Optimal Control Tracking Problem of a Hybrid Wind-Solar-Battery Energy System

Optimal Control Tracking Problem of a HybridWind/Solar/Battery Energy System

April 26, 2016

Julio [email protected]

Abstract

We consider a hybrid wind/solar/battery power system with the goal of generatingand controlling power to meet a specified power demand. Each subsystem is modeledindependently and then their separate power outputs are combined. The systems arecombined in such as way that the wind system is considered the primary source andthe solar system and battery bank are secondary. We mathematically model wind andsolar power generation systems and verify the power generation with existing literature.Once the model was verified we applied the optimal control tracking problem theoryto satisfy a desired power demand, first for a linearized time variant model and thenfor a nonlinear system with a linearized time invariant power output.

1. Introduction

Renewable energy systems are of increasing importance for sustainability and energy inde-pendence. Because each renewable energy system has its own advantages and disadvantages wecombine multiple systems which will complement each other. Managing energy production andinterfaces are the major issues for multiple hybrid renewable energy generators connected to thetraditional power source, because their connection into the existing power can lead to instabilityor even failure, if they are not properly controlled. Before control issues can be addressed wemust properly model and verify our system. Therefore, we begin by separately modeling a windand solar energy system. These separate systems will then be combined to form our hybridrenewable energy system.

In Section 2 we introduce the model of the hybrid system. Section 2.1 considers the windturbine aerodynamics, Section 2.2 introduces the dynamic model of the wind subsystem, Section2.3 considers the photovoltaic cells, Section 2.4 derives the model of the solar subsystem, andSection 2.5 considers the combined hybrid model. In Section 3 and 4 we simulate each of thesystems separately and attempt to verify them against published results.

In Section 5 will linearize the wind system using Taylor’s Series Expansion. For this purposewe will find the nonlinear operating points when the system is in equilibrium.

With the results found in Section 5 we will solve the Linear Time Variant Tracking Problemin the first part of Section 6. For the second part we will solve the tracking problem keepingthe dynamic of the system nonlinear and converting to an equivalent Linear Time Invariant justthe power output.

2. Model

We consider a hybrid wind/solar/battery bank power system. The wind system consists ofa fixed pitch wind turbine, a multipole permanent magnet synchronous generator (PMSG), arectifier, a DC/DC converter and an auxiliary battery bank, denoted as Battery Bank 1. The solarsubsystem involves a photovoltaic panel array, a DC/DC converter and a battery bank, denotedas Battery Bank 2.

Figure 1: Wind Turbine / Solar Hybrid System

2.1. Wind Turbine Aerodynamics

The mechanical power captured by a wind turbine is proportional to the swept area A, theair density ρ, the cube of the wind speed vw(t), and the power coefficient Cp. The coefficientexpresses the conversion efficiency of the turbine as a function of the tip-speed ratio, which isgiven by [1]

λts(t) =rωm(t)

vw(t)(1)

where r is the blade length and ωm(t) is the angular shaft speed. Therefore, the mechanicalpower generated by the turbine may be written as [1]

Pturbine(t) =1

2

Cp(λts(t))

λts(t)ρAvw(t)3. (2)

From (2) we can write the driving torque as [1]

Tturbine(t) =Ptωm

=1

2Ct(λts)ρArvw(t)2 (3)

where Ct(λts) = Ct(λts)λts

is the torque coefficient. The power coefficient can be expressed as [2]

Cp(λ) = c1(c21

Γ− c3θ − c4θ

x − c5)e−c6

Γ (4)

where 1Γ

= 1λ+0.08θ

+ 0.0351+θ3 . Here λ is the tip speed ratio, θ is the pitch angle, and c1, ..., c6 and

x are constants chosen for various turbines.Additionally, we can write the torque from the PMSG as [3]

TPMSG =3Vsφm2ωmL

√1− (

Vsωeφm

)2 (5)

where Vs = πvbux3√

3.

Figure 2: Per-phase circuit and phasor diagram of PMSG

2.2. Dynamic Model of Wind Turbine

The wind system consists of a fixed pitch wind turbine, a multipolar PMSG, a rectifier, aDC/DC converter, and an auxiliary battery bank, denoted as Battery Bank 1. The dynamics ofthe wind energy generation subsystem can be represented by four state equations describing thetime derivatives of the quadrature current iq, the direct current id, in a rotor reference frame,and the electrical angular speed ωe, as follows [3]:

iq(t) = −Rsw

Lwiq(t)− ωe(t)id(t) +

ωe(t)φmLw

− πvb1(t)iq(t)uw(t)

3√

3Lw√iq(t)2 + id(t)2

(6)

id(t) = −Rsw

Lwid(t)− ωe(t)iq(t)−

πvb1(t)id(t)uw(t)

3√


(7)

ωe(t) =P

2J(Tturbine(t)−

3

2

P

2φmiq(t)), (8)

˙vCb1(t) =1

Cb(π

2√

3

√iq(t)2 + id(t)2uw(t) + is(t)− iL(t)), (9)

where J is the inertia of the rotating system, P the number of poles, ωe = Pωn2

the electricalangular speed, φm the flux linked by the stator windings, and [3]

vb1(t) = Eb1 + vCb1 (t) + (π

2√

3

√iq(t)2 + id(t)2uw(t) + is(t)− iL(t))Rb1 (10)

is the voltage across the DC bus with iL(t) the load current and vCb1 the voltage across thecapacitor in Battery Bank 1.

