energies Article
A High-Performance Adaptive Incremental Conductance MPPT Algorithm for Photovoltaic Systems Chendi Li 1 , Yuanrui Chen 1 , Dongbao Zhou 1 , Junfeng Liu 2, * and Jun Zeng 1 1
2
*
School of Electric Power, South China University of Technology, Guangzhou 510640, China;
[email protected] (C.L.);
[email protected] (Y.C.);
[email protected] (D.Z.);
[email protected] (J.Z.) School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China Correspondence:
[email protected]; Tel.: +86-20-8711-4828
Academic Editor: Gabriele Grandi Received: 22 December 2015; Accepted: 1 April 2016; Published: 15 April 2016
Abstract: The output characteristics of photovoltaic (PV) arrays vary with the change of environment, and maximum power point (MPP) tracking (MPPT) techniques are thus employed to extract the peak power from PV arrays. Based on the analysis of existing MPPT methods, a novel incremental conductance (INC) MPPT algorithm is proposed with an adaptive variable step size. The proposed algorithm automatically regulates the step size to track the MPP through a step size adjustment coefficient, and a user predefined constant is unnecessary for the convergence of the MPPT method, thus simplifying the design of the PV system. A tuning method of initial step sizes is also presented, which is derived from the approximate linear relationship between the open-circuit voltage and MPP voltage. Compared with the conventional INC method, the proposed method can achieve faster dynamic response and better steady state performance simultaneously under the conditions of extreme irradiance changes. A Matlab/Simulink model and a 5 kW PV system prototype controlled by a digital signal controller (TMS320F28035) were established. Simulations and experimental results further validate the effectiveness of the proposed method. Keywords: adaptive variable step size; maximum power point tracking (MPPT); photovoltaic (PV) systems; incremental conductance (INC); adjustment coefficient; initial step sizes
1. Introduction With the increasing problem of environmental pollution and approaching depletion of conventional fossil-fuel energy sources, solar energy, as a clean, environmentally-friendly and abundant energy source is attracting more attention. An effective way of using solar energy is photovoltaic (PV) generation; however, the output characteristics of PV arrays vary with the environment (cell temperature and irradiation). The maximum power point (MPP) tracking (MPPT) techniques are thus employed to harvest the maximum power from PV arrays [1,2]. In recent years, many MPPT strategies have been proposed with differences in complexity, cost, convergence speed, and overall output efficiency [3]. Fractional open-circuit voltage (FOV) [4] and fractional short-circuit current (FSC) [5] methods take advantage of the approximate linear relationship between operating voltage or current at the MPP and open-circuit voltage or short-circuit current of PV arrays; therefore, they are simple and effective ways to track the MPP. FOV [4] and FSC [5] have already been used for PV systems of street lighting, as the precise tracking is unnecessary for it. Nevertheless, these methods have larger power
Energies 2016, 9, 288; doi:10.3390/en9040288
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losses, because the open-circuit voltage and the short-circuit current are measured by shutting down or short-circuiting PV arrays periodically. Furthermore, the operating point of FOV [4] and FSC [5] is not the real MPP, technically. Hill climbing (HC), and perturb and observe (P&O) [6–8] methods have been widely studied. The perturbation selections are different for HC and P&O: one is the duty ratio of the power converter [6] and the other one is the voltage of the PV array [7,8]. In fact, they are different realizations of the same concept. HC and P&O [6–8] have many merits, such as simple structure, ease of installation and maintenance. However, contradiction appears in choosing the perturbation parameter (duty cycle or reference voltage) in both methods. A larger parameter contributes to a better dynamic performance but excessive power loss at steady state, resulting in a comparatively low efficiency, and vice versa. In addition, fuzzy logic control [9], sliding-mode control [10], and neural-network methods [11] have also been used for MPPT over the last decades. The fuzzy logic control algorithm [9] is good at handling nonlinear problems without an accurate mathematical model, and good steady state performance can be achieved in varying atmospheric conditions. However, its effectiveness relies a lot on the experience or knowledge of the designer in determining the division of fuzzy