4 Scenario 3 Formulation Results.................................. 44
7.1 Evaluated Scenarios.............................................. 45
7.2 Scenarios Corresponding to Mode 6.............................. 46
xii


SYMBOLS AND NOMENCLATURE
Decision variables: dpv dw - Solar system DC/DC converter control signal - Wind system DC/DC converter control signal
System inputs: 5 T VdC V m - Solar irradiation - Solar panel temperature - DC bus voltage - Wind speed - Wind turbine mechanical shaft speed
Variables: fi(x) Mx) id *O-W tOJW *O-PV lojpv iq iph lrs p £PV ref MU' max *Wret Te THDiw(x) Tm Vs Wl w2 ZiU zN ^l UJe wmopi - THD objective function - Battery current objective function - PSMG direct current in the rotor reference - Wind system output current - Wind system maximum available output current - Solar system output current - Solar system maximum available output current - PSMG quadrature current in the rotor reference - Photocurrent, - Cell short circuit current - Reference power demanded from the sun - Maximum solar power point - Reference power demanded from the wind - Wind turbine electrical load torque - Total harmonic distortion - Wind turbine mechanical load torque - Wind generator terminal voltage - Weight value associated with fi(x) - Weight value associated with f2{x) - Pareto set lower bound (Utopia point) - Pareto set upper bound (Nadir point) - Wind turbine electrical shaft speed - Maximum turbine shaft speed
Xlll


Constants:
A - p-n junction ideality factor
Aw - Wind turbine swept Area
C - Battery capacity
Cb - Battery capacitance
Cp( K-&) - Power coefficient
eb - Battery counter emf voltage
eg - Band-gap energy
*6max - Battery maximum charging current
irr - Reverse saturation current
1scr - Cell reverse saturation current
J - Wind turbine inertia
k - Boltzmanns constant
kt - Short circuit current temperature coefficient
L - Wind generator per phase inductance
Lh - Stator and rotor leakage inductance
Ld - Wind generator stator direct inductance
Lq - Wind generator stator quadrature inductance
Lpv - Solar system DC/DC converter output inductor
Liu - Wind system DC/DC converter output inductor
mc - Number of cell in a battery
mp - Number of batteries in parallel
ms - Number of batteries in series
rip - Number of solar modules in parallel
ns - Number of solar modules in series
P - Wind generator number of poles
Q - Charge of an electron
rs - Wind generator synchronous resistance
R - Wind turbine radius
R, - Battery internal resistance
Tr - Cell reference temperature
Vgas - Battery gassing voltage
0 - Blade Pitch Angle
\>pt - Optimum tip speed ratio
P - Air density
4*m - Stator windings linked flux
z - Allowable depth of discharge
XIV


Abbreviations:
BPA - Blade pitch angle
DNS - Demand not served
DOE - Department of energy
emf - Electromotive force
IncCond - Incremental conductance
KKT - KarushKuhnTucker
MTBF - Meantime between failure
O&M - Operation and maintenance
P&O - Perturb and observe
PMSG - Permanent magnet synchronous generator
PV - Photovoltaic
REEPS - Renewable energy electric power systems
RMS - Root mean square
THD - Total harmonic distortion
TSR - Tip speed ratio
XV


1. Introduction
During the last few decades, renewable energy electric power systems
(REEFS) have grown significantly. Several projects have been developed world-
wide, which seek the production of clean energy. Many countries nowadays
support the implementation of REEFS. For instance, China has implemented
programs like the Golden Sun Program which consists of supporting 500 MW
or more of photovoltaic power [3]. Similarly, the U.S. government awarded 500
million dollars for clean energy projects, as stated by a press release from the
U.S. Department of Energy in September, 2009 [4],
In the U.S., REEPS correspond to 8% of the total energy consumption. The
installed capacity as of 2009 consists of 10.6% of the total electrical generation
[4]. The total generating capacity growth from 2008 to 2009 was approximately
1.5% from which wind energy represented 63.3%. Wind and solar energy repre-
sent, respectively, 9% and 1% of the U.S. renewable energy consumption accord-
ing to the 2009 Annual Energy Review [5]. These statistical analyses performed
by the Department of Energy clearly illustrate the magnitude of the growth
corresponding to REEPS.
1.1 Hybrid Systems Review
Hybrid power systems are designed for the generation and use of electrical
power from different energy sources. They are independent of a large centralized
electricity grid, incorporate more than one power source or more than one fuel
source for the same device, and are typically found in remote locations such as
rural areas or places with low economical resources [6].
1


The different generating sources are meant to complement each other. Most
of the time these hybrid systems are used as backup systems or operate in
conjunction with the grid. The main benefit of a hybrid system is to improve
reliability and efficiency. Some of the common hybrid combinations are found
in the hybrid technologies matrix published by the DOE, shown in figure 1.2.
For the most part, these combinations are dictated by the geographical
location and the average local weather conditions. In this study, we analyze the
hybrid model studied in [7]. This model consists of a stand-alone wind/solar
hybrid generation system with energy storage as shown in figure 1.1.
Figure 1.1: Wind/solar hybrid generation system
2


Storage I Fuel CeN I I I CSP I Solar I Microtuftine I Brayton I Rtnkine I Stirling Engine I 1C Engine
Figure 1.2: Energy technologies hybrid combination matrix [1]
3
Hybrid Technologies Matrix


1.2 Optimization in Power Systems
The efficient and optimal economic operation and planning of electric power
generation systems have always been important in the electric power industry.
The efficient use of fuel has become more relevant, primarily because most of the
fuel used represents costs and irreplaceable natural resources [8]. For our study,
fuel costs do not represent a concern since wind and solar energy resources are
free. Energy costs are composed of fixed and variable costs. Fixed costs account
for interests to loans, returns for investors, fixed operation and maintenance
(O&M) charges, taxes, etc. Variable costs depend mainly on the cost of fuel
and variable O&M costs [9].
Since the fuel expense is not taken into consideration, the overall cost of
wind and solar power generation systems is driven by the capital costs and
O&M expenses. Moreover, a major contribution to capital and O&M costs are
expenses related to energy storage [10, 11]. This means that we could have
economical benefits if the system is operated in such a way that the mean time
between failure (MTBF) is reduced and scheduled O&M are increased.
4


2. System Models Review
The battery system consists of a bank of batteries connected in series to
obtain the desired DC link voltage and connected in parallel to increase capacity.
Each battery contains a number of cells. For this study, 6-cell batteries are used
with 2.1 Volts per cell. The battery banks nominal voltage is 48 Volts and the
nominal capacity is 40 A-hr per battery string. Other parameters are given in
Appendix A.
The equivalent circuit of the battery bank system is shown in figure 2.1.
Here, and Vdc denote the electromotive force (emf) voltage and the terminal
voltage respectively; % is the battery bank current. The capacitance Cf, of the
batteries is neglected for this study. The resistance Rt is typically small, in the
order of tenths of an Ohm.
6 b V DC
----------------------o
Figure 2.1: Equivalent, circuit of a battery
5


The equation for the circuit in figure 2.1, ignoring the battery capacitance
Cb is
VDC £b + ibRi,
(2.1)
therefore the current following at any given time is
ib
vpc ~ eb
Rr
(2.2)
The direction of the current of the battery bank ib is dictated by the terminal
voltage. When the voltage of the battery eb is equal to the charging source
voltage VdCi no current will flow. When the voltage of the battery is less than
that of the charging source, current will flow into the battery and charge it, but
if the battery voltage is higher, current will flow out of the battery and discharge
it [12].
2.1 Wind Subsystem
The wind subsystem is formed by a permanent magnet synchronous gener-
ator (PMSG), a wind turbine, a passive rectifier and a buck DC/DC converter
as shown in figure 2.2.
Vdc
Figure 2.2: Wind subsystem block diagram
6


The mechanical power extracted from the wind turbine is directly related to
the wind velocity. This means that the wind turbine will only produce as much
power as wind availability allows. This relationship is defined as
Pw=l-Cp{\^)pAV\ (2.3)
where Cp(A, i9) is the rotor efficiency as a function of the tip speed ratio A (TSR)
and the blade pitch angle i9 (BPA). By p we denote the air density correspondent
to the altitude of the location, A = nr2 is the cross-sectional area of the turbine
with blade radius r, and V is the wind velocity. The TSR is defined as
A = r^, (2.4)
where uim is the angular shaft speed.
Figure 2.3: Analytical approximation of Cp A characteristics (blade pitch
angle $ as parameter [2])
The rotor efficiency Cp A curves from figure 2.3 may be approximated in
closed form by non-linear functions. Following the turbine characteristics of [2],
a model can be derived in the form:
Cp(A,i?) = CX
C2^-~ C30 C^y C5
Aref
Ce j
O A.
1
ref ,
(2.5)
7


where
and
1
1
0.035
Aref A + 0.08?? tf3 + 1
Ci = \ C2 = 116 C3 = 0.4
(2.6)
(2.7)
C4 = 0 C5 = 5 C6 21
are coefficients that are obtained from the model developed in [2].
For this study the turbines BPA is constant and equal to zero and therefore
CP(A) =
1
116 | j 0.035 ) 5
-21(^-0.035)
(2.8)
2.1.1 Wind Subsystem Operation
The mathematical definition of the PMSGs dynamic performance in a rotor
reference frame based on the model represented in figure 2.4 is
P TtV£)ciq 1
^ lq ri Um^d
P
Y
V
id= 4~ ~^miq
P
1
2
2 L
KVocid
3a/3 L^J~i
+ i2qdw
1