2.3. Photovoltaic Cells

The instantaneous electric energy generated by a PV cell depends on several cell parametersand on variable environment conditions such as insolation and temperature. Its electric behaviormay be modeled by a nonlinear current source connected in series with the intrinsic cell seriesresistance Rs [4]

ipv(t) = iph(t)− irs(t)(eq(vpv(t)+ipv(t)Rs)

ApvKT (t) − 1) (11)

where iph(t) is the generated current under a given insolation, irs(t) is the cell reverse saturationcurrent, ipv(t) and vpv(t) are, respectively, the output current and voltage of the solar cell, q is thecharge of an electron, K is Boltzmann’s constant and T (t) is the time-varying cell temperaturein Kelvin. This is often expressed in terms of voltage

vpv(t) = −ipv(t)Rs +ApvKT (t)

qln[iph(t)− ipv(t) + irs(t)

irs(t)]. (12)

The factor Apv considers the cell deviation from the ideal p−n junction characteristic, varyingbetween 1 and 5. The reverse saturation irs(t) and the photocurrent iph(t) depend on insolationand temperature according to the following expressions [4]:

irs(t) = ior(T (t)

Tr)3e

qEgo( 1Tr

− 1T (t)

)

ApvK , (13)

iph(t) =λ(t)(isc −Kl(T (t)− Tr))

100. (14)

where ior is the reverse saturation current at the reference temperature Tr, Ego is the band-gapenergy of the semiconductor used in the cell, isc is the short-circuit cell current at referencetemperature and insolation, Kl is the short-circuit current temperature coefficient, and λ(t) isthe insolation in mW

cm2 .In the case of PV cell arrays, the expression for the generated current is analogous to (11)

[4]:ipv(t) = npiph(t)− npirs(t)(e

q(vpv(t)+ipv(t))RsnsApvKT (t) − 1). (15)

where np represents the number of parallel modules, each on constituted by ns cells connectedin series. From (16) we obtain an expression for the power generation of the array [4]

ppv(t) = npiph(t)vpv(t)− npirs(t)vpv(t)(eq(vpv(t)+ipv(t))Rs

ApvKT (t) − 1). (16)

Figure 3: Photovoltaic Subsystem

2.4. Photovoltaic-based Generation System

Photovoltaic-based generation systems are commonly linked to a load through a solid-stateconverter. Additionally, PV arrays are usually combined with other generation subsystems and /orsome energy storage systems. The dynamic model of the hybrid system can be built around theinstantaneous switched model of the DC/DC buck converter, which is described by the followingequations [5]

vpv(t) =ipv(t)

Cs− is(t)

Csupv(t), (17)

is(t) = −vb2(t)

Ls+vpv(t)

Lsupv(t). (18)

where is(t) and vb2(t) are the current and voltage output terminals of the DC/DC converter andupv(t) is the switched control signal that can only take the discrete values 0 (switch open) or 1(switch closed).

We consider a battery bank model [1] that is comprised of an ideal voltage source Eb2 , acapacitor Cb2 , and a resistance Rb2 connected in series. Then the whole dynamic model may bewritten as (17) and (18) with the addition of [5]

vc2(t) =1

Cb2(is(t) + iw(t)− iL(t)) (19)

where vc2(t) is the voltage on Cb2 , vb2(t) = Eb1 + vc2(t) + (is(t) + iw(t)− iL(t))Rb2 , and iL(t)and iw(t) are measurable currents.

2.5. Hybrid System

The power generated by the wind subsystem can be expressed as [6]

Pw(t) = iw(t)vb1(t) (20)

whereiw(t) =

π

2√

3

√iq(t)2 + id(t)2uw(t) (21)

denotes the output current of the wind subsystem, vb1(t) is the voltage across the terminals ofBattery Bank 1 and uw(t) is the control signal (duty cycle of the DC/DC converter) [7]. Thesolar subsystem involves a photovoltaic panel array, a DC/DC converter, and a battery bank,denoted as Battery Bank 2. Two state variables, the voltage across the PV array terminals vpv(t)and the output current ipv(t) are used to characterize the dynamics (16) and (17). The powerdelivered by the solar subsystem can be computed as [6]

Ps(t) = is(t)vb2(t) (22)

whereis(t) = −vb2(t)

Ls+vpv(t)

Lsupv(t). (23)

and vb2(t) is the solar subsystem output voltage.

3. Wind Subsystem Simulations

3.1. Turbine Simulation

Here we verify the results in [3]. All simulation values can be found in the Appendix. First,we verify the power coefficient. The power coefficient can be expressed as

Cp(λ) = c1(c21

Γ− c3θ − c4θ

x − c5)e−c6

Γ

where 1Γ

= 1λ+0.08θ

+ 0.0351+θ3 . Here λ is the tip speed ratio, θ is the pitch angle, and c1, ..., c6 and

x are constants chosen for various turbines. For our simulations we choose θ = 0, c1 = 78,c2 = 9.4737, c5 = 1, c6 = 30. We also assume 1

λ>> 0.035, which results in 1

Γ= 1

λ. Fig. 4

shows a typical power coefficient for a horizontal axis turbine, which gives a result similar to[3].

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Power Coefficient for Horizontal Axis Turbine

Tip Speed Ratio, lambda

Po

wer

Co

effi

cien

t, C

p

Figure 4: Power Coefficient vs Tip Speed Ratio

Next, we simulated the turbine and PMSG torque:

Tturbine =Cp(λ)ρAv3

2ωm

TPMSG =3Vsφm2ωmL

√1− (

Vsωeφm

)2

The turbine torque is parameterised in terms of the wind speed with values from 2 to 16 m/swith an interval of 2 m/s and with the respective values of Cp obtained for each value of windspeed.

The PMSG torque is parameterised in terms of Vs, which is the line voltage on the PMSGterminal, with values from 40 to 320 v with an interval of 40 v.

For this simulation we used the values from [3], which are listed in the Appendix 1. Fig.5 and Fig. 6 show the simulations for the turbine torque and PMSG torque, respectively. Thefigures found in [3] show similar results.