field and formulating a fuzzy rule base table. Sliding-mode control [10] improves the dynamic performance greatly, as well as the robustness of the PV system, but its high complexity and implementation cost makes it be seldom applied for the practical system. The neural-network algorithm [11] increases the efficiency of the system by adopting a multilayer structure; however, each kind of PV array has to be specially trained to create the control rules; thus, its limitation is versatility. The incremental-conductance (INC) [12–18] method is also often applied in PV systems. It tracks the MPP by comparing the instantaneous and incremental conductance of the PV array. The issue of INC method is similar to P&O. The fixed step size is usually adopted, which determines the accuracy and response speed of MPPT. Thus, a tradeoff has to be made between the tracking speed and steady state performance. Such design dilemma can be settled with variable step size MPPT strategies. The derivative of power to voltage (dP/dV) is used to adjust the step size of MPPT. The step size is increased when the operating point is far from the MPP, and it is decreased gradually when the operating point gets close to the MPP [13]. The fast tracking speed and stable output can be simultaneously achieved by the adjustment of the step size. However, a scaling factor is necessary to ensure the convergence of the MPPT algorithm, and the scaling factor decreases the response speed greatly under rapid change of atmospheric conditions. An incremental-resistance (INR) MPPT algorithm is examined with the modified variable step size [14]. A threshold function is applied to shift between the mode of the fixed step size and the variable step size, and the variable step process is realized by a varying scaling factor. This method acquires fast response and accurate steady state performance, but the heavy computational load and strong non-linearity of the scaling factor restrict its application. In [15], there are two step size adjustment coefficients to reduce the effects to perturbation (duty ratio) under the extreme change of irradiation with less computation, while it does not consider the influence of the initial step size on the performance of the algorithm. In this paper, a novel INC MPPT method is proposed with the adaptive variation of step size. An adjustment coefficient is adopted to regulate the step size. Therefore, the PV system can keep a large step size when the operating point is far from the MPP and a decreasing step size when the operating point is close to the MPP, even under extreme irradiance change. The proposed method can effectively solve the problem of traditional method not taking into account the stability and dynamic response speed simultaneously when the irradiance changes tremendously. Compared with other variable step size methods, the proposed method also has less complex adjustment coefficient and computation to improve the computing speed. In addition, a method of tuning initial step size is also presented to further improve the dynamic performance. Simulation and experimental results verify the effectiveness of the proposed algorithm.
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2. Analysis Analysisof ofthe thePhotovoltaic Photovoltaic(PV) (PV)System System 2. 2. Analysis of the Photovoltaic (PV) System As shown As shown in in Figure Figure 1, 1, aa standalone standalone PV PV system system usually usually consists consists of of three three main main blocks: blocks: array, the MPPT control unit and DC-DC converter. Analysis will be conducted for them them in As shown in Figure 1, a standalone PV system usually consists of three main blocks: array, the MPPT control unit and DC-DC converter. Analysis will be conducted for in subsequent sections. array, the MPPT control unit and DC-DC converter. Analysis will be conducted for them in subsequent sections. subsequent sections.
PV PV the PV the the
Figure 1. Diagram of the photovoltaic (PV) system. Figure 1. 1. Diagram Diagram of of the the photovoltaic photovoltaic (PV) (PV) system. system. Figure
2.1. PV Model 2.1. PV PV Model Model 2.1. A PV array is a nonlinear device, which is modeled as a current source shunted with a diode. A PV PV array array is is aa nonlinear nonlinear device, device, which which is modeled modeled as as aa current current source source shunted shunted with with aa diode. diode. A Figure 2 illustrates the equivalent circuit of PVisarray. Figure 22 illustrates Figure illustrates the the equivalent equivalent circuit circuit of of PV PV array. array.
Figure 2. Equivalent circuit of PV array. Figure 2. Equivalent circuit of PV array. Figure 2. Equivalent circuit of PV array.