2 I .2 dw
d lq
(
u.
2 J
3 P
YY
O o tymiq T 2
1 Cp{X)pAVz
\
\
Um
-v
T,
(2.9)
(2.10)
(2.11)

where L represents the stator per phase inductances Ld = Lq = L; rs is the
stator per phase resistance; id and iq, are respectively, the direct and quadrature
current in the rotor reference frame; 0m is the flux linked by the stator windings;
Tt is the wind turbine torque; J is the inertia of the rotating parts; P is the
number of poles of the PMSG; and dw is the control signal (duty ratio) of the
wind system DC/DC converter [7].
8


9
Figure 2.4: Equivalent circuit and vector diagram for a PMSG [2]
For steady state operation iq 0, id 0, and oje 0 which result in
P T^^Dciq 1
P
T
n ^rrdd 4
2 L
3V3
rs P .
' 4 ^ ^mdq
3 P
KVoCld
1
3^3 L^jil
v 2 dit
= 0,,
= 0,
2 2 ^ + 2
1 Cp(X)PAV3
= 0.
LU-n
(2.12)
(2.13)
(2.14)
The expression for the output current of the wind subsystem DC/DC con-
verter is:
in
7T

l
(2.15)
3V3V9 v dw
In order to calculate the output current, dw, id, iq, V and u)e must be
known. The control signal dw is adjusted by the control system depending on
the rotor shaft speed um. If the wind velocity V is known, then um may be
calculated using the steady state set of equations (2.12) to (2.14). However, if
u>m is measured, the wind velocity measurement is not necessary since id and iq
9


may be calculated using equations (2.12) and (2.13) only.
The power demanded to the wind subsystem is dictated by the following
reference:
PwTef Vdc(}L T ib io-pv)) (2.16)
where is the load demand current, and iopv is the current contribution from
the solar system.
In order to determine if the wind system is able to sustain the power demand
PWre( we need to know the maximum power that can be extracted from the
generator. Due to the concavity of the Cp A curves, it is possible to calculate
the value of the TSR (Aopt) in which the rotor efficiency Cp is at its maximum.
Figure 2.5 shows the optimum TSR for a BPA of 0 where Aopt = 7.954.
Figure 2.5: Aopt at a blade pitch angle d 0
10


Given that Aopt is known, the maximum power that can be extracted from
the wind generator is
With equation (2.17) we may obtain an expression for the optimum shaft
speed where the power extraction is at its maximum [7]. Solving equation (2.17)
for u)m we obtain
This relationship allows us to compare the measured shaft speed with the
optimum shaft speed and determine the value of the maximum available power
at any given time.
In order to control the amount of power extracted from the wind turbine,
the machine torque must be adjusted. Adjusting the machines terminal voltage
magnitude |us| will adjust the electrical load torque using
where the stator resistance rs is neglected; Ls is the stator and rotor leakage
inductance; and uie = |-uim is the electrical shaft speed. This can be achieved
by adjusting the control signal dw of the DC/DC converter from
(2.17)
(2.18)
(2.19)
(2.20)
11


2.2 Solar Subsystem
The Solar subsystem consists of an array of solar panels and a DC/DC buck
converter as illustrated in figure 2.6.
dpv
Figure 2.6: Solar subsystem block diagram
The solar panels are formed by photovoltaic cells, which are p n semicon-
ductor junctions that convert solar energy into electricity. Figure 2.7 illustrates
the equivalent circuit of a photovoltaic cell.
------O
^ ph (3
f'ish U px
Figure 2.7: Solar cell equivalent circuit
12


The current extracted from the photovoltaic array is defined as
Ipy --- H-pZp/j Tlplrs
tpr + *pt R-s
nsAKT
(2.21)
where ns is the number of cells connected in series; np accounts for the modules
connected in parallel; ry is the output voltage at the PV terminals; q is the
charge of an electron; Rs is the intrinsic cell series resistance (for simplicity the
series and shunt resistance, Rs and Rsh are neglected); A is the cell deviation
from the ideal p n junction; K is the Boltzmanns constant; T is the cell
temperature.
Moreover, iph and irs are defined as
Iph Pscr
Zrft --- %T
~T_
Tr
1
e\ KA l7v
5 Too (2.22)
(2.23)
where kt is the short circuit current temperature coefficient; Tr is the cell refer-
ence temperature; iscr is the short circuit current at reference temperature and
radiation; S is the solar radiation in [inW/crn2]; irr is the reverse saturation
current; and Eq is the band-gap energy of the semiconductor used.
2.2.1 Solar Subsystem Operation
The dynamic, mathematical model of the solar system is:
v,
^ojpv
pv
VDC
T T
J-jpv ^pv
Vpv ~ ^
pv
^o_;
pv
(2.24)
(2.25)
'pv ^pv
where Lpv and Cpv are the solar system DC/DC converter pole inductance and
throw capacitance respectively. For steady state operation i0Jpv = 0 and iy, = 0.

13


Thus, the expression for the output current of the solar subsystem is:
i
O-PV
1
(lyj
(2.26)
(a) Typical I-V characteristic curves
(b) Typical P-V characteristic curves
Figure 2.8: I-V characteristic curves
In order to calculate amount of power injected by the photovoltaic array, the
solar irradiation S and the cell temperature T must be measured. The current
14


available from the solar panel is given by one of the I-V curves in figure 2.8
which depends on these parameters.
The power demanded to the solar subsystem is dictated by the following
reference:
Ppvief vdc(iii T i0-w)- (2.27)
In order to determine if the solar system is able to sustain the power de-
manded from equation (2.27) we need to know the maximum power that can
be extracted from the photovoltaic array. The power demand is driven by the
load demand and the battery charging current demand as illustrated in equa-
tion (2.27).
The maximum solar power available is calculated using a technique called
IncCond (Incremental Conductance) developed in [13]. This technique is an
improved algorithm of the conventional Perturb and Observe (P&O) method.
This technique is used to constantly track the maximum power available from
the solar system at any time. The photovoltaic array output power, ignoring Rs
and Rsh may be calculated using equation (2.21) as follows:
Ppv Upv^piph
VpyTlplj's
(2.28)
Given the characteristics of the I-V and P-V curves of the solar panel in
figure 2.8, the maximum power that can be extracted from the solar panels is
obtained from
5 Ppv
SVpv
5(ipvVpv) Si
pv
Sv
pv
Sv<
Vpv + ipv d.
pv
(2.29)
resulting in
p
P^max
5 ipv o Q
Vpv = n~kTA
Sv
2 fa_rE!L)
lrsVpue''kTA >
pv
(2.30)
15


(a) I-V Curves maximum available current
(b) P-V Curves maximum available power
Figure 2.9: PV Curves maximum values at 25C
Subsequently, the set of curves from figure 2.9, obtained using equa-
tion (2.30), show the maximum available power at any given time independent
of the solar irradiation S.
Once the maximum power for both systems is known, we are able to simulate
modes of operation as outlined in chapter 4. The following chapter prepares this
discussion by analyzing the individual operation of each subsystem described in
this chapter.
16


3. System Operational Contingencies Analysis
Power systems economics play a very important role in the engineering and
planning portion of the design. As mentioned before, power systems take into
consideration two major economical factors: fixed and variable costs. Fixed
costs account for interests on loans, returns to investors, fixed operation and
maintenance (O&M) charges, taxes, etc. Variable costs depend mainly on the
cost of fuel and actual O&M of the plant [9].
Power systems optimization aims at operating the system at the lowest pos-
sible cost without violating established constraints. For wind and solar systems
fuel expenses do not exist and therefore the economical analysis takes into ac-
count other variable costs such as O&M. For this reason, it is important to
analyze each systems individually to identify the operational disturbances that
could cause malfunctions and thereby reduce the mean time between failure
(MTBF).
3.1 Battery Potential Hazards
The proper operation of the batteries is critical to the performance of the
system. Batteries represent a significant part of the capital/O&M costs in the
entire system. In a study composed of a PV systems with energy storage, bat-
teries accounted for more than 40% of the capital cost [11]. For this reason it is
important to address the scenarios where the lifespan of the batteries could be
compromised.
Some of the scenarios that could potentially decrease the lifespan of batteries
are the following [14]:
17


discharging the batteries below the allowable percentage
overcharging the batteries
undercharging
extreme operating temperatures
Most of these scenarios are able to be reduced by properly managing the
state of charge of the batteries.
3.1.1 Batteries Discharging Mode
For the extent of this study, the DC bus voltage is the measuring point of
the battery voltage. This is illustrated in the battery equivalent circuit from
figure 2.1. When the batteries are committed, the following criteria must be
taken into consideration:
Here vdc is the voltage at the DC link, £ is a constant that represents the
allowable depth of discharge (typically 80%-90% of the nominal voltage [15]),
and ib is the battery discharge current. For this study the allowable depth of
discharge £ is set to 85%.
When the batteries are being discharged, the current demanded by the load
will dictate the discharge time.
3.1.2 Batteries Charging Mode
Similar to the discharge state of the batteries, the charging state of the bat-
teries could become a life-reducing factor for the batteries if these become over-
charged. The batteries specifications dictate the maximum amount of current
if vdc > £ eb,
if vDC < £ eb.
18