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

Angular Shaft Speed, omegam

(rad/s)

To

rqu

e, N

m

Turbine Torque vs Angular Shaft Speed

4 m/s6 m/s8 m/s10 m/s12 m/s14 m/s16 m/s

Figure 5: Turbine Torque

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

omegam

, rad/s

torq

ue,

Nm

PMSG Torque Shaft Speed Curve

4080120160200240280320

Figure 6: Wind Velocity

The wind turbine verification simulations approximately match those shown in [3]. Whileequation (4) is given in [2], we use simulation values given to us from the authors of [3] toapproximately replicate the power coefficient curve in [3].

3.2. Dynamic Wind Subsystem Simulation

This simulation attempts to replicate the results found in [3]. The Fig. 7a through 7d showthe input signals to the dynamic wind system: wind velocity v, solar subsystem current is, loadcurrent iL, and the control signal uw, respectively. The values for these input signals were takenfrom [3] and they were replicated using a code written in Matlab.

Time0 20 40 60 80 100 120 140 160

v, (

m/s

8.5

9

9.5

10

10.5

11

11.5

12

12.5

13Wind Velocity, v

(a) Wind Velcocity v, m/s

Time0 20 40 60 80 100 120 140 160

i s, (A

)

10

11

12

13

14

15

16

17

18

19

20Solar Subsystem Current, i

s

(b) Solar Subsystem Current is, A

Time0 20 40 60 80 100 120 140 160

i L, (

A)

20

30

40

50

Load Current, iL

(c) Load Current iL, A

Time0 20 40 60 80 100 120 140 160

u

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6Control Law, u

(d) Control Signal uw

Figure 7: System Inputs

Fig. 8 to 11 show the output from each state. We use the initial values iq0 = 0, id0 = −0.001,ωe0 = 530, vCb10

= −1 and the nonlinear state space model:

iq(t)

id(t)

ωe(t)

˙vCb1(t)

=

−Rsw

Lwiq(t)− ωe(t)id(t) + ωe(t)φm

Lw− πvb1 (t)iq(t)uw(t)

3√

3Lw√iq(t)2+id(t)2

−RswLw

id(t)− ωe(t)iq(t)−πvb1 (t)id(t)uw(t)

3√

3Lw√iq(t)2+id(t)2

P2J

(12CtρArvw(t)2 − 3

2P2φmiq(t))

1Cb

( π2√

3

√iq(t)2 + id(t)2uw(t) + is(t)− iL(t))

(24)

p(t) = (Eb1 + vCb1(t) + ( π2√

3

√iq(t)2 + id(t)2uw(t) + is(t)− iL(t))Rb1)( π

2√

3

√iq(t)2 + id(t)2uw(t) + is(t))

(25)where p(t) (maximum power available) is the output, Ct is the torque coefficient of the turbine,

ρ is the air density, A is the swept area, r is the blade length, P is the number of poles, J isthe inertia of the rotating system, vb1(t) is the DC bus voltage, and vw(t) is the wind velocity.

To replicate these results we ran simulations using Matlab (ode15s) and Matlab code (Runge-Kutta). In the begining of the simulation there is a discrepancy between both methods. We chosethe results obtained from Matlab code because they are more accurate and match very well withthe results from [3].

Time0 20 40 60 80 100 120 140 160

i q, (

A)

0

5

10

15

20

25Quadrature Current, i

q

Figure 8: State: iq

Time0 20 40 60 80 100 120 140 160

i d, (

A)

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5Direct Current in the Rotor Reference Frame, i

d

Figure 9: State: id

Time0 20 40 60 80 100 120 140 160

ωe, (

rad/

s)

300

350

400

450

500

550Electrical Angular Velocity, ω

e

Figure 10: State: ωe

Time0 20 40 60 80 100 120 140 160

v cb1, (

F)

-1

-0.995

-0.99

-0.985

-0.98

-0.975

-0.97

-0.965

-0.96Voltage in Capacitor, v

cb1

Figure 11: State: vCb1

Finally, Fig. 12 shows the maximum power output available from the hybrid wind/solar system.

Time0 20 40 60 80 100 120 140 160

Pow

er, (

w)

0

1000

2000

3000

4000

5000

6000

7000Maximum Power Available

Figure 12: Generated Power

The dynamic wind subsystem simulations match the results in [3].

4. Solar Subsystem Simulations

4.1. Photovoltaic Cell Simulation

Here we verify the results in [5]. All simulation values can be found in the Appendix. Wesimulate the following system:

ipv(t) = iph(t)− irs(t)(eq(vpv(t)+ipv(t)Rs)

ApvKT (t) − 1)

with

irs(t) = ior(T (t)

Tr)3e

qEgo( 1Tr

− 1T (t)

)

ApvK ,

iph(t) =λ(t)(isc −Kl(T (t)− Tr))

100.

Using the model above we get the following I-V and Power-V results.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

3.5

4

Terminal Voltage vpv

Cel

l Cur

rent

ipv

, A

Current − Voltage Curve of PV Cell

100 / 50100 / 2580 / 5080 / 2560 / 5060 / 2540 / 5040 / 2520 / 5020 / 25

Figure 13: I-V Graph for Modified System

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Terminal Voltage vpv

Cel

l Pow

er p

pv, W

Power − Voltage Curve of PV Cell

100 / 50100 / 2580 / 5080 / 2560 / 5060 / 2540 / 5040 / 2520 / 5020 / 25

Figure 14: Power-V Graph of Modified System

Our simulation results for the photovoltaic cell matched those given in [5]. For this simulationwe used a slightly modified version of their equation for ipv. Using the modified equation, takenfrom [4], and using the simulation values from [5], we were able to very closely replicate theresults given in [5]. This was done because we were not able to match the results using boththe model and simulation values given in [5].