The output I-V characteristic is given as: The output I-V characteristic is given as: The output I-V characteristic is given as: q
V IR I I ph I s exp[ q (V IRs )] 1 V IRss , , I I ph I s!exp[ AkT (V IRs )] )1 V `RIR sh q Rsh s , I “ I ph ´ Is expr AkT pV ` IRs qs ´ 1 ´ AkT
3
Rsh
qEg 1 1 ˜ T¸3 I s I sref TT 3 exp[qE ( 11 )] , qE gg 11 Tref expr Ak T )] p ( Tref ´ qs,, Is “I s I sreIf sref T T exp[ Ak TTrereff T T Ak re ref f
(1) (1) (1) (2) (2)
” ı IIph “ I sreff ` (3) K KIIpT (T´25q 25) λ{100, / 100 , ph Isre I ph I sref KI (T 25) / 100 , (3) where is the where II is is the the output output current current of of the the PV PV module module in in A, A, and and V V is is the the output output voltage voltage in in V. V. IIph ph is the light-generated current in A, I is the diode reverse saturation current in A, q is the electron charge, where I is the output current of the PV module in A, and V is the output voltage in V. I ph is the light-generated current in A, Iss is the diode reverse saturation current in A, q is the electron charge, ´ 19 ´ 23 1.602׈1010 isinthe constant, k = 1.381 10 A stands J/K, stands for thefactor ideality light-generated current A, Boltzmann’s Is is the diode reverse saturation in A,Aqfor is the the ideality electron charge, −19 C, qq == 1.602 kC,iskthe Boltzmann’s constant, k = 1.381 × 10−23ˆcurrent J/K, of −19 −23˝J/K, factor of P-N junction, 1 ď A ď 2, T is the cell temperature in C, R is the intrinsic series resistance qP-N = 1.602 × 10 C, k is the Boltzmann’s constant, k = 1.381 × 10 A stands for the ideality factor s junction, 1 A 2 , T is the cell temperature in °C, Rs is the intrinsic series resistance in Ω, Rsh of is in Ω, Rsh isresistance the1shunt in Isrefreverse is the cell reverse current Tref , Egap band P-N junction, , Ω, T isIsref theiscell temperature in °C, Rs is saturation the intrinsic series at resistance Ω, Rgap sh is A in 2resistance g isin the shunt theΩ, cell saturation current at T ref, Eg is band of silicon, ˝ C), 2, K ofg silicon, E 25 Iscref is the shortcurrent circuit at current 25 Crefand short the resistance in(at Ω, Isrefis isthe the cell reverse saturation current at˝1000 T , EW/m g 1000 is band gap ofI is silicon, 2, W/m g= E =shunt 1.12 eV (at1.12 25 eV °C), Iscref short circuit 25 °Catand KI is short circuit 2 2 circuit current temperature in A/C, and λ is solar irradiation in W/m . E g = 1.12 eV (at 25 °C), I screfcoefficient is the short circuit current at 25 °C and 1000 W/m , K I is short circuit 2 current temperature coefficient in A/C, and λ is solar irradiation in W/m . current temperature coefficient in A/C, and λ is solar irradiation in W/m2. 2.2. MPPT Control Unit 2.2. MPPT Control Unit 2.2. MPPT Control Unitoutput differences of the MPPT algorithm, structures of INC MPPT control unit According to the According to the output differences of the MPPT algorithm, structures of INC MPPT control can be divided into two groups. As shown in 3,algorithm, the MPPTstructures algorithmof generates a reference to the output of shown theFigure MPPT INC MPPT controla unit According can be divided into twodifferences groups. As in Figure 3, the MPPT algorithm generates signalcan for be thedivided outer control loop groups. in structure 1, and the reference signal is eitheralgorithm a voltage generates or a current unit into two As shown in Figure 3, the MPPT a reference signal for the outer control loop in structure 1, and the reference signal is either a voltage reference signal for the outer control loop in structure 1, and the reference signal is either a voltage or a current reference. A comparator is utilized to calculate the error signal of voltage or current, or current comparator is utilized Integral to calculate error signal of voltage or current, anda this errorreference. signal is A utilized by Proportional (PI) the controller to acquire the duty cycle of and this error signal is utilized by Proportional Integral (PI) controller to acquire the duty cycle of
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reference. is utilized to calculate the error signal of voltage or current, and this 4error Energies 2016,A 9, comparator 288 of 17 Energies 2016, 9, 288 by Proportional Integral (PI) controller to acquire the duty cycle of power converter. 4 of 17 signal is utilized Instead of selecting a PV array voltageaor as the output variable, theoutput duty cycle of the power converter. Instead of selecting PVcurrent array voltage or current as the variable, thepower duty power converter. Instead of selecting PV array voltageinorstructure current as the output variable, the duty converter is obtained directly by the aMPPT algorithm 2, which results in a simplified cycle of the power converter is obtained directly by the MPPT algorithm in structure 2, which cycle of the power converter directly by the the MPPT algorithm in structure which structure to the absence of is PI obtained controller. controller leads to2,a higher results indue a simplified structure due to the However, absence of PIadditional controller.PIHowever, the additional PI results in a simplified structure due to theinabsence of1 PI controller. the PI efficiency a faster dynamic performance [10]. The MPPTHowever, algorithm willadditional be1 discussed controllerand leads to a higher efficiency and a structure faster dynamic performance in structure [10]. The controller to 3. a higher efficiency and a faster dynamic performance in structure 1 [10]. The in detail inleads Section MPPT algorithm will be discussed in detail in Section 3. MPPT algorithm will be discussed in detail in Section 3.