that can be injected into the batteries. This current is 4max and it is typically
dictated by the manufacturer.
Figure 3.1: Battery charge characteristic curve
From chapter 1 we know that the batteries charging current depends on
the batteries terminal voltage v^c following equation (2.2). During the early
charge stage, when the emf voltage is very low, the batteries require higher
currents to recover from the discharge event. The charging current is limited to
the maximum allowable current 4max during this period of bulk charging current.
Once 95% of the battery voltage is recovered, the batteries will continue to be
charged with the recommended trickle charge current. This method allows to
maintain the batteries fully charged at a constant current without overcharging
them. The trickle charge current is generally dictated by the manufacturer in
19


the batteries specifications. If a recommended trickle charge current is not
provided, a conservative value of ^ is used, where C is the capacity of the
batteries [16]. For example, if the batteries were rated as 100 Ahr, the trickle
charge current would be 1A. Figure 3.1 illustrates the charge characteristic of
the batteries.
With the previous information we may now determine a suitable charging
function for the batteries:
^frspec *
vpc-e b
Ri
C
100
(3.2)
4max if 0< vDC if eb Ri ibmax < Vdc < 95% Vdc,
if 95% vDC < vDC < vgas,
o if Vgas < Vdc,
where vgas is the gassing voltage. This is the voltage at which a battery will
begin to produce hydrogen and oxygen, which reduces the amount of water in
the battery [16]. This phenomenon causes the lifespan of the battery to be
shortened.
3.2 Wind Subsystem Potential Hazards
Wind turbines are currently designed to last around 20 years [17]. This
is achieved by diligent maintenance and operation of the turbine. The most
common damaging factors of wind turbines are stresses in the drive train of the
generator and overheating.
Stresses on the shaft of the generator are directly related to the torques that
are subject to fluctuation due to periodic and aperiodic processes such as [2]
changes in wTind speed;
20


tower shadow or tower-occasioned upwind overpressure;
blade asymmetry;
blade bending and skewing;
tower oscillation.
From an electrical point of view, we only consider the changes in wind
speed as a threatening factor from the list above since all other factors are
mechanical/structural-related. The maximum stress occurs at the surface of the
shaft when the radial position is at rest. This means that, in order to avoid
stress that could potentially reduce the lifespan of the wind turbine, it is desired
to maintain any speed greater than zero.
Another perturbation that reduces the life of the wind turbine is heat. This
phenomenon occurs on any PMSG due to the current flow through resistive
elements such as the armature windings. The armature winding losses, the core
losses, and the stray losses are frequency-dependant and are directly related with
the output current of the turbine. If this current is distorted with harmonics
produced by the passive rectifier, these will create more losses in the system.
An illustration of the relationship between harmonic currents Ia and the turbine
losses (heat generated in the turbine) is given by
00 OO 00
Pw\oss = 'y 1 ^Ooss = y y ^an-^ln ~ mlR-ldc y ^an^lRn (3-3)
n= 1 n= 1 n=l
where n = 5, 7, 11, ..., which represent the odd harmonics that are not
multiples of three. This expression is in terms of the frequency-dependent losses
21


of a generator with a passive rectifier attached downstream [18] similar to this
studys application.
Furthermore, if harmonics produced by the wind generator are not controlled
or suppressed, they could create damage in other components of the systems such
as the batteries, the solar panel, or even the load. Additionally, enhancing the
power quality in this manner would also increase efficiency.
3.3 Solar Subsystem Potential Hazards
The power output from a PV array is directly related to the solar irradiation
and the cell temperature. The PV system will output a DC current that is
essentially a current source with a few non-idealities. This means that this
current will be free of current harmonics for the most part. Additionally, the
output of the PV module is managed by the DC/DC converter that links the
PV array to the system.
There are potential events that will create a malfunction in the solar panel
such as hot spot, thermal cycling, moisture, or mechanical loads. These condi-
tions are not part of the systems operation and thus nothing that can be done to
prevent these threats by means of operation. There has been research regarding
hot-spot analysis and prevention [19]; however, so far a solution has not been
established prevent such phenomenon from an operational perspective. Alter-
natively, hot spot heating and the other mentioned threats may be prevented
from a fabrication perspective.
22


4. System Modes of Operation
The systems modes of operation vary over time with respect to the varia-
tions in the load demand. The system is analyzed in steady state, considering
the present value of the load at any given time not taking into consideration
previous or future load demands.
Because the load is repeatedly changing and the wind and solar energy
sources change as well, the nature of the system is to cycle. These cycles have
to do with natures day and night cycles: during the day, the solar energy
may be the predominant energy source and, similarly, the wind system may be
predominant during the night time. Furthermore, the battery will be in either
state of charge or disconnected. For this reason, the modes of operation may
be represented by a number that would represent the ON/OFF state of each
energy source.
For instance, 1 would represent the ON state when the unit is commit-
ted; 0 would represent the OFF state when the unit is uncommitted; and
using a special notation for the battery, 1 would represent its charging state.
Table 4.1 contains the proposed modes of operation using the format mentioned
above.
4.1 Operation Mode 1
This mode of operation corresponds to the unlikely situation in which the
wind and solar sources are off and the batteries are responsible for handling the
23


Table 4.1: Modes of Operation
Modes of Unit Combination
operation Pw Ppv Pb
Mode 1 0 0 1
Mode 2 0 1 -1
Mode 3 0 1 1
Mode 4 1 0 -1
Mode 5 1 0 1
Mode 6 1 1 -1
Mode 7 1 1 1
Mode 8 0 0 0
entire load. This mode of operation is best described as follows:
if

Ppv = 0
=> Mode 1
Wind Uncommitted, Pw = 0
Solar Uncommitted, Ppv = 0
Battery Committed, P), = vdc h
(4.1)
In this case the batteries are susceptible to fast discharge if the energy is
not managed properly using the procedures described in chapter 2.
If this mode remains for a long time, the batteries must be disconnected
for their protection and the power will not be restored until the wind and solar
sources become available and the batteries are completely recharged.
Once the monitoring equipment detects voltage at the DC link bus, the bat-
tery bank is immediately re-connected to be charged again. When the batteries
voltage raise above £udc the load may be restored using an appropriate load
24


shedding procedure.
4.2 Operation Mode 2
This mode of operation occurs when the solar panel is responsible of sup-
plying the load, including the load of the batteries when they are being charged.
The wind turbine is unavailable during this mode. This situation may be en-
countered in weather conditions with high solar intensity and a low wind speed
profile or if the wind turbine is disconnected for maintenance. Depending on
how the system is sized and the load shedded, this system may or may not be
functional.
This mode of operation is illustrated by the following definitions:
4.3 Operation Mode 3
This mode of operation will be encountered if mode 2 is in operation, and
the load increases above f^,, so that the battery may function as a regulator
for the power that the solar system cannot support Since the batteries are now
committed, the batteries must be protected against deep discharge as it is done
in mode 1. This is illustrated as follows:
if
Wind Uncommitted, Pw = 0
if
=> Mode 3 : < Solar Committed (MPPT), Ppv = (4-3)
Battery Committed, Pi, vpc ib
25


4.4 Operation Mode 4
Mode 4 is similar to mode 2. In this case the load is supplied by the wind
turbine. The batteries are part of the total load in this scenario since they are
being charged. The solar system is unavailable. This occurs at night time and
whenever the solar system is disconnected for maintenance. This scenario, as in
mode 2, may not be functional if the wind system is undersized or if the load is
not shedded properly.
^Vn wm0 Pt
Ppv = 0
=> Mode 4 :
Wind Committed, Pw = + ib)
Solar Uncommitted, Ppv = 0
Battery Charging, Pf, = voc h
(4.4)
4.5 Operation Mode 5
Mode 5 is an analogous representation of mode 3, where the load is being
supplied by the wind system and the batteries. This occurs if the load increases
above Pw while the system is in mode 4. Similar to mode 3, the batteries
function to compensate for the demand not served (DNS) by the wind turbine.
The batteries must be protected from being discharged excessively.
if
< WTTiot,t
P = 0
1 pv w
=$ Mode 5 :
Wind Committed (MWPG), Pw Kovt^m P,,-lo
Solar Uncommitted, Ppv = 0 (4-5)
Battery Committed, Pi, = vdc ib
4.6 Operation Mode 6
During this mode of operation the wind and solar systems work in conjunc-
tion to supply the load P& + Pl- This mode of operation is the most desirable
26


because there is more than one source available. Combining these sources en-
ables the opportunity to cover the demand with enough flexibility to apply
optimization techniques.
if
u"> ^ u"i0Pt I , c
> => Mode O :
- ^Plref J
Wind Committed, Pw = vdc{il + ib io.pv)
Solar Committed, Ppv = voc{iL + *6 io.w)
_ Battery Charging, P\, = vdc ib
(4.6)
4.7 Operation Mode 7
This is the mode of operation where all of the sources are committed. This
will occur if the load increases above + Pw while mode 6 is in operation.
The batteries will compensate for DNS during this mode. If the battery drops
to a value below the allowable depth of discharge then the entire system would
shutdown to protect the batteries.
{n I Wind Committed, Pw = Kaptw^ Pwtm
P / => Mode 7 : < Solar Committed, Ppv = T^vpv (4-7)
ax < 7pt>rf ) _ .
I Battery Committed, Pb = vdc lb
4.8 Operation Mode 8
This mode is encountered when the system is not in operation. This mode of
operation illustrates an initial condition where all components are off. This could
also occur after a forced shutdown occurs. For instance, this mode would be
committed if the batteries voltage drops below the allowable depth of discharge
£ in modes 3, 5, and 7.
27


