DEM: Direct Estimation Method for Photovoltaic Maximum Power Point Tracking

Available online at www.sciencedirect.com

Procedia Computer Science 17 (2013) 537 – 544

Information Technology and Quantitative Management , ITQM 2013

DEM: Direct Estimation Method for Photovoltaic Maximum Power Point Tracking Jieming Maa,b , K.L. Manb , T.O. Tingb,∗, N. Zhangb , Sheng-Uei Guanb , Prudence W.H. Wonga b Xi’an

a University of Liverpool, UK Jiaotong-Liverpool University, P.R.China

Abstract Since the electrical characteristics of a Photovoltaic (PV) Module vary with the changing atmospheric conditions, researchers have shown an increasing interest in Maximum Power Point Tracking (MPPT) approaches. This paper presents a direct Maximum Power Point (MPP) estimation method derived from the mathematical expressions of the Current-Voltage (I-V) characteristics of a PV module. Simulation results demonstrate that the proposed approach is sufficiently accurate for practical applications.

© 2013 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of the organizers of the 2013 International Conference on Information Technology and Quantitative Management Keywords: Photovoltaic (PV), Maximum Power Point (MPP), MPP estimation

1. Introduction Owing to the sustained growing concern about the global energy crisis and environmental issues, the interest in sustainable energy source is progressively increasing. Photovoltaic (PV) generation known as one of the promising green alternative energy sources serves to generate electricity for a wide range of applications. In recent years, the PV industry has experienced phenomenal growth, with market demand expanding at an annual rate in excess of 40% [1]. PV cells are the elementary components of a PV generator and their electrical properties are exhibited nonlinear in light of recent research results [2]. A commercial PV module is composed of a series of PV cells connected electrically. In view of the fact that the power generated by PV modules is heavily dependent on a number of atmospheric conditions (e.g. temperature and solar irradiance), the efficiency of energy conversion has drawn the most attention in PV system design [3, 4, 24]. The term Maximum Power Point (MPP) is used to refer to the operating point where a PV module produces the maximum power under a specific environmental condition. Loosely speaking, the better a PV system tracks MPPs, the higher energy conversion efficiency it can achieve. Maximum Power Point Tracking (MPPT) algorithms provide the theoretical means to constrain operating points to work on the MPPs by detecting of several input signals. Over the years, a considerable number of MPPT algorithms have been proposed to improve the efficiency of energy conversion. Characterized by the function of control strategies, MPPT algorithms can be grouped as “offline” and “on-line” [2, 10]. With the aid of some numerical approximation methods or mathematical expressions ∗ Corresponding

author Email address: [email protected] (T.O. Ting)

1877-0509 © 2013 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of the organizers of the 2013 International Conference on Information Technology and Quantitative Management doi:10.1016/j.procs.2013.05.069

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Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

of PV electrical characteristics, off-line algorithms search MPPs based on the measures of an array of parameters such as temperature, irradiance, operating voltage and current of a PV module. On-line MPPT algorithms, however, are independent of the prior MPP prediction and require lower computational load in most cases. After acknowledging the operating current and voltage, on-line methods enable the system to obtain an optimum operating point. Either on-line or off-line approach has its disadvantages. Although off-line algorithms have shown their capabilities of optimizing the operating points under rapid changing environmental conditions in many cases, their tracking performance depends not only on the convergence speed of the algorithm, but also on the accuracy of the mathematical model and the quality of the sensors applied in measures [5]. On-line algorithms are feasible for implementation. However, they are usually insensitive to the changing non-linear electrical characteristics of a PV module [4, 17]. MPP estimation wards off the iterative MPP seeking process and this form of off-line methods is therefore an increasing concern [8, 19, 20]. The main disadvantages of the existing estimation methods lie in the fact that they are developed on the basis of simplified PV models and their estimation quality is limited. Without overlooking the effect of series resistance (Rs) and shunt resistance (Rp), a high-accuracy estimation method of MPPs is proposed in this paper. By only using the parameters in datasheet, the proposed Direct Estimation Method (DEM) is capable of predicting the upper bound and lower bound of MPPs. The significance of the new estimation method lies not only in its accurate prediction capacities, but also in its great auxiliary aids in MPPT. Initiated with the MPP DEM results, an on-line MPPT algorithm can avoid a wide range of seeking processes and thus the number of iterations tends to be dramatically reduced. 2. PV Module Model




