Figure 3. Diagram of incremental conductance (INC) maximum power point tracking (MPPT) Figure 3. Diagram of of incremental incremental conductance conductance (INC) (INC) maximum maximum power power point point tracking tracking (MPPT) (MPPT) Figure 3. Diagram control unit. control unit. unit. control
2.3. DC-DC Converter Analysis 2.3. DC-DC Converter Analysis Generally, a boost converter is utilized as the power processing unit. As shown in Figure 4, the Generally, aa boost boost converter converterisisutilized utilizedasasthe thepower power processing unit. shown in Figure 4, processing unit. As As shown in Figure 4, the boost converter consists of an inductor, a diode and two capacitors, as well as a metallic oxide the boost converter consists of an inductor, a diode andtwo twocapacitors, capacitors,asaswell wellas as aa metallic metallic oxide boost converter consists of an inductor, a diode and semiconductor field effect transistor (MOSFET) switch. semiconductor field effect transistor (MOSFET) switch. switch.
Figure 4. The proposed maximum power point tracking (MPPT) system. Figure 4. The proposed maximum power point tracking (MPPT) system. Figure 4. The proposed maximum power point tracking (MPPT) system.
To design the voltage control loop of the PV system, further studies are needed to analyze how To design the voltage control loop of the PV system, further studies are needed to analyze how variation in the duty cycle d(t) affects the input voltage VPV. The state-space averaging method [19] To design control loop the voltage PV system, studies are neededmethod to analyze variation in the the dutyvoltage cycle d(t) affects theof input VPV. further The state-space averaging [19] is applied next to derive the low-frequency small-signal model and transfer function for the how variation duty the cycle d(t) affects the input voltage VPVand . The state-space averaging is applied nextintothe derive low-frequency small-signal model transfer function for the boost converter. method [19] is applied next to derive the low-frequency small-signal model and transfer function for boost converter. It is assumed that the boost converter operates in continuous current mode, and the natural the boost converter. It is assumed that the boost converter operates in continuous current mode, and the natural frequency of the converter is far less than the switching frequency. The input voltage VPV , inductor It is assumed that the is boost converter operates in continuous current mode, andV the natural frequency of the converter far less than the switching frequency. The input voltage PV, inductor current i L, and output voltage VO are chosen as state variables, that is, x = [VPV(t), iL(t), VO(t)]T. The frequency the converter is far the as switching frequency. V VO, (t)] inductor T. The current iL, of and output voltage VOless arethan chosen state variables, thatThe is, input x = [Vvoltage PV(t), iL(t), PV T input variable isoutput input current iPV , namely u = iPVas (t).state Taking output voltage VO=as the(t), output variable, current i , and voltage V are chosen variables, that is, x [V i (t), V L O PV L O (t)] . input variable is input current iPV, namely u = iPV(t). Taking output voltage VO as the output variable, that is, y = V PV(t). According to KCL (Kirchoff's Current Law) and KVL (Kirchhoff’s Voltage Law), The input is input current , namely u = iPV (t). Taking voltage VO Voltage as the output that is, y = variable VPV(t). According to KCLiPV (Kirchoff's Current Law) andoutput KVL (Kirchhoff’s Law), the state equation is given as: variable, that is, y =isVgiven According to KCL (Kirchoff's Current Law) and KVL (Kirchhoff’s Voltage PV (t).as: the state equation Law), the state equation is given as: # ¨ x Ax`Bu Bu x (4) x“Ax Ax Bu ,,, (4) T Tx x (4) y y“ CC T y C x
where A dAd (1 d) A1 d where A dAd (1 d) A1 d
0 0 1/ L 1/ L 0 0
1/ C1 1/ C1 R/ L R/ L (1 d) / C2 (1 d) / C2
0 1 / C1 1 0 1 / C1 1 (d 1) / L , B 0 ,, C 0 .. (d 1) / L , B 0 0 C 00 1/ ROC2 1/ ROC2 0 0
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» fi fi » fi 1 1{C1 0 — ffi ffi — ffi pd ´ 1q{L fl, B “ – 0 fl, C “ – 0 fl. 0 0 ´1{RO C2
»
0 — where A “ dAd ` p1 ´ dqA1´d “ – 1{L 0
´1{C1 ´R{L p1 ´ dq{C2
^ ^ ^ ^
Introducing small perturbation p x, u, y, dq around the steady state points, and neglecting the quadratic term, the dynamic equation of boost converter is then described as: #
¨
^
^
^
x “ A x ` Bu ` E ˚ d ^
^
y “ CT x
,
(5)
»
fi » fi 0 0 0 VPV — ffi — ffi where E “ pAd ´ A1´d qx “ – 0 0 1{L fl – i L fl “ r0 VO {L ´ i L {C2 sT . 0 ´1{C2 0 VO From the Laplace Transform, the state equation becomes: $ ^ ^ ^ & psI ´ Aqxpsq “ Bupsq ` D dpsq . ^ ^ % ypsq “ C T xpsq
(6)
The transfer function of control to PV voltage can be represented as: ^
^
Gvd psq“
VPV psq ^
dpsq
“
ypsq ^
dpsq
where a1 “ Vo {LC1 , a0 “
VO 1 s´ “ C T psT ´ Aq´1 D “ |sT´ ˆ p´ LC A| 1 Vo ` Ro p1´dqi L , b2 Ro LC1 C2
“
1 R o C2
` RL , b1 “
VO ` Rp1´dqi L q“ RLC1 C2
C2 `C1 p1´dq2 LC1 C2
`
R Ro LC2 ,
` a0 ´ s3 `ba1s2s` b s`b 2
1
0
(7)
and b0 “ 1{Ro LC1 C2 .
3. MPPT Algorithm 3.1. Variable Step Size Method For fixed step size INC method, a larger step size contributes to a faster response, while more power losses are caused in steady state, thus resulting in a comparatively low efficiency. This situation is the opposite for small step size. Hence, contradiction occurs between the tracking speed and steady state performance. Such a design dilemma can be settled with a variable step size algorithm. The fixed step size is replaced by a function that depends on the derivative of power to voltage (dP/dV), and the algorithm is given by: Vre f pkq “ Vre f pk ´ 1q ˘ N ˆ |dP{dV| , (8) where Vref is the reference voltage, k and k ´ 1 are the present and previous time interval, N is the scaling factor. Variable step size methods can also be realized through the slope of P-D curve [20], and the update rule of the MPPT algorithm is presented as: Dpkq “ Dpk ´ 1q ˘ N ˆ |dP{dV| ,
(9)
where D(k) is the duty cycle of power converter at time interval k. To guarantee the convergence of the MPPT algorithm, the scaling factor must obey the following inequality [13]: N ă ∆Dmax { |dP{dV| ,
(10)
where ∆Dmax is the upper limiter of step size. If Equation (10) is satisfied, the system will be working in variable step size mode; otherwise, the system will be operating with a fixed step size of ∆Dmax .