5. Problem Formulation
The base for the formulation of the problem is operation mode 6 since it
provides the most flexibility in terms of the power distribution to the load.
During this scenario, as mentioned before, both sources (wind and solar) are
committed to fulfill the load. In this case, the power extraction from both
sources depends on the load demand. The supply of the load may be achieved
with an infinite number of combinations of energy drawn by the two sources;
however, some combinations may not be optimal.
In order to create a mathematical formulation, it is necessary to understand
the systems behavior in steady state. It is also important to know the reaction
of the different components of the system to certain pollutants such as the
harmonics injected into the system or surges as explained in chapter 3. These
harmonics are created from the power electronics attached to each source and
the load.
As described in chapter 3, the wind turbine is affected by the constant
additional heat produced by the current harmonics. This heat creates damage
in the components of the generator in the long run. Minimizing these harmonics
will also minimize the damage to the wind turbine. Additional benefits are
obtained from minimization of the harmonics such as reducing the losses from
the generator and enhancing efficiency. It is not in the scope of this project
to analyze the efficiency of the generator, but it is important to mention the
achieved benefits.
28


The wind turbine is directly connected to a passive three phase rectifier.
The output of this converter is uncontrolled, and therefore the analysis is done
downstream of the rectifier. The DC current from the rectifier is then controlled
using a buck DC/DC converter.
The output current of the wind subsystem is a DC current previously defined
in equation (2.26). This current may contain current harmonics generated by
the DC/DC converter and the wind generator resulting in produced heat at the
generator core and stator windings. For this reason it is important to quantify
the amount of current harmonics in the output current waveform.
5.1 Total Harmonic Distortion Index
Various harmonic components of a periodic waveform of the form /(f) with
a period 7r may be readily conducted using a Fourier analysis of the waveform
based on time domain integration [20]. This is because a general noil-sinusoidal
waveform f(t) repeating with an angular frequency ojsw can be expressed as [21]
and Xa(n) and X^n) are respectively, the odd and even components of the har-
monic components:
For our study the average value from equation (5.2) is non-zero because it
the waveform is lion-symmetrical. The expressions of the odd (cosine) and even
n=1 n=l
where F0 = |a0 is the average value defined as
(5.2)
xa(n) = ^ Jo f{t) cos{nujswt) dt 71 = 0, 1, . OO
xb(n) = l Jo* f(t) Sin{nuswt) dt n = 0, 1, ..., oo.
(5.3)
29


(sine) components of the nth harmonic of the wind system output current io w
are
W = i jfmT" io.w cos(naW) dt =
isin = jt SqwTsw io.w sm(nu>swt) dt = iQ_w cos(2/wnd<") )
(5.4)
where TSU) is the switching period and cjsui is the angular frequency defined as
kSSW 27T F'sw
(5.5)
where
fstu = (5.6)
J 5U>
The complex phasor form of the nth harmonic of io w, for all positive integer
values of n is
lWvect *cos Rsin
(5.7)
Due to symmetry, if the function is odd, the odd component of the nth
harmonic will be zero for all values of n. If the function is even, it would
similarly result in zeros for the even component of the nth harmonic for all
values of n.
Moreover, the magnitude of the complex phasor from equation (5.7) is
I Wet I = \j + *L = sin(7rn4,). (5.8)
The amount of distortion in the voltage or current waveforms is quantified by
means of an index called the Total Harmonic Distortion (THD) [21]. This THD
index is defined as the ratio of the RMS of the harmonic content to the RMS
value of the fundamental quantity, expressed as a percent of the fundamental [22]
OC 9
E |w
THDiw{x) = ^
n1_______
Nvect.oP
(5.9)
30


After substituting the correspondent variables and constants, the expression
becomes
THDlw(x)
(5.10)
5.2 Mathematical Optimization Model
One of the objectives of the optimization problem is to minimize the amount
of harmonics produced by the wind generator ( THDiw). Now that these har-
monics have been quantified and represented by a mathematical expression, it
is possible to formulate the optimization problem.
It should be noted that the expression of the THD in equation (5.10) as well
as the equality constraints and some of the inequality constraints are 11011-linear.
Furthermore, the battery system requires the charging and discharge current
to match the manufacturers charging and discharge recommended parameters
as closely as possible. This means that ideally it, = f&spec. In order to ensure such
expression to be fulfilled, the absolute or squared residual could be minimized
and become a second objective function.
31


The resulting optimization model is given as
E
2
sin( nndw)
7T71
~dT
miux h(x)= THDtJ =^~
lllinj; f2{x) [^bsper *b] = [^bspec ~ ^L *OJpV
subject, to
La-4 M jj -I- CJm<^>T7
llq 2uJm ' 2 L
0 < i 0 < h < ^9max
0 < We < wemx Upper bound
0 < dw < 1 and
0 < ipv < *P> mx 1 lower bound
0 < Vpv < ^Wmax inequality
0 < dpi? < 1 constaints
0 < ib <
0 < Vdc < Vgaft <
KVDCtQ i -n
La. 4 -L f.i 4 ^ypcid_1_ n
L Id + 2 Umtq 3V5l^-5_^ ^ U
_3£a j 1 CfPAV3 _0
2 2( -Cp + i [116 (i 0.035) 5] e21(^- 035) = 0
-A + = 0
-to
-io.w + (2^^) d! 0 ^
V [npiph TipirB (e(X) ljj =0
iph + [tscr + Tr)] = 0
-t + trr [£]3e(^[^~*]) =0
Wp-WDCaJ-=0
-2,
OJpV + ipv d
= 0
Wind
steady state
equations
Solar
steady state
equations
iL+ib- io^pv ~ io.n, = 0 } Power balance equation
vdc -eb- ibB.i = 0 } Battery model equation
(5.11)
32


The vector containing the problems design variables is
x =
id.
iq
dw
i-pv
Vpv
dpv
ib
VDC
(5.12)
where the actual decision variables are the control signals of the DC/DC convert-
ers dw and dpy. However, these are also interrelated by the rest of the variables
in vector x.
Note that the objective function is non-linear, and that the model includes
both linear and non-linear constraints. The upper and lower bound constraints
of the form lb < x practical and feasible values within the capabilities of the system. Similarly, the
non-linear equality constraints are necessary for maintaining the power balance
in the system.
The inputs to the problem are measurements of different parameters that
dictate the operation mode of the system. The vector containing the problems
inputs is denoted by
V
T
u =
S
(5.13)
e-b
33


6. Problem Solution
The problem formulated in chapter 5 is a multi-objective optimization prob-
lem. These type of problems typically have conflicting objectives, such that a
gain in one objective may negatively affect the other objective(s). Therefore the
definition of optimality is not obvious. For this reason the decision maker must
choose a solution based on experience or trade-off analysis.
The solution to these problems is approached using Pareto optimality tech-
niques such as the weight method or the e-constraint method. Using these
techniques, the problem may have an infinite number of efficient points that
constitute a Pareto front curve (also called Pareto frontier or Pareto set) as
illustrated in figure 6.1a.
The weight method is generally used when the objective functions form a
convex Pareto frontier; the objective functions are not always required to be
convex in order to create convex Pareto frontier. However, a multi-objective
optimization problem is said to be convex if the feasible objective region is convex
or if the feasible region is convex and the objective functions are quasiconvex
with at least one quasiconvex function [23]. If the functions are non-convex then
the e-constraint method is the preferred method between the two.
The two methods mentioned above are solved by scalarization. This means
that the multi-objective problem is converted into a single objective function to
find a solution that will minimize this single objective function while maintaining
the physical constraints of the system [23].
34


(a) Convex Pareto objective space
Figure 6.1: Pareto
(b) Non-convex Pareto objective space
frontiers illustration
Furthermore, the solutions obtained are normalized to be consistent with
the weights assigned to the objective functions. The functions are normalized
using the form
0 < Mn] Ziu < 1
ZiN-ZiU
(6.1)
where fi{x) is the ith objective function; zf is the lower bound (Utopia point)
of the Pareto set, normally infeasible because of the conflicting nature of the
individual objectives; and zf is the upper bound of the Pareto optimal set,
obtained from the components of the Nadir point zN [24].
35


6.1 Solution Approach
For this systems study, both objective functions were evaluated using the
weight method, which resulted in a convex feasible objective region of the shape
shown in figure 6.1a. The weighted objective function is of the form
f(x) = Wlf1(x) + W2f2(x), (6.2)
where
£
Mx) = THD,(xf =^i-
irn
sin(7rndU))
dl
f2{x) = A ib
1
lbs.
1>L T ipv j T
pv
7r
2\/3
v^+3)z:l
and W\ 0, 0.01, 0.02, ..., 1 and \V2 1 \\\.
(6.3)
(6.4)
The problem was solved using MATLABs fmincon function. The code is
found in Appendix B and a description of the fmincon function is given in
appendix C. The problem is initialized by means of a warm start iteration since
the problem solution does not change significantly from one iteration to another.
This process consists of using the solution of the previous iteration xr~l to set
the initial conditions for the new iteration r.
As mentioned in chapter 5 the base for the formulation of the problem
focuses on mode 6 since it provides the most flexibility in terms of the power
distribution to the load and the batteries. Based on this presumption three
scenarios were analyzed:
Scenario 1: i0_w + i0Jpv > iL + ib where iQ_w 36