(a)














 

 



(b)




 

(c)

Fig. 1. Different mathematical models for a PV module: (a) ideal model, (b) Rs-model and (c) Rp-model

A precise model is the basis of the high-performance MPP estimation. For the sake of studying the forward Current-Voltage (I-V) and Power-Voltage (P-V) characteristics, many mathematical models have been developed in the past decades [12]. As shown in Figure 1, depending on the quantity of effects taken into account in modeling, PV module models may be classified into three types: (a) ideal model; (b) Rs-model and (c) Rp-model. An ideal PV module can be simply expressed as a linear independent current source in parallel to a diode [6]. When the p-n junction of a PV module is exposed to incident light, a reverse current, which depends linearly on the solar irradiance intensity (G) and is also slightly influenced by the temperature (T ) [12, 18], is generated across the junction. This current is known as photocurrent I pv and it can be mathematically expressed as: I pv = (I pv,S TC + Ki ΔT )

G GS TC

(1)

where I pv,S TC is the photocurrent at Standard Test Conditions (STC) (T=25◦C, G = 1000W/m2 ), Ki is a constant named short circuit current coefficient, ΔT = T − T S TC (T S TC is the nominal temperature), and GS TC is the solar irradiance at STC. Eliminated the effect of photocurrent, a PV module behaves like a conventional diode and the I-V characteristics of an ideal PV module can be derived by Equation (2) :

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Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

Rp−model Rs−model ideal model exprimental data MPPs (Rp−model) MPPs (Rs−model) MPPs (ideal model)

70 60

Power(W)

50 40 30 20 10 0 4 25

3 20

2

15 10

1

5 0

Current(A)

0

Voltage(V)

Figure 2. MPPs and characteristic curves of a PV module

V

I = I pv − Io (e nNs VT − 1)

(2)

In this equation, n is the diode ideality constant and varies between 1 and 2 depending on the material and the physical structure of the diode. The number of series connected cells in the module is denoted by N s . VT is a constant called the thermal voltage, whose value is a function of temperature (T) that can be written as: VT = kT/q

(3)

where k is the Boltzmann constant (1.380650 × 10−23 J/K) and q is the electron charge (1.602176 × 10−19 C). Io denotes the saturation current and its value can be described as a nonlinear function [12]: Io =

(I sc,S TC + Ki ΔT ) e(Voc,S TC +Kv ΔT )/(nNs VT ) − 1

(4)

In Equation (4), the short circuit current coefficient Ki and the open circuit coefficient Kv , as well as the short circuit current and the open voltage at STC (I sc,S TC , Voc,S TC ), are normally available in the datasheet. Rs-model improves the ideal model by recognizing the series resistive losses in practical solar modules, and has been widely applied in a wide range of research activities like modeling, simulation and estimation [7, 8, 9, 10]. The values of PV current and voltage forming I-V curve are given in Equation (5): V+IR s

I = I pv − Io (e nNs VT − 1)

(5)

According to [11], the accuracy of Rs-model deteriorates at high temperature. Rp-model further improves the modeling accuracy by identifying a host of shunt resistive losses, such as perimeter shunts along cell borders. The refined I-V relationship is expressed as: V+IR s

I = I pv − Io (e nNs VT − 1) −

V + IR s Rp

(6)

The approximation value of series and shunt resistance can be obtained from an iterative approach as described in [12]. Figure 2 provides an intuitive view of the I-V and P-V characteristics of a PV module at STC. The four curves shows the solar P-V relationships, while the corresponding I-V characteristics of the PV module can be presented by their projections in the base. Some of the critical observations made from the study of these curves are threefold as follows: 1. Only a single MPP exists in the P-V characteristic curve under a steady environmental condition. It may vary around the range of 73% – 80% of the open-circuit voltage Voc as the external conditions change [13, 14].