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However, the scaling factor N and the upper limiter of step size ΔDmax cannot be altered once it However, the scaling factor N and the upper limiter of step size ∆Dmax cannot be altered once it is is tuned at design time. The selection of N and ΔDmax determines which region the working point of tuned at design time. The selection of N and ∆Dmax determines which region the working point of the the system is located in. An optimal speed factor N failed to satisfy the need of maximum power system is located in. An optimal speed factor N failed to satisfy the need of maximum power tracking tracking under the condition of intense irradiation change. under the condition of intense irradiation change. As shown in Figure 5, curve P1 and P2 are the output power of a PV array under different As shown in Figure 5, curve P1 and P2 are the output power of a PV array under different irradiation levels. The scaling factor N1 and upper limiter step size ΔDmax1 are chosen by reference to irradiation levels. The scaling factor N1 and upper limiter step size ∆Dmax1 are chosen by reference P1; in this case, fast dynamic response and good steady performance are achieved simultaneously. to P1 ; in this case, fast dynamic response and good steady performance are achieved simultaneously. However, when irradiation changes greatly, the same parameters always make the system operate However, when irradiation changes greatly, the same parameters always make the system operate within the variable step size mode for P2 curve, which increases the start-up time, as well as the within the variable step size mode for P2 curve, which increases the start-up time, as well as the response time. If the scaling factor N2 and upper limiter of step size ΔDmax2 are selected according to response time. If the scaling factor N2 and upper limiter of step size ∆Dmax2 are selected according to power curve P2, the variable step size area of the system that worked for P1 curve becomes too small, power curve P2 , the variable step size area of the system that worked for P1 curve becomes too small, which incurs severe oscillations at steady state and continuous power loss. All in all, the parameters which incurs severe oscillations at steady state and continuous power loss. All in all, the parameters have a significant effect on the system performance, and a poor choice may lead to inefficiency or have a significant effect on the system performance, and a poor choice may lead to inefficiency or failure during start-up or dynamic tracking. It is then impossible to find suitable scaling factor and failure during start-up or dynamic tracking. It is then impossible to find suitable scaling factor and upper limiter of step size that satisfy the requirements of the MPPT system under enormous upper limiter of step size that satisfy the requirements of the MPPT system under enormous irradiance irradiance changes. Furthermore, manual tuning of these parameters for different kinds of PV changes. Furthermore, manual tuning of these parameters for different kinds of PV arrays restricts arrays restricts its application. its application.
Figure under different different irradiation irradiation levels. levels. Figure 5. 5. Normalized power, power, abs(dP/dV) abs(dP/dV) under
To improve the the problem problem above, above, an an incremental incremental resistance resistance MPPT MPPT method method is is examined examined with To improve with a a modified variable step size [14], and a threshold function is introduced to switch the step size modified variable step size [14], and a threshold function is introduced to switch the step size modes modes of the algorithm: MPPT algorithm: of the MPPT F“ FP ˆ P |dP{dI| dP / dI. .
(11) (11)
The threshold function F has two extreme points at the two sides of MPP. The system works in bof The threshold function F has two extreme points at the two sides MPP. The system works in the variable step size mode, with a proportionality factor of |dP{dI| { 1 ` |dP{dI|2 to2 adjust the step the variable step size mode, with a proportionality factor of dP / dI / 1 dP / dI to adjust the size when the operating point is located between the two extreme points. Otherwise, it operates in the step the operating point is located adjusts between extremestep points. Otherwise, it fixed size step when size mode. This method automatically thethe areatwo of variable size mode and the operates in the fixed step size mode. This method automatically adjusts the area of variable step step size to track the MPP as irradiation changes. The dynamic speed and steady-state performance size mode and the step size tothe track the as irradiation The dynamic and are improved as well. However, value of MPP the threshold functionchanges. is very large, and, morespeed than once, steady-state performance improved as well.points However, theasvalue of the threshold function is derivatives are needed to are calculate the extreme as well the proportionality factor, which very large, and, more than once, derivatives are needed to calculate the extreme points as well as generates pretty heavy computational loads. Furthermore, the expression of the proportionality factor the proportionality factor,non-linearity which generates computational loads. Furthermore, the is very complex and strong exists.pretty MPPTheavy algorithms based on the current-mode feedback expression of the proportionality factor is very complex and strong non-linearity exists. MPPT control are less stable than the voltage-mode feedback control, especially when irradiation drops algorithms sharply [21].based on the current-mode feedback control are less stable than the voltage-mode feedback control, especially when irradiation drops sharply [21].