0 10 20 30 40 50 60 70 80 90 10060

65

70

75

80

85

90

95

100

Time (s)

Inso

lati

on

(m

W/c

m2 )

Insolation vs Time

(a) Insolation

0 10 20 30 40 50 60 70 80 90 100312

314

316

318

320

322

324

Time (s)

Tem

per

atu

re (

K)

Temperature vs Time

(b) Temperature (K)

0 10 20 30 40 50 60 70 80 90 10041

42

43

44

45

46

47

48

Time (s)

Cu

rren

t (A

)

Load Current vs Time

(c) Load Current iL

0 10 20 30 40 50 60 70 80 90 10048

50

52

54

56

58

60

Time (s)

Cu

rren

t (A

)

Wind Subsystem Current vs Time

(d) Wind Current iW

Figure 15: System Inputs

4.2. Dynamic Solar Subsystem Verification Simulation

Here we verify the results in [5]. We simulate the following dynamic system

vpv(t) =ipv(t)

Cs− is(t)

Csupv(t),

is(t) = −vb2(t)

Ls+vpv(t)

Lsupv(t)

vc2(t) =is(t) + iw(t)− iL(t)

Cb2

with the initial conditions vpv0= 47V , is0 = 11A, and Vc2 = 0V . The simulation uses the

following inputs: λ (insolation), T (temperature), iL (load current), iW (wind system current),and us (solar subsystem control signal). For this simulation us = 1 for the entire duration. Beloware plots of the time varying input signals [5].

Below we show the results of our simulation.

0 10 20 30 40 50 60 70 80 90 10042

44

46

48

50

52

54

56

Time (s)

Vo

ltag

e (A

)

Photovoltaic Voltage vs Time

(a) Photovoltaic Voltage vpv

0 10 20 30 40 50 60 70 80 90 1008

9

10

11

12

13

14

15

16

Time (s)

Cu

rren

t (A

)

Solar Subsystem Current vs Time

(b) Solar Subsystem Current is

Figure 17: States

0 10 20 30 40 50 60 70 80 90 100450

500

550

600

650

700

750

800

Time (s)

Po

wer

(W

)

Photovoltaic Power vs Time

(a) Photovoltaic Power ppv

0 10 20 30 40 50 60 70 80 90 10048.2

48.25

48.3

48.35

48.4

48.45

48.5

48.55

48.6

Time (s)

Vo

ltag

e (V

)

X Y Plot

(b) Battery Voltage vb

0 10 20 30 40 50 60 70 80 90 10010

11

12

13

14

15

16

Time (s)

Cu

rren

t (A

)

Photovoltaic Current vs Time

(c) Photovoltaic Current ipv

Figure 18: (a) Photovoltaic Power, (b) Battery Voltage, and (c) Photovoltaic Current

0 10 20 30 40 50 60 70 80 90 1002500

3000

3500

4000

Time (s)

Po

wer

(W

)

Power vs Time

Figure 16: Combined Wind and Solar Power

The simulation of the dynamic model for the photovoltaic-based generation system gives an

output power of approximately 3100 W whereas [5] shows an output of approximately 3500 W.

5. Wind System Linearization

The nonlinear model for a wind/solar system is shown below

iq(t) = −Rsw


ωe(t)φmLw


3√


(26)

id(t) = −Rsw


πvb1(t)id(t)uw(t)

3√


(27)

ωe(t) =P

2J(Tturbine(t)−

3

2

P

2φmiq(t)), (28)

˙vCb1(t) =1

Cb(π

2√

3


p(t) = (Eb1 + vCb1(t) + ( π2√

3

√iq(t)2 + id(t)2uw(t) + is(t)− iL(t))Rb1)( π

2√

3

√iq(t)2 + id(t)2uw(t) + is(t))

(30)where

vb1(t) = Eb1 + vCb1 (t) + (π

2√

3


is the voltage across the DC bus with iL(t) the load current and vCb1 (t) the voltage across thecapacitor in Battery Bank 1 and iq is the quadrature current, id the direct current in a rotorreference frame, ωe the electrical angular speed, J the inertia of the rotating system, P thenumber of poles, and φm the flux linked by the stator windings,p(t) (power generated) is theoutput, Ct is the torque coefficient of the turbine, ρ is the air density, A is the swept area, r isthe blade length, vb1(t) is the DC bus voltage, and v(t) is the wind velocity. We used the initialvalues iq0 = 0, id0 = −0.001, ωe0 = 530, vCb10

= −1 to solve the nonlinear system.For linearization purposes we need to find first the operating points. These are the points

around which, the linear system behavior is similar than the original nonlinear system, thismeans that in these points the system is in equilibrium. Mathematically expressed the system isin equilibrium when

0 = −Rsw

Lwiq − ωeid +

ωeφmLw

− πvb1(t)iquw

3√

3Lw

√iq

2+ id

2(32)

0 = −Rsw

Lwid − ωeiq −

πvb1(t)iduw

3√

3Lw

√iq

2+ id

2(33)

0 =P

2J(Tturbine(t)−

3

2

P

2φmiq), (34)

0 =1

Cb(π

2√

3

√iq

2+ id

2uw + is(t)− iL(t)), (35)

whereTturbinne(t) =

CtρArvw(t)2

2(36)

Now we have to find the values of each state and control that satisfies this system of equations.Since we have four equations and five unknowns, we have infinite number of solutions, in otherwords infinite nonlinear operating points. If we let id to be the free variable, solving for the restof states and control we find

iq =2CtρArvw(t)2

3φmP(37)

ωe = 0 (38)

¯vcb1 =3Rs((

2CtρArvw(t)2

3φmP)2 + id

2)

2(is(t)− iL(t))− Eb (39)

uw = − 2√

3(is(t)− iL(t))

π√

(2CtρArvw(t)2

3φmP)2 + id

2(40)