Scenario 2. io_w T k-pv ^ fl T fb wliere k-w ^ fz, T k nncl io^pv ^ fz, T fbi
Scenario 3: i0_w + i0_p > z'l + 4 where z0_, + 4;
where iOJW and io pil respectively are the wind and solar maximum available
output currents. The voltage of the batteries determines the amount of charging
current demand and therefore each scenario is analyzed with three different
voltage levels:
Completely discharged > 85% of nominal voltage;
Partially charged - 92% of nominal voltage;
Fully charged > 100% of nominal voltage.
6.1.1 Scenario 1
Table 6.1 includes the systems inputs used to model Scenario 1. The Pareto
sets for this scenario are shown in figure 7.5. Since the feasible region is convex
the problem is suitable to be solved by the weighted method. The convex shape
of this Pareto frontier illustrates the conflicting nature of the two objective
functions for this scenario. If we choose to improve A if,, THDiw would be
worsened; similarly, if THDiu, is improved, A % is worsened.
Table 6.1: Scenario 1 Inputs
Scenario V [m/s] S [mW/cm2] T [K] eb [Volts] k [Amperes] ^bpec [Amperes] Iqjw |Amperes] to.pv [Amperes]
7 60 298 40.8 30 12.0 22.13 20.94
1 7 60 298 44 30 6.66 20.55 19.44
7 60 298 48 30 4.0 18.84 17.81
37


I et = 40.8 Volts!
fi(x) = Ai2 [Amperes2
(a) f\{x) vs f'/i-x) for ej, = 40.8 Volts
cs
J!:
C
Cl
h.
E no
c
/2 (c) /i(.x') vs /2(.t) for et = 48 Volts
Figure 6.2: Scenario 1 Pareto set
38


(a) Te as a function of dw
(b) io.w as a function of dw
Figure 6.3: Te and i0_w as a function of dw for different wind velocities V
39


Figure 6.4: fi{x) as a function of dw
This conflict exists because THDiw is minimized when dw = 100% as shown
in figure 6.4. Furthermore, figure 6.3 shows that the output current of the wind
system iow and the electrical torque Te approach zero as dw approaches one.
Since the solar system is not able to supply the load independent of the wind
system, iow must be greater than zero in order to satisfy which creates a
conflict between both objectives.
For this scenario minimizing Aq, is more critical than minimizing THDiw
because the power balance equality constraint II + h io.w io-pv = 0 must be
satisfied. For this reason, more weight should be given to A % when choosing
an optimal solution. Giving more weight to THDiw would expose the batteries
to undercharging. Considering the fact that the batteries could represent up to
40% of the total cost of the system [11], the opportunity cost of giving more
weight to Aib is well justified.
40


6.1.2 Scenario 2
The inputs used to simulate scenario 2 are included in table 6.2. Figure 6.5
shows the Pareto set for scenario 2, which forms a convex Pareto frontier. Similar
to scenario 1 the shape of this curve indicates a conflict between THDiw and
A ib.
Table 6.2: Scenario 2 Inputs
Scenario V [m/s] S |m\V/cm2] T IK] e-b [Volts] h [Amperes] ^6pec [Amperes] io.w [Amperes] lojpv [Amperes]
9 60 298 40.8 20 12.0 47.36 20.94
2 9 60 298 44 20 6.66 43.42 19.44
9 60 298 48 20 4.0 39.86 17.81
In this case THDiw is significantly less than the results obtained in scenario
1. This is because the amount of available wind power is large so the control
signal dw does not need to be as small as in scenario 1. This situation allows
the weight of THDiw to be greater than the previous scenario, thus improv-
ing THDiw. Nevertheless, Aib requires more weighing factor to ensure proper
charging of the batteries.
41


0.1 02 02 0.4 0.5 0.6 0.7 0.8 0.9
h(x) = A[Amperes2]
(a) f\{x) vs h(x) for eb = 40.8V Volts
(b) fi(x) vs f2{x) for eb = 44 Volts
(c) /i (x) vs f2(x) for eb = 48 Volts
Figure 6.5: Scenario 2 Pareto set
42


6.1.3 Scenario 3
This scenarios simulation inputs are depicted in table 6.3. In this case the
objectives do not conflict with one another. This is because both objectives may
be minimized at the same time since io pv > il + b>, allowing THDiw and Aib to
be minimized at all times.
Table 6.3: Scenario 3 Inputs
Scenario V [m/s] S [mW/cm2] T IK] et [Volts] U [Amperes] [Amperes] lo.w [Amperes) 2o_pr [Amperes]
5 100 298 40.8 20 12.0 8.11 36.50
3 5 100 298 44 20 6.66 7.52 33.83
5 100 298 48 20 4.0 6.89 31
Since the objectives do not conflict with one another, for all cases when the
solar system is able to fulfill the entire load independent of the wind system,
the load will be supplied by the solar system only by setting dw 100%. This
operation strategy ensures both functions to be minimized. Using the parame-
ters established in table ?? we obtain the results from table 6.4. It is important
to note that even though the current is very small, the rotor shaft speed is not
zero. This is because there is available power from the wind turbine that is not
being used. It is not desired to halt the machine because stress occurs at the
surface of the shaft if the machine is started when the radial position is at rest
as mentioned in chapter 3.
43


Table 6.4:
Scenario 3 Formulation Results
X (et, = 40.8 Volts) (et = 44 Volts) (et, 48 Volts)
id [Amperes] 0.00 0.00 0.00
iq [Amperes] 0.1 0.16 0.26
ujm [rad/s] 6.32 6.73 7.31
dw [%] 100 100 100
ipv [Amperes] 16.14 16.27 16.28
Vpt, [Volts] 82.79 72.70 70.572
dpv [%] 50.6 61.3 68.5
if, [Amperes] 12.00 6.66 4.00
Vdc [Volts] 41.90 44.61 48.36
44


7. Discussion
Different scenarios were analyzed to verify the performance of the optimiza-
tion formulation. These scenarios are evaluated using high, average, and low
values of various systems inputs. The combinations are shown in table 7.1.
The cell temperature T is fixed to 298 K (25C) for all scenarios and the objec-
tives respective weights are \\\ = 0.05 and W2 = 0.95, giving more weight to
A if, as explained in chapter 3. Since the number of evaluations resulting from
these scenarios is quite large only a few scenarios were illustrated in this section.
Nevertheless, the results of all evaluations correspondent to mode 6 of operation
are shown in appendix D.
It is important to note that all cases from table 7.1 where i0_w + io.pv < tL + h
do not correspond to mode 6 of operation and therefore are ignored. Table 7.2
shows all scenarios corresponding to mode 6 represented by 6 and all other
modes of operation by X.
Tab le 7.1: Evaluated Scenarios
Case V S eb lL
Study [Volts] [mW/cm2] [Volts] [Amperes]
LOW O 5 20 40.8 10
AVERAGE( = ) 7 60 44 30
HIGH n 10 100 48 60
45


Table 7.2: Scenarios Corresponding to Mode 6
V V V
s 5 s s s S S S S
4 4 X 6 6 6 6 6 6 6 6
4 X X 6 X 6 6 6 6 6
4 X X X X X X X 6 6
4 4 X 6 6 6 6 6 6 6 6
4 X X 6 X 6 6 6 6 6
4 X X X X X X X 6 6
4 4 X 6 6 6 6 6 6 6 6
4 X X 6 X 6 6 6 6 6
4 X X X X X X 6 6 6
Since the main objective of any electric power system is to serve the load,
our system is mostly sensitive to changes in the load demand, more than any
other system input. For this reason, it is important to illustrate the system
response to changes in the load demand.
In figure 7.1 the objective functions show no conflict while io pv > + 4-
However, when the load increases so that io w < II + ib and 4_pu < 4 + 4
then THDiw begins to increase and A4 remains at zero. These characteris-
tics are consistent with the control signals behavior against the load demand
increments as shown in figure 7.2. Moreover, the wind turbines direct and
quadrature axis currents approach zero as dw approaches one, and start increas-
ing as dw decreases. The same behavior is expected from the rotor shaft speed
46


u>e, corroborating the results described in chapter 6 and table 7.2.
The solar systems current changes abruptly with a small change on the
control signal dpv. The voltage vpv is high when the current demand is low and
falls to the voltage which maximizes the power output of the solar system once
the demand increases to that level. This is illustrated in figures 7.2b and 7.4.
0.05
0.04
f. i, 0.03

5. 0.02
< 001
^-o o
<

II
-0.02
H
'~7^ -0.03 L

-004 -
-o.os >
ii, [Amperes]
(b) Aib vs iL
Figure 7.1: Objective functions variation with respect to il
47


Furthermore, the batteries current % and voltage Vdc remain constant since
the batteries are assumed to be fully charged for this case study and Aif, = 0
for all values of Ll-
110
100 t-
M
00
70
^ 60
a
40
30
20
10
0 >
il [Amperes]
(b) dpv vs iL
]
J
J
Figure 7.2: Control signals variation with respect to II
48


(b) iq vs iL
(c) u)m vs iL
Figure 7.3: Wind system design variables variation with respect to iL
49