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Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

2. The accuracy of the Rp-model is prior to other analytical models, namely, the ideal model and the Rs-model [12]. This is because series and shunt resistive effects are taken into account in the Rp-model. The DEM described in this paper is therefore based on the Rp-model. 3. The intrinsic error of the characteristic expressions of a PV module affects the prediction performance for MPPs. Even though the figure size is relatively limited, significant deviations can be identified in the MPP positions of the ideal and the Rs-model. 3. Proposed MPP DEM MPP estimation, which extracts an approximation of MPPs by a number of atmosphere parameters, is capable of providing forecasts for MPPT. To the best of our knowledge, most of the existing MPP estimation methods apply the ideal or the Rs-model and perform the predictions by ignoring series or shunt resistive effect. As mentioned in Section 2, the estimation performance is mainly dependent upon the accuracy of the mathematical models. Estimation will never achieve high degree precision on condition that it is based on a coarse model. Consequently, Equation (6), namely, the mathematical expression of I-V characteristic for the Rp-model, is applied for studying in this paper, and its circuit model is already shown in Figure 1 (c). Assume that the potential difference between the output terminals of a PV module is V. The voltage drop across the diode VD can be written as the following equation by means of Kirchhoff’s Voltage Law (KVL): VD − V − IR s = 0

(7)

Obviously, the operating current I can be expressed by V and VD after reformulating (7): VD − V (8) Rs Equation (8) is not the only equation that can be obtained to express current I. Remember that the total current flowing towards junction D is equal to the total current flowing away from D. Another mathematical expression for I can be written by means of Kirchhoff’s Current Law (KCL) as: I=

VD

I = I pv − Io (e nNs VT − 1) −

VD Rp

(9)

Taking into account the direction of the current flow in Figure 1(c), dc power generated from the solar module is calculated by: P = VI = V · or

VD − V Rs VD

P = V I = V · [I pv − Io (e nNs VT − 1) −

(10) VD ] Rp

(11)

As from Figure 2, the P-V characteristics of a PV module exhibit nonlinear convex curves with hill shapes. According to the theory of Fist Derivative Test for Local Extreme Values [21], the first derivative of a continuous and derivable function is equal to 0 in its local maximum point. Intuitively, the power tangent at a MPP is horizontal. By using mathematical knowledge, two equations for the slope of operating power can be derived from Equations (10) and (11). By applying derivative to (10), we get: 1 ∂VD ∂P = (VD + V − 2V) ∂V Rs ∂V and applying derivative to (11), we obtain: VD VD ∂P VD  Io V V  ∂VD = I pv − Io (e nNs VT − 1) − − (e nNs VT ) + ∂V Rp nN s VT R p ∂V Since

(12)

(13)

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Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

 ∂P   =0 ∂V V=VMP

(14)

the power slope ∂P/∂V at a MPP can be further written as:   1 ∂VD   − 2V MP = 0 VD,MP + V MP Rs ∂V V=VMP and

 I V VD,MP VD,MP VD,MP V MP  ∂VD  o MP  (e nNs VT ) + − I pv + Io (e nNs VT − 1) + =0 nN s VT R p ∂V V=VMP Rp

(15)

(16)

where VD,MP is the voltage drop across the diode at the MPP. V MP is the potential difference between the output terminals of the PV module at the MPP. From Equation (15), ∂P/∂V at a MPP can be expressed as a simple rational function:  2V MP − VD,MP ∂VD   = (17) ∂V V=VMP V MP After substituting Equation (17) into (16), a new relationship between V MP and VD,MP can be derived: Io (e

VD,MP nN s VT

)=

I pv + Io −

2V MP Rp

(18)

2V MP −VD,MP nN s VT

1+

Based on Shockley diode equation [1], the left-hand side of Equation (18) is the current across the diode at the MPP (ID,MP ). Thus, ID,MP is described by a rational function replaced with the nonlinear mathematical expression. For simplicity, α is applied to represent the difference between VD,MP and V MP , namely, the voltage drop across the series resistance. Then, VD,MP can be expressed by the following equation after some simplification: VD,MP = nN s VT ln

I pv + Io − Io (1 +

2V MP Rp

(19)

V MP −α nN s VT )