Using Taylor’s series expansion around the operating points given by x = [iq; id; ωe; ¯vcb1]T weobtain a model as follows

δ(t) = A(t)δx(t) +B(t)δuw(t) (41)

δp(t) = C(t)δx(t) +D(t)δuw(t) (42)

where δx(t) = x(t) − x(t), δuw(t) = uw(t) − uw(t) and δp = p(t) − p(x, uw) are variations ofthe variables in the neighborhood of the operating points and A(t), B(t), C(t), and D(t) arethe Jacobian matrices of the system defined as

A(t) =

∂iq∂iq

∂iq∂id

∂iq∂ωe

∂iq∂vcb1

∂id∂iq

∂id∂id

∂id∂ωe

∂id∂vcb1

∂ωe∂iq

∂ωe∂id

∂ωe∂ωe

∂ωe∂vcb1

∂ ˙vcb1∂iq

∂ ˙vcb1∂id

∂ ˙vcb1∂ωe

∂vcb1∂ ˙vcb1

(43)

B(t) =

∂iq∂u

∂id∂u∂ωe∂u∂ ˙vcb1∂u

(44)

C(t) =[∂p∂iq

∂p∂id

∂p∂ωe

∂p∂vcb1

](45)

and

D(t) =[∂p∂uw

](46)

where all the elements of the Jacobian matrices are evaluated in the operating points.

Since the operating points are functions of time, the linearized system will be a Linear TimeVariant (LTV) system where the Jacobian matrices after their evaluation in the operating pointsare

A(t) =

A11(t) A12(t) −id + φm

LA14(t)

A21(t) A22(t) −2CtρArvw(t)2

3φmPA24(t)

−3φmP8J

0 0 0

− 2CtρArvw(t)2(is(t)−iL(t))

3φmPCb((2CtρArvw(t)2

3φmP)2+id

2)− id(is(t)−iL(t))

Cb((2CtρArvw(t)2

3φmP)2+id

2)0 0

(47)

B(t) =

−πCtρArvw(t)2Rs

√(

2CtρArvw(t)2

3φmP)2+id

2

3√

3LφmP (is(t)−iL(t))+ 2πCtρArvw(t)2Rb(is(t)−iL(t))

9√

3LφmP

√(

2CtρArvw(t)2

3φmP)2+id

2

−πidRs

√(

2CtρArvw(t)2

3φmP)2+id

2

2√

3L(is(t)−iL(t))+

√3πidRb(is(t)−iL(t))

9L

√(

2CtρArvw(t)2

3φmP)2+id

2

0

π2√

3Cb

√(2CtρArvw(t)2

3φmP)2 + id

2

(48)

C(t) =

[(−3Rs

2− RbiL(t)(is(t)−iL(t))

(2CtρArvw(t)2

3φmP)2+id

2)(2CtρArvw(t)2

3φmP) (−3Rs

2− RbiL(t)(is(t)−iL(t))

(2CtρArvw(t)2

3φmP)2+id

2)id 0 iL(t)

](49)

and

D(t) =

[π√

3Rs((2CtρArvw(t)2

3φmP)2+id

2)3/2

4(is(t)−iL(t))+ πRbiL(t)

2√

3

√(2CtρArvw(t)2

3φmP)2 + id

2

](50)

where

A11(t) = −Rs

L+

6id2Rs − 4Rb(is(t)− iL(t))2

6L((2CtρArvw(t)2

3φmP)2 + id

2)

+2id

2Rb(is(t)− iL(t))2

3L((2CtρArvw(t)2

3φmP)2 + id

2)2

(51)

A12(t) = − 2CtρArvw(t)2idRs

3LφmP ((2CtρArvw(t)2

3φmP)2 + id

2)− 4CtρArv(t)2idRb(is(t)− iL(t))2


3φmP)2 + id

2)2

(52)

A14(t) =4CtρArvw(t)2(is(t)− iL(t))


3φmP)2 + id

2)

(53)

A21(t) = − 2idCtρArvw(t)2Rs


3φmP+ id

2)− 4idCtρArvw(t)2Rb(is(t)− iL(t))2


3φmP)2 + id

2)2

(54)

A22(t) = −RsL

+4(CtρArvw(t)2)2Rs − 6φ2mP

2Rb(is(t)− iL(t))2

9Lφ2mP2(( 2CtρArvw(t)2

3φmP)2 + id

2)

+8(CtρArvw(t)2)2Rb(is(t)− iL(t))2

27Lφ2mP2(( 2CtρArvw(t)2

3φmP)2 + id

2)2

(55)

A24(t) =2id(is(t)− iL(t)

3L((2CtρArvw(t)2

3φmP)2 + id

2)

(56)

Since we know how the wind velocity, the current from the photovoltaic panel, and the loadcurrent change with time, in order to have our LTV system fully defined we only have to choosea reasonable value of id. In the next section we will choose this value.

6. Optimal Control Tracking Problem

In this section we will find an optimal control law that forces the plant to track a desiredpower demand trajectory pd(t) during some time interval.

6.1. LTV Optimal Control Tracking Problem

In the last section we linearized our model. Now we can write it in the form

x(t) = A(t)x(t) +B(t)u(t) (57)

p(t) = C(t)x(t) +D(t)u(t) (58)

with initial conditions x(0) = x0 where x(t) = [iq; id;ωe; vcb1]T , u(t) = uw(t), and x0 =[0; 0; 0; 70]T .

Because of space reasons sometimes we will not write the matrices that are part of thefollowing analysis as function of time but we have to remember that they are time dependent.