[Amperes]
(a) ipv vs iL
Figure 7.4: Solar system design variables variation with respect to ii
50


(a) ib vs iL
it [Amperes]
(b) vDC vs iL
Figure 7.5: Battery system design variables variation with respect to ii
51


8. Conclusion
This study has analyzed a stand-alone hybrid wind/solar system with energy
storage. The subset of systems has been thoroughy researched and evaluated in
order to model the system accurately. Special attention was given to the sen-
sitivity of the equipment against potential operational disturbances in order to
determine and protect against any life-shortening operational conditions. Eight
modes of operation were defined and mode 6 of operation was analyzed in detail
because this mode of operation is the most suitable for optimization.
The optimization was formulated as a multi-objective function with linear
and non-linear constraints. Given that the objective space formed a convex set,
the problem was solved using the weighted objective function method.
Three scenarios from operation mode 6 were analyzed. These scenarios
result in conflicting objectives when the solar system may not supply the loads
independent of the wind system. When the solar system may supply the loads
without contribution from the wind, the load is supplied by the solar system
only and no optimization is required.
This studys research demonstrate that the lifespan of the equipment may be
enhanced by reducing the operational disturbances in the system that shorten
the equipments lifespan. The results presented in this study demonstrate that
these operational disturbances may be reduced using the solutions implemen-
tation and therefore the proposed formulation extends the lifespan of the equip-
ment. This achievement is valuable because it provides an economical benefit
52


to the systems owner. This study proposes concepts that could be introduced
in the economic dispatch optimization of wind and solar REEPS.
Moreover, wind and solar REEPS are typically part of the base load of the
system in larger-scale power systems, which enables the flexibility to perform
optimization techniques similar to those applied in mode 6 of this study. How-
ever, the application of these formulations into an actual system depends heavily
on the proper operation of a custom-designed control system.
Finally, further improvements may be done by designing and implementing
a robust controller that would merge the concepts introduced in this study along
with the model referenced in [7]. Additionally, a real-life implementation of the
system would provide more accurate results using real-time information and
measurements.
53


APPENDIX A. Simulation Parameters
Battery system:
C = 40 A-hr
e;, = 2.0 Volts
*iwx = 7 AmPs
mc 6
mp = 10
ms = 4
R, = 0.022 il
vgas = 2.4 Volts/cell
^ = 85 %
Wind system:
Aw 10.636 m2
Cp( V) = 0.382 at Aopt = 7.954
L/j = 3.55 mH
P = 28 Poles
rs = 0.3676 Q
R=1M m
i? = 0 degree
Aopt = 7.954 at d = 1
p = 1.225 kg/m3
(f>m = 0.2867 Wb
54


Solar system:
A = 1.60
Eg = 1.10 Volts
iscr = 3.27 Amps
irr = 2.0793x10-6 A
k = 1.3805xl023 Nrn/K
kt = 0.0017 A/C
np = 5
ns = 200
q = 1.6xl0~19 C
Tr = 301.18 K
System initial conditions:
id = 1.61412
iq = 11.395
ujm = 20
dw = 0.3
ipv = 15.436
Vpv = 96.00
dpv 0.3
ib = 10
vdc 48
55


APPENDIX B. Computational Code
B.l Scenarios Generator
'/.Scenario parameters
v = [5 7 10] ;
v_text = strvcat(v.LOVP v_hVG, v.HIGH);
S_pv = [20 60 100];
S_pv_text = strvcat(,S_pv_L0W>, ,S_pv_AVG> S_pv_HIGHJ);
T = [273 273+25 273+50];
T_text = strvcat(T_L0W, T.AVG, 'T_HIGH);
eb_par = [0.85 0.9167 1];
eb_par_text = strvcat(,e_b_L0W, 'e.b.AVG', 'e_b_HIGH');
i_L = [10 30 60];
i_L_text = strvcat(i_L_L0W', 'i_L_AVG\ iJL_HIGH);
'/.Variables call function
S = Mejia_variables(S);
'/.Warm start initial conditions
S.Xin(l,:)
S.Xin(2,:)
S.Xin(3,:)
S.Xin(4,:)
S.X_init
S.X_init;
S.Xin(l,:)
S.Xin(2,:)
S.Xin(3,:)
S.XinCl,:)
iter = 0;
count = 3;
for a = 1:1:3
for b = 1:1:3
for d =
for
1:1:3
e = 1:1:3
S.Wind.v = v(a);
S.Solar.S_pv = S_pv(b);
S.Solar.T = 298;
S.Batt.SOC = S0C(d);
S.W1 = 0.05;
S.W2 = 1-S.W1;
S.i_L = i_L(e);
S = Mejia_variables(S);
S.X_init = S.Xin(count-2,:);
S = Mejia_Solve_0PT(S);
iter = iter+1;
ss(iter,:) = [strvcat(v_text(a,:))
strvcat(S_pv_text(b,:))
strvcat(eb_par_text(d,:))
56


strvcat(i_L_text(e,:)) ];
flag(iter,l) = S.exitflag;
xhistCiter,:) = S.X;
id = S.X(l);
iq = S.X(2);
we = S.X(3);
dw = S.X(4);
ipv = S .X(5);
vpv = S.X(6);
dpv = S.X(7);
ib = S.X(8);
Vb = S.X(9);
io_w = pi/(2*sqrt(3))*sqrt(iq~2+id~2)/dw;
io_pv = ipv/dpv;
x = [] ;
for ii = 1:1:50
x(ii) = (2*io_w*sin(pi*ii*dw)/(pi*ii))~2;
end
fl(iter,l) = (sum(x)/(2*dw*io_w)~2);
f2(iter,l) = ((S.Batt.ib_spec-ib)~2);
S.Xin(count+l,:) = S.X;
count = count+1;
end
end
end
end
B.2 Optimization Solving Function
function S = Mejia_Solve_OPT(S)
Slower bound
LB = zeros(9,1);
"/.upper bound
UB = [Inf;Inf;Inf;1;Inf;Inf;1;S.Batt.ib_max;S.Batt.Vgas];
"/.Initial conditions
X_init = S.X_init;
"/.Linear inequality constraints
A = [];
b = [];
"/.Linear equality constraints
Aeq = [] ;
beq = [] ;
"/, Defines optimset
OptOpf = optimsetCDisplay,iter,Diagnostics,on....
DerivativeCheck,on,LargeScale,off,GradConstr,...
57


'off','GradObj','off','Hessian','off' ,'Maxlter',1000,...
'MaxFunEvals',1000);
[X,fval,exitflag,output,lambda] = ...
fmincon(@Mejia_0F,X_init,A,b,Aeq,beq,LB,UB,@Mejia_NLC,OptOpf, S);
if exitflagcl
return
end
S.exitflag = exitflag;
S.X = X;
B.3 Variables Definition Function
function S = Mejia_variables(S)
"/.System initial conditions
S.X_init_id
S.X_init_iq
S. X_init_we
S.X_init_dw
S.X_init_ipv
S.X_init_vpv
S.X_init_dpv
S.X_init_ib
S.X_init_Vb
= 1.6412;
= 11.3951;
= 479.7681;
= 0.22;
= 15.436;
= 96;
= 0.3;
= 10;
= 48;
S.X_init = [S.X_init_id;
S.X_init_iq;.
S.X_init_we;.
S.X_init_dw;.
S.X_init_ipv;
S.X_init_vpv;
S.X_init_dpv;
S.X_init_ib;.
S.X_init_Vb];
0//0/0//0/0/0/0/0/0/0/0/0/0//0/0/0/0/0/0//0/0/0/0/0/0/0/0/0/0/0/0/0/0/O/O/O/O/O/O/0/0/0//0/0/0/0//0/0/0/0/0/0/0/0/0/0/0//0/0/0/0/0/0/0/0/0/0/
/o /o /o /p / /o /o /o /o /o /o /p /o /o /o /o /p /o /o /o /o /o /o /p /o /o /p /p /p /p /p /o /p /o /p /p /p /p /p /p /p /p /p /p /p /p /p /p /p /p /p /P /p /p /p /p /p /P /o /p /P /p /P /p /P /p /o /p /P /p /p /P /p /p
/p /p /p /o /p /p /p /p /p /p /o /p /p /e /p /p /p /p /p /o /p /p /p /p /p /p /p /p /p /o /p /p /o /o /o /o /o /p /o /o /p /p /o /p /p /o /p /p /p /p /p /p /o /p /p /p /p /p /p /p /o /p /p /o /o /p /o /p /p /p /p /p /o
0/ 0/ P/ 0/ P/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/
/o /p /p /p / /o /p /o /p / /p /p /o /p /p /o
Battery system parameters
0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/
/o /o /o /o /p /p /p /p /p /p /p /p /p /o /p /P /p /p /o /p /P /o
0/ 0/ 0/0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ P/0/ 0/ 0/ 0/ 0/ o/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ / 0/ o/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ / 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/
/o /p / /o /o /o /p /o /o /o /o /p /o /p /o /p /p /p /p /p /p /P /p /p /P /p /p /p /p /p /p /p /p /o /o /o /o /p /p /p /p /p /p /o /p /p /p /p /o /P /p /p /p /o /P /P /p /o /o /p /o /p /p /o /o /p /P /p /o /o / /p / /o
0/ P/ 0/ 0/0/0/ 0/ 0/0/0/ 0/ 0/ 0/ 0/ 0/ 0/0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/
/o /o /o /o /o /o /o /o /o / /o /o /o /o /o /o /o /o /o /o /o /o /p /o /o /o /o /o /o /o /o /o /o /p /p /o / /p /o /p /o /o /p /o /p /o /o /o /o /o /p /o /o /o /o /p /p /o /o /p /o /o /p /o /o /o /p /p /o /p /o /o /o /p
'/.Battery system constants
58