According to Ohm’s Law, α is the product of operating current I and series resistance R s . Since I varies from 0 to I pv , α is in the range 0 ≤ α ≤ I pv R s . By substituting the upper bound and lower bound of α into Equation (19), the maximum and minimum values of VD,MP can be obtained by Equations (20) and (21) respectively.  VD,MP 

= nN s VT ln

α=I pv R s

 VD,MP 

α=0

= nN s VT ln

I pv + Io − Io (1 +

V MP −I pv R s nN s VT )

I pv + Io − Io (1 +

2V MP Rp

2V MP Rp

V MP nN s VT

)

(20)

(21)

It is easy to understand that the MPP is also an operating point at I-V characteristic curve of a PV module. Therefore, I MP and V MP must satisfy the relationships given by Equations (8) and (9). I MP =

VD,MP − V MP Rs VD,MP

I MP = I pv − Io (e nNs VT − 1) −

(22) VD,MP Rp

(23)

After eliminating I MP , observe that V MP can be expressed in terms of VD,MP by equating Equations (22) and (23): VD,MP

V MP = [Io (e nNs VT − 1) +

VD,MP − I pv ]R s + VD,MP Rp

(24)

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Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

60

Initiate with DEM results

59.76

59.5

Power(W)

59

58.5

58.25

58

P&O [17]

57.5

FSM [18] DEM − FSM

57 0

2

4

6

8

10

12

14

16

Iteration

Figure 3. Estimated upper bound and lower bound of the MPP at STC Figure 4. Seeking process of different MPPT algorithms under STC

As being discussed in Section 2, n and N s are constants. VT , I pv , R s and R p can be calculated by the given environment parameters (T and G). Finally, after substituting Equation (20) into (24), the variable VD,MP is eliminated and we can derive the upper bound of MPPs (V MP,U ) directly by a simple root-finding algorithm (e.g. Newton-Raphson [23]). The lower bound of MPPs, defined as V MP,L , can be estimated in a similar way. For a clear interpretation, the estimation results can be alternatively presented as an approximation value with tolerances: V MP =

V MP,U + V MP,L V MP,U − V MP,L ± 2 2  

 

approximation value

(25)

tolerance

In summary, the proposed estimation method is derived from the mathematical expression of the Rp-model and only two parameters, namely temperature and solar irradiance intensity, are required in calculation. With the well-known range of operating current, the proposed estimation method is able to predict MPPs with a relatively small tolerance. According to the relationships between V MP and atmospheric conditions, the tolerance may be effected by the tolerance of meters used in the practical measurements. Pseudocodes are given to summarize the process of proposed MPP DEM: Algorithm 1 Pseudocodes of proposed MPP DEM 1: Collect the parameters in datasheet 2: Calculate R s and R p by the method in [12] 3: Substitute Equation (20) into (24), and obtain the estimated V MP,U by a root-finding algorithm 4: Substitute Equation (21) into (24), and obtain the estimated V MP,L by a root-finding algorithm 5: Obtain estimation results by Equation (25)

4. Validation of Proposed MPP DEM The proposed estimation method was implemented in MATLAB/SIMULINK. Figure 3 shows the estimated upper bound and lower bound of the MPP at STC. The square, which is at the P-V curve, denotes the MPP of the Rp-model (V MP,S ). It is obtained by the analytical model in [12] with an optimization algorithm (tolerance = 10−4 ). The upper bound and the lower bound of MPPs are estimated after some mathematical calculations. The circle near the lower bound is the MPP provided in datasheet. It is worth pointing out that the estimation method described in this paper is based on the Rp-model. Although the accuracy of the Rp-model has been proven by [12, 13, 14] and the mathematical approach is sufficiently accurate to predict MPPs, the experimental value may be outside of the interval due to the intrinsic error of the mathematical expressions of the Rp-model.

