Power Optimization for Photovoltaic Micro ... - Semantic Scholar
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Power Optimization for Photovoltaic Micro-Converters using Multivariable Newton-Based Extremum-Seeking Azad Ghaffari, Miroslav Krsti´c, and Sridhar Seshagiri Abstract— Extremum-seeking (ES) is a real-time optimization technique that has been applied to maximum power point tracking (MPPT) design for photovoltaic (PV) micro-converter systems, where each PV module is coupled with its own DCDC converter. However, most existing designs are scalar, i.e., employ one ES MPPT loop around each converter, and all current designs, whether scalar or mutivariable, are gradientbased. The convergence rate of gradient-based designs depends on the Hessian, which in turn is dependent on environmental conditions such as irradiance and temperature. Consequently, when applied to large PV arrays, the variability in environmental conditions and/or PV module degradation result in non-uniform transients in the convergence to the maximum power point (MPP). Using a multivariable gradient-based ES algorithm for the entire system instead of a scalar one for each PV module, while decreasing the sensitivity to the Hessian, does not eliminate this dependence. We present a recently developed Newton-based ES algorithm that simultaneously employs estimates of the gradient and Hessian in the peak power tracking. The convergence rate of such a design to the MPP is independent of the Hessian, with tunable transient performance that is independent of environmental conditions. We present simulation results that show the effectiveness of the proposed algorithm in comparison to multivariable gradientbased ES.
I. I NTRODUCTION a) Motivation: Maximum power point tracking (MPPT) algorithms for extracting the maximum achievable power from a photovoltaic (PV) system have been studied by several researchers [4], [13], [15], [16], [17], with detailed comparisons presented in [5], [9], [10]. Several recent works [2], [3], [12], [14] have focused on the application of gradient-based extremum-seeking (ES) [1] to MPPT design. In [8], we presented a multivariable gradient-based ES MPPT design for the micro-converter architecture, where each PV module is coupled with its own DC-DC converter. The design reduced the balance-of-system cost by reducing the number of required sensors (hardware reduction), and was shown to result in more uniform transients under sudden changes in solar irradiance and environmental temperature in comparison to a scalar gradient-based ES for each PV module. However, the convergence to maximum power point (MPP) is dependent on the unknown Hessian, which varies Azad Ghaffari is with Joint–Doctoral Programs (Aerospace and Mechanical) between San Diego State University and University of California at San Diego, La Jolla, CA 92093-0411, USA, [email protected]. Miroslav Krsti´c is with Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA, [email protected]. Sridhar Seshagiri is with Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA 92182-1309, USA, [email protected].
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
with irradiance, temperature, and module degradation and mismatch. In order to alleviate the issue of unknown Hessian dependent convergence of a gradient-based ES algorithm, we presented a multivariable Newton-based ES design for general nonlinear systems in [6]. In comparison with the standard gradient–based multivariable extremum seeking, the algorithm in [6] removes the dependence of the convergence rate on the unknown Hessian and makes the convergence rate of the parameter estimates user–assignable. In particular, all the parameters can be designed to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets. When applied to the MPPT problem in PV systems, the method offers the benefit of uniform convergence behavior under a wide range of working conditions, that includes temperature and irradiance variations, and the non-symmetric power generation of the neighboring PV modules as a result of module degradation or mismatch. b) Our contribution: We present a multivariable Newton-based ES scheme with the following features: (1) It is applied to micro-converter systems, and hence deals with the case of non-unimodal power characteristics, and deals specifically with the issue of module mismatch (for example, possibly different irradiance levels as a result of partially shaded conditions). (2) The use of the non-model-based ES technique makes the design robust to partial knowledge of the system parameters and operating conditions. (3) As opposed to gradient-based designs, our proposed Newtonbased design removes the dependence of the transient on the working condition and potential mismatch in PV modules. (4) Lastly, the multivariable design requires fewer sensors, which reduces the hardware cost compared to scalar designs. c) Organization: The rest of this paper is organized as follows: The mathematical model of a PV module, along with a discussion of the DC/DC converter power electronics, is briefly presented in Section II. This section is based on our previous work, and is reproduced here for the sake of completeness/clarity. Section III summarizes the multivariable gradient-based ES scheme in [8], and allows us to present key distinctions with the proposed design. Our proposed multivariable Newton-based ES is presented and discussed in Section IV. Some simulation results are presented in Section V, and our work is summarized and some concluding remarks made in Section VI.
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S is
+ vd −
ic
Rp
(a)
(+)
Current(A)
Rs
vc (-)
6
5
5
4
4
3
3
2 1 0 0
Fig. 1.