S.Batt.num_p = 10;
S.Batt.num_s = 4;
S.Batt.num_cell = 6;
S.Batt.eb = S.Batt.eb_per*(2*S.Batt.num_s*S.Batt.num_cell);
S.Batt.Vgas = 2.4*S.Batt.num_s*S.Batt.num_cell;
S.Batt.RI = 0.023*S.Batt.num_s;
S.Batt.C = 40;
S.Batt.ib_trickle = S.Batt.OS.Batt.num_p/100;
if S.Batt.C*S.Batt.num_p*0.2*(1-S.Batt.eb_per)>S.Batt.ib_trickle
S.Batt.ib_max = S.Batt.OS.Batt.num_p*0.2*(1-S.Batt.eb_per);
else
S.Batt.ib_max = S.Batt.ib_trickle;
end
if 0< = S.Batt.eb && (48-S.Batt.ib_max*S.Batt.RI)>S.Batt.eb
S.Batt.ib_spec = S.Batt.ib_max;
elseif (48-S.Batt.ib_max*S.Batt.RI)< = S.Batt.eb && 48*0.95>S.Batt.eb
S.Batt.ib_spec = (48-S.Batt.eb)*S.Batt.num_p/S.Batt.RI;
elseif 48*0.95< = S.Batt.eb && S.Batt.Vgas>S.Batt.eb
S.Batt.ib_spec = S.Batt.ib_trickle;
elseif S.Batt.eb> = S.Batt.Vgas
S.Batt.ib_spec = 0;
end
o/ y#/ V y y v y y y yyy y y y y y y y y y y y y y y y y y y y y y y y y y#/ y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y
to A A A A AAA A A A A A A A AA AA A A A A A A A A A A A A A A A /o A /o A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A
0/ 0/ 0/ 0/ Of 0/ 0/ 0/ 0/ Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of Of
A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A
mnmnmm
Solar system parameters
mmmnrammn
0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/0/ 0/ 0/ 01 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0.
'o to 10 to 10 /o to !o to to to 10 to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to
%Solar system variables
l S.Solar.T = 25+273;
% S.Solar.S_pv = 100;
7,Solar system constants
S.Solar.n_p = 5;
S.Solar.n_s
S.Solar.q =
S.Solar.A =
S.Solar.B =
S.Solar.k =
200;
1.6*10''19;
1.6;
S.Solar.A;
1.3805*10~-23;
S.Solar,I_scr = 3.27;
S.Solar.k_t = 0.0017;
S.Solar.I_rr
S.Solar.T_r = 301.18;
S.Solar.E_G = 1.10;
2.0793*10~-6;
o to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to
59


0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 9/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 9/ 0/ 0/ 0/ / 0/ 0/ 0/ 0/ 0/ 9/ 0/ 9/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 0/ 9/ 0/ 0/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 0,
9/ 9/ 9/ 9/ 9/ 9/ 9/9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/
A A /O A AA A /O A A A A A A A A
Wind system parameters
rarammmnra 7.7.
9/ 0/0/9/0/9/9/0/9/9/9/0/9/9/0/9/0/0/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 0/9/9/9/0/9/9i
9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/9/ 9/ 9/9/ 9/ 9/9/ 9/ 9/ 0/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 0/9/ 9/ 9/ 9/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 0/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/ 9/9/ 9/ 9/9/
A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A AAA A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A
%Wind system variables
l S.Wind.v = 14;
/ S. Wind, we = 479.768;
7,Wind system constants
S.Wind.rs = 0.3676;
S.Wind.L = 0.00355;
S.Wind.phi = 0.2867;
S.Wind.rho = 1.225;
S.Wind.BPA = 1;
S.Wind.P = 28;
S.Wind.Cl = 0.5;
S.Wind.C2 = 116;
S.Wind.C3 = 0.4;
S.Wind.C4 = 0;
S.Wind.C5 = 5;
S.Wind.C6 = 21;
S.Wind.y = 2;
S.Wind.r = 1.84;
S.Wind.A = pi*S.Wind.r~2;
S. Wind. v_cutin = 3;
S.Wind.v_cutout = 20;
B.4 Objective Function
function f = Mejia_0F(X,S)
7Decision variables
S.Wind.id = X(l);
S.Wind.iq = X(2);
S.Wind.we = X(3);
S.Wind.dw = X(4);
S.Solar.ipv = X(5);
S.Solar.vpv = X(6);
S.Solar.dpv = X(7);
S.Batt.ib = X(8);
S.Batt.Vb = X(9);
7Wind system output current
io_w = pi/(2*sqrt(3))*sqrt(S.Wind.iq~2+S.Wind.id~2)/S.Wind.dw;
7,Solar system output current
io_pv = S.Solar.ipv/S.Solar.dpv;
7,Sum of the vectorized wind system current
60


X = [] ;
for ii = 1:50
x(ii) = (2*io_w*sin(pi*ii*S.Wind.dw)/(pi*ii))~2;
end
7,Objective functions for weighted objectives
fl = (sum(x)/(2*S.Wind.dw*io_w)~2);
f2 = (S.Batt.ib_spec-S.Batt.ib)~2;
f = S.Wl*fl + S.W2*f2;
B.5 Non-linear Constraints Function
function [C, Ceq] = Mejia_NLC(X,S)
'/.Decision variables
S.Wind.id = X(l);
S.Wind.iq = X(2);
S.Wind.wm = X(3);
S.Wind.dw = X(4);
S.Solar.ipv = X(5);
S.Solar.vpv = X(6);
S.Solar.dpv = X(7);
S.Batt.ib = X(8);
S.Batt.Vb = X(9);
'/.Wind system parameters
S.Wind.lambda = S.Wind.r*S.Wind.wm/S.Wind.v;
S.Wind.lambda_ref = 1/(S.Wind.lambda+0.08*S.Wind.BPA)
0.035/(S.Wind.BPA~3+1);
S.Wind.Cp = S.Wind.Cl*(S.Wind.C2*S.Wind.lambda_ref-...
S.Wind.C3*S.Wind.BPA-S.Wind.C4*S.Wind.BPAS.Wind.y-...
S.Wind.C5)*exp(-S.Wind.C6*S.Wind.lambda.ref)+0*0.0068*S.Wind.lambda
io_w = pi/(2*sqrt(3))*sqrt(S.Wind.iq~2+S.Wind.id~2)/S.Wind.dw;
'/.Solar system parameters
S.Solar.i_rs = S.Solar.I_rr*(S.Solar.T/S.Solar.T_r)~3*exp(S.Solar.q*...
S.Solar.E_G/(S.Solar.k*S.Solar.A)*(1/S.Solar.T_r-1/S.Solar.T));
S.Solar.i_ph = (S.Solar.I_scr+S.Solar.k_t*(S.Solar.T-S.Solar.T_r))*...
S.Solar.S_pv/100;
io_pv = S.Solar.ipv/S.Solar.dpv;
'/.Inequality constraint
C = [];
'/.Equality constraints
Ceq = [-S.Wind.rs*S.Wind.iq/S.Wind.L-...
61


S.Wind.P/2*S.Wind.wm*S.Wind.id+...
S.Wind.P/2*S.Wind.wm*S.Wind.phi/S.Wind.L-...
pi*S.Batt.Vb*S.Wind.iq/(3*S.Wind.dw*...
sqrt(3)*S.Wind.L*sqrt(S.Wind.id~2+S.Wind.iq~2));...
-S.Wind.rs*S.Wind.id/S.Wind.L+...
S.Wind.P/2*S.Wind.wm*S.Wind.iq-pi*...
S.Batt.Vb*S.Wind.id/(3*S.Wind.dw*sqrt(3)*S.Wind.L*.
sqrt(S.Wind.id~2+S.Wind.iq'2));...
-3*S.Wind.P*S.Wind.phi*S.Wind.iq/4+S.Wind.Cp*...
S.Wind.rho*S.Wind.A*S.Wind.v~3/(2*S.Wind.wm); . .
S.Solar.ipv-(S.Solar.n_p*S.Solar.i_ph-S.Solar.n_p*...
S.Solar.i_rs*(exp(S.Solar.q*S.Solar.vpv/(S.Solar.k*
S.Solar.T*S.Solar.A*S.Solar.n_s))-l)); . .
S.Solar.vpv-S.Batt.Vb/S.Solar.dpv;...
S.i_L+S.Batt.ib-io_pv-io_w;...
-S.Batt.Vb+S.Batt.eb+S.Batt.ib*S.Batt.RI
'/, -S. Batt. ib_spec+S. Batt. ib
];
B.6 Weighted Objectives Solver Function
clear all
format short g
clc
'/.System Inputs
S.Batt,eb_per = 0.85;
S.Wind.v = 14;
S.Solar.S_pv = 100;
S.Solar.T = 298;
S.i_L = 10;
S = Mejia_variables(S);
S = Mejia_Solve_OPT(S);
S.XinCl,:) = S.X_init;
S.Xin(2,:) = S.XinCl,:);
S.Xin(3,:) = S.Xin(2,:);
S.Xin(4,:) = S.Xin(3,:);
S.X_init = S.XinCl,:);
iter = 0;
iterl = 0;
for ii = 0:1:60
iterl = iterl+1;
iter2 = 0;
count = 2;
for y = 0:0.01:1
S.W1 = y;
S.W2 = 1-S.W1;
62