Jieming Ma et al. / Procedia Computer Science 17 (2013) 537 – 544

Table 1. MPP DEM results of different PV modules under STC MonoModule Thin-film Multi-crystalline crystalline Type Module SP-70 ST-40 MSX60 SM55 KG200GT V MP,U a

16.5649

16.6916 17.1548 17.4573

26.3598

V MP,L b

16.4237

16.3321 17.0979 17.3806

26.2297

c

16.5484

16.6457 17.1220 17.4490

26.3497

V MP,S

V MP,D d a b c d

16.5

16.9

17.1

17.4

26.3

Operating voltage of the estimated MPP upper bound ( V ) Operating voltage of the estimated MPP lower bound ( V ) Operating voltage of the MPP in Rp-model ( V ) Operating voltage of the MPP in datasheet ( V )

Table 2. MPP DEM results and their relative errors under different atmospheric conditions. Environ. Ga

Tb

V MP,S

Estimation Results [22]

Proposed V MP

Relative Errors [22]

V MP,U

V MP,L

200

0 18.0965 18.4611 18.0933±0.0046 2.0148% 0.0070% 0.0429%

400

0 18.7271 19.1315 18.7197±0.0091 2.1594% 0.0083% 0.0886%

600

0 19.0015 19.5243 18.9897±0.0136 2.7514% 0.0096% 0.1339%

800

0 19.1436 19.8031 19.1272±0.0183 3.4450% 0.0099% 0.1814%

1000 0 19.2157 20.0194 19.1950±0.0231 4.1825% 0.0125% 0.2282% 200 25 15.9156 16.2609 15.9116±0.0057 2.1696% 0.0102% 0.0611% 400 25 16.5899 16.9848 16.5811±0.0112 2.3804% 0.0145% 0.1203% 600 25 16.8947 17.409 16.8809±0.0168 3.0441% 0.0175% 0.1812% 800 25 17.0593 17.7105 17.0407±0.0225 3.8173% 0.0228% 0.2414% 1000 25 17.1499 17.9445 17.1264±0.0285 4.6333% 0.0288% 0.3032% a b

Solar irradiance intensity ( W/m2 ) Temperature ( ◦ C )

The estimation method has been applied to a variety of PV modules, including mono-crystalline, multicrystalline and thin film types. Table 1 shows the simulation results of MPP DEM under STC. All the values of V MP,S , even the ones in datasheet (V MP,D ), are in the range that is identified by estimation. Moreover, the approximation values are relatively close to their reference values. In order to further study the accuracy of the proposed DEM, the relative error e, which is expressed as e = (|p − p|)/p ˆ in [23], is introduced. pˆ denotes an approximation value and p is the true value that is never equal to 0. With different atmospheric conditions, Table 2 illustrates the estimation results and their corresponding relative errors. Meanwhile, the method presented in [22], which predicts the MPPs by ignoring the series and shunt resistive effect, has been used as a reference model in validation. Simulation results exhibit that the proposed estimation approach constrains the relative error within 0.5% and reduces the error at least 4 times with respect to the method in [22]. Since the new method can provide either an approximation value or a specified range of MPPs, it can aid on-line algorithms for tracking MPPs. Figure 4 shows the seeking process of different MPP algorithms under STC. One iteration refers to a sampling period. Since Perturbation and Observe (P&O) [15] is still by far the most commonly used on-line algorithm in commercial converters, it is presented as a reference method. P&O perturbs the operating point and determines the change of direction by comparing the power with the historical reference value. As seen in Figure 4, initiated with the power of 58.25W, the output power cannot achieve 59.76 W until after 15 iterations. Fibonacci Search Method (FSM)-based MPPT algorithm [16] uses variable step sizes replacing the monotonic fixed increments, and it requires only 10 iterations to approach the MPP. With the help of

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MPP estimation, the tracking efficiency of DEM-FSM can be improved significantly. It starts at the same initial value as P&O and FSM. After only one sampling period, DEM-FSM already approaches the MPP. Obviously, the convergence speed is greatly improved. 5. Conclusions Based on the fact that the tangent of power is horizontal at a MPP, this paper has presented a new MPP DEM derived from the mathematical expressions of the electrical characteristics of the Rp-model. Since temperature and solar irradiance intensity have been taken into account in the estimation process, the method works for a variety of atmospheric conditions. By careful inspection of simulated values and theoretical values, the proposed method has shown its high-accuracy in off-line prediction performance. Moreover, off-line algorithms can accelerate the convergence speed by using the estimation results as initial values. The hybrid approach tends to have higher tracking efficiency and stability. References [1] A. 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