To keep the power output close to the desired trajectory lets define the cost function as

J =1

2< Cx(tf ) +Du(tf )− pd(tf ), S(Cx(tf ) +Du(tf )− pd(tf ) >

+1

2

∫ tf

t0

[< Cx+Du− pd, Q(Cx+Du− pd) > + < u,Ru >]dt (59)

where the operator < ·, · > represents the dot product of vectors. Q is a real symmetric1 × 1 matrix that is positive semi-definite. The element of Q is selected to weight the relativeimportance of the output and to normalize the numerical values of the deviations. R is a realsymmetric positive definite m ×m, S is a real symmetric positive semi-definite n × n matrix,tf is the final time, pd(t) is the desired trajectory, and the final state x(tf ) is free.

Defining the Hamiltonian we have

H(x, u, ψ, t) =1

2[< Cx+Du− pd, Q(Cx+Du− pd) > + < u,Ru >]+ < ψ,Ax+Bu > (60)

where ψ(t) are the costates which are deducted from the minimization problem solved usingthe approach of Calculus of Variations. They represent the Lagrange multipliers used to solvesuch problem.

Using the Pontryagin’s minimum principal we find the adjoint equation

ψ(t) = −C(t)TQ(C(t)x(t) +D(t)u(t)− pd(t))− A(t)Tψ(t) (61)

with ”initial” conditions ψ(tf ) = C(t)TS(C(t)x(tf ) +D(t)u(tf )− pd(tf ))and the algebraic relation that must be satisfied is given by

D(t)TQ(C(t)x(t) +D(t)u(t)− pd(t)) +Ru(t) +B(t)Tψ(t) = 0 (62)

Now the problem became a two point boundary problem.Solving the last equation for u(t) we have

u(t) = (D(t)TQD(t) +R)−1[−D(t)QC(t)x(t) +D(t)TQpd(t)−B(t)Tψ(t)] (63)

Lets assume that the solution of ψ(t) has the form ψ(t) = P (t)x(t) − η(t), where P (t) is asymmetric matrix of dimensions n×n and η(t) has dimensions n× 1. Taking its derivative andreplacing both, ψ(t) and ψ(t) in the two point boundary problem deducted above, after somematrix algebra we have

P = −PA−ATP +PB(DQC +R)−1(DQC +BTP ) +CTQD(DTQD+R)−1(DQC +BTP )−CQC (64)

with ”initial” condition P (tf ) = C(t)TS(C(t)x(tf ) +D(t)u(tf )) and

η = −(AT − (PB +CTQD)(DTQD+R)−1BT )η+ (−CTQ+ (PB +CTQD)(DTQD+R)−1DTQ)pd (65)

with ”initial” condition η(tf ) = C(t)TSpd(tf ).The above differential equations have to be solved backwards.Replacing the assumed solution of the adjoint equation in the expression of u(t) found before

we have the final feedback control law u(t) as

u(t) = −(DTQD+R)−1(BTP+DQC)x+(DTQD+R)−1BTη+(DTQD+R)−1DTQpd (66)

and the states of the closed loop system will be found using the values of P (t) and η(t) calculatedpreviously to solve the differential equation

x(t) = [A−B(DTQD +R)−1(BTP +DQC)]x+B(DTQD +R)−1(BTη +DTQpd) (67)

In order to solve all the differential equations deducted above we will choose id = −4 to beused in the time variant matrices A, B, C, and D.

Since the Riccati equation is symmetric and P (t) has dimension n × n we need to find 10solutions for it. η(t) has four solutions.

Using a code written in Matlab , we simulated and found the results for the desired trajectory.In our case we chose the trajectory provided as power demand by [3].

To solve Riccati system of differential equations we used Runge-Kutta 4th order with stepsize h = 0.001. We used the same numerical method to solve the extra system of differentialequations that involve η(t) and the state space equations.

For simulations we used the following weight matrices:S is a matrix of dimensions 4 by 4 with all its elements equals to zero. Choosing these values

for S our cost function becomes

J =1

2

∫ tf

t0


notice that now the cost function has free terminal values and also the ”initial” conditions ofthe Riccati equation and the differential equation that involve η are equals to zero.R is a matrix of dimensions 1 by 1 with its only element equals to one.Q is a matrix of dimensions 1 by 1 and its only element is equals to one.

The solutions of the State Space system of differential equations are shown from Fig.19 toFig.22.

Time0 20 40 60 80 100 120 140 160

i q, (

A)

-80

-60

-40

-20

0

20

40

60

80

100

120Tracking Problem i

q

Figure 19: Quadrature Current

Time0 20 40 60 80 100 120 140 160

i d, (

A)

-200

-100

0

100

200

300

400

500


d

Figure 20: Direct current in a rotor reference frame

Time0 20 40 60 80 100 120 140 160

ωe, (

rad/

s)

-150

-100

-50

0

50

100Tracking Problem ω

e

Figure 21: Electrical angular speed

Time0 20 40 60 80 100 120 140 160

v cb1, (

v)

70

70.005

70.01

70.015

70.02

70.025

70.03

70.035

70.04

70.045

70.05Tracking Problem v

cb1

Figure 22: Voltage in the capacitor

The control law that forces the output to track the desired trajectory is shown in Fig.23.

Time0 20 40 60 80 100 120 140 160

u

0

5

10

15

20

25

30

35

40

45

50Control Law, u

Figure 23: Control Law, uw

The power generated to supply the power demand is shown in Fig.24.

Time0 20 40 60 80 100 120 140 160

Pow

er, (

w)

1000

1500

2000

2500

3000

3500Desired Output Trajectory and Tracking Problem

desired Powercontrolled Power

Figure 24: Controlled Power tracking Desired Power Demand

6.2. LTI Optimal Control Tracking Problem

In the previous section we found a control law that forced the power output to track a desireddemand using as dynamic model a LTV form of the original nonlinear model. As we will describein the conclusions at the end of this paper, the controller designed for the above LTV systemis theoretically correct but it looks like is unlikely to implement due to the high values of thedirect current in the rotor reference frame among other reasons. So we have decided to designa controller for an LTI system based in the above linearization. For this purpose we will choosea value of id = −2.10481 and we fixed the values of wind velocity, current from photovoltaicpanel, and load current as vw = 11, is = 19.2, and iL = 18.