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Two step size adjustment coefficients are introduced to eliminate the differences caused by various irradiation levels [15]. The adjustment coefficients are as follows: $ ˇ ˇ ˇ ˇ & F ” 1 ´ ˇˇ I ˇˇ { ˇˇ dI ˇˇ ˇ V ˇ dV ˇ ˇ % G ” 1 ´ ˇˇ dI ˇˇ { ˇˇ I ˇˇ V dV
when when
dI dV dI dV
` `
I V I V
ă0 ă0
.
(12)
This method judges firstly whether the work point is located on the left side or the right side of the MPP and then selects the corresponding formula to calculate. It is based on dI/dV and V/I, which can reduce the computation burden effectively. However, this scheme does not take into consideration that the initial step size has an effect on the performance of the system. Lastly, a novel concise adaptive variable step size INC algorithm is proposed for the maximum power harvest under the conditions of enormous irradiance changes. 3.2. Proposed Methods Conventional algorithms of variable step size usually regulate step size through the derivative of power to voltage (dP/dV) of the PV array; however, as shown in Figure 5, derivative curves differ greatly under different irradiation levels. The derivative of power to voltage of PV array is given as: dP dpV ˆ Iq dI “ “ I`Vˆ . dV dV dV
(13)
Equation (12) indicates that the difference of derivative curves mainly depends on the output current, while output current relies on the irradiation as shown in Equation (3). Hence, a novel adjustment coefficient S(k) is presented to eliminate the difference caused by the output current under various irradiation levels: ˇ ˇ ˇ ˇ V 1 ˇ dP ˇˇ ˇˇ dI ˇˇ 1 ` Spkq “ ˆ ˇˇ “ ˆ , (14) I dV ˇ ˇ I dV ˇ where V/I represents instantaneous resistance and dI/dV represents incremental conductance. Figure 6 shows that, as output voltage V increases, V/I increases from zero, while dI/dV is almost zero when the operation voltage is located in the left side of MPP, where the current change is almost zero, and dI/dV decreases along with V increases. Therefore, (V/I) ˆ (dI/dV) decreases negatively, and its practical value is ´1 at the MPP (curves shown in Figure 6 is normalized value), which can be also validated as follows: ˇ dP ˇˇ dI “ I`Vˆ “ 0, (15) ˇ dV MPP dV ˇ V dI ˇˇ ˆ “ ´1. (16) I dV ˇ MPP As shown in Figure 7, S1 and S2 are the adjustment coefficient curves corresponding to P1 and P2 , respectively. Value of the adjustment coefficient S(k) decreases with the operating point getting close to the MPP, and it becomes zero when the system arrives at the MPP. Change trends and value ranges of S(k) are roughly the same under different irradiation levels. Furthermore, compared with the dP/dV, the adjustment coefficient S(k) varies more smoothly since its value is relatively smaller. Therefore, the adjustment coefficient S(k) is better suited for regulating the step size to track the MPP.
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Figure 6. Normalized V/I, dI/dV and and (V/I) (V/I) ׈ (dI/dV) (p.u.). Figure 6. Normalized V/I, dI/dV (dI/dV) (p.u.). Figure 6. Normalized V/I, dI/dV and (V/I) × (dI/dV) (p.u.). Figure 6. Normalized V/I, dI/dV and (V/I) × (dI/dV) (p.u.).
Figure 7. Normalized power, abs(dP/dV) and S(k).