end
end
S.X_init = S.Xin(count-l,:);
S = Mejia_Solve_OPT(S);
iter = iter+1;
iter2 = iter2+l;
xhist(iter,:) = S.X;
id = S.X(l);
iq = S.X(2);
wm = S.X(3);
dw = S.X(4);
ipv = S.X(5);
vpv = S.X(6);
dpv = S.X(7);
ib = S.X(8);
Vb = S.X(9);
io_w = pi/(2*sqrt(3))*sqrt(iq~2+id~2)/dw;
io_pv = ipv/dpv;
x = [] ;
for ii = 1:1:50
x(ii) = (2*io_w*sin(pi*ii*dw)/(pi*ii))"2;
end
fl(iter2,iterl) = (sum(x)/(2*dw*io_w)~2);
f2(iter2,iterl) = ((S.Batt.ib_spec-ib)~2);
flag(iter,l) = S.exitflag;
S.Xin(count+l,:) = S.X;
count = count+1;
flmin = min(fl)
flmax = max(fl)
f2min = min(f2)
f2max = max(f2)
flnorm = (f1-flmin)/(flmax-flmin);
f2norm = (f2-f2min)/(f2max-f2min);
63


APPENDIX C. MATLABs fmincon Function
MATLAB fmincon function is designed to find the minimum of a con-
strained non-linear multi-variable function. MATLAB solves and optimization
problem of the form
niinj f(x)
subject to
c(x) < 0
ceq(x) = 0
(C.l)
A x beq
lb < x < ub
where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x)
are functions that return vectors (@nonlincon), and fix) is a function that
returns a scalar (@fun).
The function is defined in the MATLAB environment as
[X.fval.exitflag.output .lambda] = fmincon(@fun,xO,A,b,Aeq,beq,lb,ub,@nonlcon,options) (C.2)
where xO sets the initial conditions; X is the optimized value obtained from the
function @fun subject to linear inequalities Ax = beq, non-linear inequalities
c(x) < 0, or equalities ceq(x) = 0, in the range lb < x < ub. Further,
lambda is a structure containing the Lagrangean multipliers at the solution x;
exitflag is an integer identifying the reason the algorithm terminated; output
is a structure containing information about the optimization, and fval contains
the value of the objective function @fun at the solution X.
64


APPENDIX D. System Case Studies
Wind Speed (m/*| Solar irradiation Battery Voltage f*l load Currant [Am per as] THDiVj Ai6 lAMparM) Flag id (AMparaa) i* (Awpene) wm (radfc) dw ipv [AMtMTWl Vpv [>Mb] dpn it [ANparae] VDC tVata]
Spl'AVQ e-frtcw LU* 91.75% 0X0 4 0.10 2.78 18.26 35.1% 5.71 10856 386% 12.00 41.90
C_t>*vc Lln 0.00% 0X0 1 0.00 0.17 874 iaao% 6.98 105.49 42.3% 6.66 44.62
1,1,0* 0.00% 0X0 1 0.00 0.27 731 100.0% 6.19 10734 45.0% 4.00 4837
S_p\ HIGH C-biow O.OOK 0X0 5 0.00 an 632 1000% 1832 5825 74.5% 12X0 41.90
i_Uvo 95.63% 0X0 5 0.10 2.78 18.76 34.1% 15.79 91.87 45.6% 12X0 41.90
e_6*vc tluw 0.00% 0X0 1 0.00 ai7 6.74 100.0% 6.36 115.91 383% 6.66 44.62
i_U 88.28% 0X0 5 0.10 2.77 18.96 36.0% 12.17 10870 41.0% 6.66 44.61
UUA, 0.00% 0X0 1 0.00 0.27 731 100.0% 5.72 116.42 413% 4.00 48.37
LUw 38.59% 0X0 5 0X5 2.09 1315 56.2% 15.44 95.95 50.4% 4.00 4837
Vmg SPViON C_&LOW i_U* 251.71% 0X0 5 0.11 2.93 38.86 16.4% 2.58 93.89 44.6% 12X0 41.90
ei>AVG Lko* 251.94% 0X0 5 0.06 2.20 41.31 16.4% 3.25 61.41 72.7% 6.66 44.61
e_f>HK3H LU* 71.33% 0X0 s 0.20 3.99 18.15 41.0% 2.72 92.04 52.6% 4.00 48.37
S-.pVwc e_6io i_l.oa 40.31% 0X0 5 0.01 0.96 1133 SS.2% 9X7 94.38 44.4% 12X0 41.90
LUs 183.23% 0.00 5 0.32 5.04 30.24 21.2% 9.05 94.59 44.3% 12X0 41.90
0_i>Ave Lko 0.00% 0X0 5 0.00 aoi 6.72 1000% 7.05 105.29 42.4% 6.66 44.62
LU* 128.48% 0X0 s 0.37 5.43 24.72 273% 852 9899 45.1% 6.66 44.62
i_U 0.00% 0.00 1 0.00 0.05 739 100.0% 6.29 10730 4S.1% 4.00 4837
LUe 148.34% 0X0 s 0.33 313 29.66 25.0% 9.71 76.41 63.3% 4.00 48.37
S_pV HUH e_bLOW Lko* 0.00% 0.00 s 0.00 0.01 631 100.0% 16.32 56.48 74.2% 12X0 41.90
LUs 7235% 0.00 5 0.11 2.96 1SX3 40.6% 15.41 96.20 43.6% 12.00 41.90
e_bAVB LU. 0.00% 0X0 1 0.00 002 6.72 100X% 16.32 4S.S0 98.1% 6.66 44.62
LU 67.04% 0X0 5 0.12 3.09 16.10 423% 16.14 83.15 53.7% 6.66 44.61
e_bHWH LU 0.00% 0.00 1 0.00 0.05 739 100.0% S.80 11635 41.6% 4.00 4837
LU* 4232% 0.00 5 0.05 1.91 13.68 54.0% 14.96 99.52 46.6% 4.00 48.37
Vw S-pl'XON e_blOW Lkow 11436% 0.00 5 0.34 325 21.48 30.1% 2.98 86.56 48.4% 12.00 41.90
LU 16234% 0.00 5 1.08 9.27 28.14 23.3% 820 74.09 56.6% 11.99 41.90
e_6AvG Lkow 98.75% 0.00 S 0.26 4.61 20.57 33.4% 1.84 99.88 44.7% 6.66 44.61
LUe 14232% 0.00 5 0.96 8.74 27.04 25.8% 3.09 82.73 53.9% 6.66 44.61
LU 79.33% 0.00 5 0.17 3.73 19.31 38.5% 2.73 91.83 52.7% 4.00 4837
LU 128.13% 0X0 S 0.96 8.77 27.12 27.9% 2.82 90.25 53.6% 4.00 48.37
S_pVwa p-biow LU 120.93% 0.00 S 0.42 383 22.32 29.1% 1.37 11433 36.6% 12.00 41.90
i_Ue 12933% 0.00 5 035 6.65 23.52 27.6% 9.36 90.06 463% 12.00 41.90
>_U, 208.46% 0.00 5 133 11.00 34.31 19.1% 9.56 84.92 49.3% 12.00 41.90
e_b avc LU 0.00% 0.00 1 0.00 aoo 6.72 100.0% 7.06 105.26 42.4% 6.66 44.62
Ll 113.42% 0X0 5 0.47 312 22.74 30.4% 806 10132 43.9% 866 44.61
LU 186.73% 0.00 5 1.49 10.88 33.44 20.9% 867 97.97 453% 6.66 44.61
e-btMH LU 0.00% 0.00 4 0.00 aoo 739 100.0% 9.76 69.37 69.7% 4.00 4837
LU* 99.74% 0X0 5 0.45 6.00 2236 33.2% 9.28 91.46 52.9% 4.00 4837
LU 220.66% 0X0 1.44 10.70 41.24 183% 4.54 11034 43.7% 4.00 4837
S_pVHK e_t>LOMr LU 0.00% 0X0 0.00 aoo 631 ioox% 8.06 114.37 36.6% 12.00 41.90
IJUc 162.32% 0X0 1.07 9.24 2807 23.4% 2.08 1183S 35.3% 12X0 41.90
LU 16438% 0.00 1.10 9.38 28.38 23.1% 15.68 93.39 44.9% 12.00 41.90
e_t>AV6 Lko* 0.00% 0X0 0.00 aoo 6.72 100.0% 6.42 115.86 383% 6.66 44.62
LU 144.07% 0X0 0.98 885 27.27 25.6% 1.90 11895 373% 6.66 44.61
LU 149.50% 0X0 1.07 9.24 2807 24.9% 15.71 93.03 48.0% 6.66 44.61
e.bwoH LU* 0.00% 0X0 0.00 aoo 739 100.0% 5.81 11635 41.6% 3.98 48.37
LUs 74.67% 0X0 0.13 324 1857 39.9% 11.77 109.47 44.2% 4.00 4837
LU 139.61% 0X0 1.16 9.61 28.94 282% 1539 94.44 51.2% 4.00 48.37
65


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