Evaluating the values chosen above in the Jacobian matrices we have

A =

−101.0165 16.0995 82.8654 20.801

16.0995 −2.5659 −13.2067 −3.3152

−10.7294 0 0 0

−6.1536e− 07 9.8074e− 08 0 0

(69)

B =

−1.1054e− 04

1.7617e− 03

0

6.7379e− 05

(70)

C =[−7.3101 1.165 0 18

](71)

and

D =[800.4166

](72)

The optimal control tracking problem theory deducted for LTV systems can be applied forLTI systems.

Using initial conditions x0 = [0;−0.001; 530;−1]T and the values of id, is, and iL fixed above,we found an equivalent LTI system which gives us a similar maximum available power than thenonlinear original system. The comparison between both outputs is shown in Fig.25.

Time0 20 40 60 80 100 120 140 160

Pow

er, (

w)

0

1000

2000

3000

4000

5000

6000


Nonlinear PowerLinear Power

Figure 25: Comparison between Nonlinear and Linearized maximum power available

The values of matrices S, Q, and R are the same used for the LTV problem.The solutions of the State Space system of differential equations are shown from Fig.26 to

Fig.29.

Time0 20 40 60 80 100 120 140 160

i q, (

A)

-80

-60

-40

-20

0

20

40

60

80

100


q


Time0 20 40 60 80 100 120 140 160

i d, (

A)

-20

-15

-10

-5

0

5

10


d


Time0 20 40 60 80 100 120 140 160

ωe, (

rad/

s)

150

200

250

300

350

400

450

500

550Tracking Problem ω

e


Time0 20 40 60 80 100 120 140 160

v cb1, (

v)

-1

-0.995

-0.99

-0.985

-0.98

-0.975

-0.97

-0.965Tracking Problem v

cb1


The control law that forces the output to track the desired trajectory is shown in Fig.30.

Time0 20 40 60 80 100 120 140 160

u

1

1.5

2

2.5

3

3.5

4

4.5Control Law, u



Time0 20 40 60 80 100 120 140 160

Pow

er, (

w)

1000

1500

2000

2500

3000

3500Desired Output Trajectory and Tracking Problem

desired Powercontrolled Power


6.3. Nonlinear Optimal Control Tracking Problem

In this section we will design a control law that forces a linearized wind system power outputto track a desired power demand trajectory keeping the state space equations nonlinear.

Our original nonlinear system has the form x(t) = f(x, u, t) with an output p(t) = g(x, u, t).Now we want to express this system with an equivalent one written in the form x(t) =

f(x, u, t) with an output p(t) = Cx(t) +Du(t) where the matrices C and D are 1×n and 1×1respectively.

In order to find the elements of the matrices C and D we will use the formulas obtainedin section 5. Replacing id = −2 in those formulas we find iq = 13.2067, ωe = 0, and ¯vcb1 =33.9822. These values were used to evaluate the Jacobian matrices for C and D and we foundC = [−7.3045; 1.1062; 0; 18] and D = 996.1588.

Then the equivalent system that we will use to design the control law which satisfies thetracking problem will be

iq(t) = −Rsw


ωe(t)φmLw


3√


(73)

id(t) = −Rsw


πvb1(t)id(t)uw(t)

3√


(74)

ωe(t) =P

2J(Tturbine(t)−

3

2

P

2φmiq(t)), (75)

˙vCb1(t) =1

Cb(π

2√

3


p(t) =[−7.3045 1.1062 0 18

]x(t) + 996.1588u(t) (77)

where

vb1(t) = Eb1 + vCb1 (t) + (π

2√

3


with initial conditions x0 = [0;−0.001; 0;−49]T . A comparison between the nonlinear maximumpower available and the linearized maximum power available is shown in Fig.32

Time0 20 40 60 80 100 120 140 160

Pow

er, (

w)

0

1000

2000

3000

4000

5000

6000


Nonlinear PowerLinear Power

Figure 32: Comparison between Nonlinear and Linearized maximum power available

To keep the linearized output close to a desired power demand we can specify the cost functionas

J =1

2< Cx(tf ) +Du(tf )− pd(tf ), S(Cx(tf ) +Du(tf )− pd(tf ) >

+1

2

∫ tf

t0


where Q is a real symmetric 1× 1 matrix that is positive semi-definite. The element of Q isselected to weight the relative importance of the output and to normalize the numerical valuesof the deviations. R is a real symmetric positive definite m×m, S is a real symmetric positivesemi-definite n×n matrix, tf is the final time, pd(t) is the desired trajectory, and the final statex(tf ) is free.

Defining the Hamiltonian we have

H(x, u, ψ, t) =1

2[< Cx+Du− pd(t), Q(Cx+Du− pd(t)) > + < u,Ru >]+ < ψ(t), f(x, u, t) > (80)

where ψ(t) are the costates which are deducted from the minimization problem solved usingthe approach of Calculus of Variations. They represent the Lagrange multipliers used to solvesuch problem.

Using the Pontryagin’s minimum principal we find the adjoint equation

ψ(t) = −C(t)TQ(C(t)x(t) +D(t)u(t)− pd(t))− (∂f(x, u, t)

∂x)Tψ(t) (81)

with ”initial” conditions ψ(tf ) = C(t)TS(C(t)x(tf ) +D(t)u(tf )− pd(tf ))here we will choose S as a matrix of dimensions 4 by 4 with all its elements equals to zero.

Choosing these values for S our cost function becomes

J =1

2

∫ tf

t0


notice that now the cost function has free terminal values and also the ”initial” conditions ofthe costates differential equations are equals to zero.