Figure 7. Normalized power, abs(dP/dV) and S(k). Figure 7. Normalized power, abs(dP/dV) and S(k). Figure 7. Normalized power, S(k) abs(dP/dV) and S(k).on the right hand side of It should be noted that the adjustment coefficient rises rapidly the MPP, which may incur instability of MPPT algorithm. Hence, S(k)onmust meet the following It should be noted that the adjustment coefficient S(k) rises rapidly onthe theright righthand hand sideofofthe It should be noted that the adjustment coefficient S(k) rises rapidly side It should be noted that the adjustment coefficient S(k) rises rapidly on the right hand side of inequality on the right hand side the of MPP: S(k) algorithm. ≤ 1. FigureS(k) 7 shows the normalized S(k)following without thewhich MPP,may which may incur instability MPPT Hence, S(k) must the MPP, incur instability ofof MPPT algorithm. Hence, must meet themeet following inequality the MPP, which may incur instability of MPPT algorithm. Hence, S(k) must meet the following regulation, andthe theright practical with constraint are7illustrated Figure 8. Value of the inequality on hand curves side of of theS(k) MPP: S(k) ≤ 1. Figure shows thein normalized S(k) without on the right hand side of the MPP: S(k) ď 1. Figure 7 shows the normalized S(k) without regulation, inequality thethe right hand side of MPP: S(k)constraint ≤ 1. Figure 7 shows normalized S(k) without adjustment coefficient S(k) stays at the 1of when the operating point is fartheaway from 8. the MPP, regulation,onand practical curves S(k) with are illustrated in Figure Value ofand the andregulation, the practical curves of S(k)curves with constraint are illustrated in Figure 8. of the adjustment and theoperating practical of1 S(k) with illustrated in Value Figure Value of and the decreases with the point at getting close toconstraint the MPP.are adjustment coefficient S(k) stays when the operating point is far away from8.the MPP, coefficient S(k)coefficient stays at 1 when the operating point is far away from the MPP, and decreases with the adjustment S(k) stays 1 when thetooperating decreases with the operating pointatgetting close the MPP.point is far away from the MPP, and operating point getting close to the MPP. decreases with the operating point getting close to the MPP.
Figure 8. Adjustment coefficient S(k) versus voltage. Figure 8. Adjustment coefficient S(k) versus voltage.
Figure S(k)versus versusvoltage. voltage. Figure8.8.Adjustment Adjustment coefficient coefficient S(k)
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improved variable variable step step size size INC INC algorithm algorithm is is thus thus obtained obtained as: as: The update rule of the improved Vref (k) Vref (k 1) S(k) Vref Vre f pkq “ Vre f pk ´ 1q ˘ Spkq ˆ ∆Vre f,,
(17) (17)
where Vref(k) is the reference voltage at time k, Vref(k − 1) is the reference voltage at time k − 1, and where Vref (k) is the reference voltage at time k, Vref (k ´ 1) is the reference voltage at time k ´ 1, and ΔVref(k) stands for the initial perturbation step size. ∆Vref (k) stands for the initial perturbation step size. The flowchart of the proposed method is shown in Figure 9, the fundamental INC strategy is The flowchart of the proposed method is shown in Figure 9, the fundamental INC strategy is used used to judge whether the operating point is located on the right or left hand side of the MPP. to judge whether the operating point is located on the right or left hand side of the MPP. When the When the system operates on the right hand side of the MPP and S(k) ≥ 1, the proposed method is system operates on the right hand side of the MPP and S(k) ě 1, the proposed method is forced to forced to operate in fixed step mode with step size ΔVref. Otherwise, it operates in variable step size operate in fixed step mode with step size ∆Vref . Otherwise, it operates in variable step size mode mode with step size S(k) × ΔVref. The proposed method provides a simple and effective way to with step size S(k) ˆ ∆Vref . The proposed method provides a simple and effective way to harvest the harvest the maximum power. maximum power. Sample V(k)、I(k) ΔV(k)=V(k)-V(k-1)，ΔI(k)=I(k)-I(k-1) P(k)=V(k)*I(k)，ΔP(k)=P(k)-P(k-1) G(k)=I(k)/V(k)+ΔI(k)/ΔV(k) S(k)=abs(ΔP(k)/ΔV(k))/I(k)=abs(1+(V(k)/I(k))*(ΔI(k)/ΔV(k))) Yes
ΔV(k)=0
Yes ΔI(k)=0
No
Yes ΔI(k)>0 Vref(k)=Vref(k-1)
Yes
G(k)=0
G(k)>0
No
No
Yes
S(k)