The algebraic relation that must be satisfied is given by

D(t)TQ(C(t)x(t) +D(t)u(t)− pd(t)) +Ru(t) + (∂f(x, u, t)

∂u)Tψ(t) = 0 (83)

where

∂f(x, u, t)

∂x=

∂iq∂iq

∂iq∂id

∂iq∂ωe

∂iq∂ccb1

∂id∂iq

∂id∂id

∂id∂ωe

∂id∂vcb1

∂ωe∂iq

∂ωe∂id

∂ωe∂ωe

∂ωe∂vcb1

∂ ˙vcb1∂iq

∂ ˙vcb1∂id

∂ ˙vcb1∂ωe

∂vcb1∂ ˙vcb1

(84)

∂f(x, u, t)

∂u=

∂iq∂u

∂id∂u∂ωe∂u∂ ˙vcb1∂u

(85)

Now the problem became a two point boundary problem which can be solved using a numericalmethod, in this case we used the gradient method to solve it with an accuracy between theminimum values of the cost function, ∆J = 0.01, after 160 iterations.

The states to satisfy the nonlinear tracking problem are shown from Fig.33 to Fig.36

Time0 20 40 60 80 100 120 140 160

I q, (

A)

0

2

4

6

8

10

12

14

16

18Stator current in q axis, i

q


Time0 20 40 60 80 100 120 140 160

I d, (

A)

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0Stator current in d axis, i

d


Time0 20 40 60 80 100 120 140 160

ωe, (

rad/

s)

0

2

4

6

8

10

12

14

16

18Electric rotor speed, ω

e


Time0 20 40 60 80 100 120 140 160

v c (v)

-49.002

-49

-48.998

-48.996

-48.994

-48.992

-48.99

-48.988

-48.986

-48.984

-48.982Voltage in capacitor, v

c


The feedback control law that forces the output to track the desired power demand is shown

in Fig.37.

Time0 20 40 60 80 100 120 140 160

u

2

2.5

3

3.5

4

4.5Control Law, u



Time0 20 40 60 80 100 120 140 160

Pow

er (

w)

1000

1500

2000

2500

3000

3500Desired Trajectory and Power

Desired PowerControlled Power


7. Conclusion

The results of the wind turbine simulations and dynamic wind subsystem match those of ourreferences and, thus, we can conclude that we have correctly verified our model.

The above results show that our simulations of the photovoltaic cells and solar subsystem matchthose of our references. Therefore we can conclude that we have verified the solar subsystem inits entirety.

The results found for the LTV Tracking Problem look theoretically satisfactory but becauseof the high values of the control law and the stator current id these results could not be able tobe implemented.

The results obtained for the LTI tracking problem are theoretically correct and also they looklike able to be implemented. The only restriction with these results is the limitation of theirapplication around the operating points which range of stability was not calculated in this paper.

The results of the Nonlinear tracking problem look theoretical and practical satisfactory. Thevalues of the states and the control law are, based on the results from [3], seem acceptable to beimplemented. In this case the operating points are not a big limitation since they were chosenjust to write the power output as linear combination of the states and control but the dynamicof the states remain nonlinear.

For future work we can solve the nonlinear tracking problem when the output is LTV andwhen the output is nonlinear.

8. Appendix 1

Parameter ValueP 28Rs 0.3676 ΩL 3.55 mHφm 0.2867 WbJ 7.856 Kgm2

r 1.84 mRb1 14 mΩRb2 14 mΩCb1 180000 FCb2 180000 FEb1 48 VEb2 48 Vnp 5ns 200q 1.6×10−19 CApv 1.60K 1.3805×10−23 Nm/KIsc 3.27 A (@ 25 C and 100 mW/cm2)Kl 0.0017 A/CEgo 1.10 VCs 0.001 FLs 0.004 Hθ 0c1 78c2 9.4737c5 1c6 30

References

[1] B.S. Borowy and Z.M. Salameh, “Dynamic response of a stand-alone wind energy conversion system withbattery energy storage to a wind gust,” Energy Conversion, IEEE Transactions on, vol. 12, no. 1, pp. 73 –78,mar 1997.

[2] Sung-Hun Lee, Young-Jun Joo, J. Back, and Jin Heon Seo, “Sliding mode controller for torque and pitchcontrol of wind power system based on pmsg,” in Control Automation and Systems (ICCAS), 2010 InternationalConference on, oct. 2010, pp. 1079 –1084.

[3] F. Valenciaga, P.F. Puleston, P.E. Battaiotto, and R.J. Mantz, “Passivity/sliding mode control of a stand-alonehybrid generation system,” Control Theory and Applications, IEE Proceedings -, vol. 147, no. 6, pp. 680 –686,nov 2000.

[4] K.H. Hussein, I. Muta, T. Hoshino, and M. Osakada, “Maximum photovoltaic power tracking: an algorithmfor rapidly changing atmospheric conditions,” Generation, Transmission and Distribution, IEE Proceedings-,vol. 142, no. 1, pp. 59 –64, jan 1995.

[5] F. Valenciaga, P.F. Puleston, and P.E. Battaiotto, “Power control of a photovoltaic array in a hybrid electricgeneration system using sliding mode techniques,” Control Theory and Applications, IEE Proceedings -, vol.148, no. 6, pp. 448 –455, nov 2001.

[6] F. Valenciaga and P.F. Puleston, “Supervisor control for a stand-alone hybrid generation system using windand photovoltaic energy,” Energy Conversion, IEEE Transactions on, vol. 20, no. 2, pp. 398 – 405, june 2005.

[7] W. Qi, J. Liu, and P. D. Christofides, “Distributed supervisory predictive control of distributed wind and solarenergy systems,” Control Systems Technology, IEEE Transactions on, vol. PP, no. 99, pp. 1 –9, 2012.

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