Equivalent circuit of a PV cell. Power(W)
200
II. P HOTOVOLTAIC M ODULES AND P OWER E XTRACTION Our design and analysis are based on the standard PV cell model described for example in [18], and shown schematically in Fig. 1. The PV cell is modeled as an ideal current source of value is in parallel with an ideal diode with voltage vd . Electrical losses and contactor resistance are accounted for by the inclusion of the parallel and series resistances Rs and Rp respectively. The amount of generated current is is dependent on the solar irradiance S and the temperature T through the following equation S , (1) is = (Is + ki (T − Tr )) 1000 where Is is a reference short-circuit current, Tr a reference temperature, and ki the short-circuit temperature coefficient. The diode models the effect of the semiconductor material and its I − V characteristics are given by vd id =i0 exp −1 , (2) N Vt 3 T kT Eg T (3) − 1 , Vt = exp i0 =I0 Tr N Vt Tr q where I0 , Eg and N are respectively the diode reference reverse saturation current, the semiconductor bandgap voltage (barrier height), and the emission coefficient, all three being cell material/construction dependent, Vt is the thermal cell voltage, and k = 1.38 × 10−23 J/K and q = 1.6 × 10−19 C are Boltzman’s constant and the charge on an electron respectively. The cell model described by the above equations along with KCL/KVL: ic = is −id −vd /Rp , vd = vc +ic Rs , is then scaled to represent a PV module by considering ns cells in series (each having thermal voltage Vt ), so that the terminal I − V relationship for the PV module is given by # v " v ( ns + iRs ) ns + iRs i = is−i0 exp −1 − . (4) N Vt Rp For the sake of model development and performing simulations, we pick the PV module 215N from Sanyo, with the following numerical values derived from the manufacturer’s datasheet [7]: Eg = 1.16 eV, N = 1.81, I0 = 1.13×10−6 A, Is = 5.61 A, ki = 1.96 mA/K, Tr = 298.15 K, Rs = 2.48 mΩ, Rp = 8.7 Ω, and the number of PV cells connected in series is ns = 72. The resulting I −V and P −V curves are shown in Fig. 2. As is clear from Fig. 2(b,d), the power-voltage (P–V) characteristic has a unique but (T, S) dependent peak (v ∗ , p∗ ). It is the job of the MPPT algorithm to automatically track this peak. In many grid-tied PV systems (including our current
(c)
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Fig. 2. Characteristic (a) I−V and (b) P−V for varying irradiance, T=25◦ C. Characteristic (c) I−V and (d) P−V for varying temperature, S=1000W/m2 .
work), this is done by means of a separate DC/DC power electronics stage that serves two functions: (i) regulating the output DC voltage at a (near) constant value, and (ii) extracting maximum power by forcing the PV module output v to equal v ∗ . Fig. 3 shows this setup for a DC/DC boost converter stage, whose output voltage is maintained constant as vdc . The ratio between the input voltage v and output voltage vdc can be controlled by changing the duty cycle of the transistor switch, which serves as the control input d. Under the assumption that the boost converter is working in Continuous Current Mode (CCM), and that the switching Pulse Width Modulation (PWM) frequency fc is significantly higher than the bandwidth of the control loop, the boost converter input-output voltage relationship is given by the following (averaged) relations: v
= =
idc
(1 − d)vdc (1 − d)i.
(5) (6)
From (4), (5) and Fig. 2(b,d), it follows that at the MPP def (v ∗ ,p∗ ), the power p = iv = f (v)v = J(v), satisfies g
=
h
=
∂J ∗ (v ) = 0 ∂v 2 ∂ J ∗ (v ) < 0. ∂v 2
(7) (8)
Also we have ∂v/∂d = −vdc then g¯ = ¯ h
=
∂J ∗ (d ) = −gvdc = 0 ∂d 2 ∂ J ∗ 2 (d ) = hvdc < 0. ∂d2 Diode
L
i + PV
v
Module T, S −
Fig. 3. bus.
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(9) (10)
idc +
C′
d
Q C iq
vdc
DC Bus
−
DC/DC boost converter for PV module supplying power to DC
Converter I1
(T1 , S1 )
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Fig. 4. A cascade PV system including n PV module. Each module has a separate DC/DC converter.
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Many MPPT techniques, including the classical perturb-andobserve (P&O) class of methods, and extremum-seeking (ES) techniques, are based on detecting the sign of the power gradient. The next section discusses gradient-based ES MPPT design for a cascade configuration of PV modules. III. M ULTIVARIABLE G RADIENT-BASED E XTREMUM S EEKING A cascade PV system is shown in Fig. 4. A DC/DC boost converter is assigned to each PV module to extract maximum power from the PV system. The output side of the converters are connected in series. The whole system is connected to the AC grid through a DC/AC inverter. It is assumed that the DC voltage at the input side of the inverter is held constant at VDC . Assume that the voltage and current ripple at the output side of converters are negligible. Applying electrical rules on the input side of the inverter gives n X
Voj
=
VDC
Ioj
=
IDC ,
(11)
j=1
∀j ∈ {1, 2, · · · , n}.
(12)
From (5), (6), and the I–V functional dependence Ij = fj (Vj ), the relation between the voltage V = [V1 V2 · · · Vn ]T of PV modules and the pulse duration D = [D1 D2 · · · Dn ]T is defined by n independent equations n X
Vj 1 − Dj
=
VDC
(13)
(1 − Dj )fj (Vj )
=
IDC ,
∀j ∈ {1, 2, · · · , n}. (14)
j=1
This means that for each set of pulse duration we have a unique set of voltages for PV modules. Assuming lossless converters results in Poj = Voj Ioj = Pj for all j ∈ {1, 2, · · · , n}. Using (12) we obtain Pj Voj = Pn
j=1
Pj
VDC ,
(15)
Fig. 5. Variation of the power of a cascade PV system including two Sanyo PV modules versus pulse duration, D = [D1 D2 ]T . Level sets show the power in Watt. S1 = S2 = 1000 W/m2 , T1 = T2 = 25◦ C, and VDC = 200 V.
which means that the share of the output voltage of each converter from DC bus voltage is defined by the generated power of its relevant PV module. The variation of the generated power of a PV system including two series module from Sanyo connected to a DC bus with VDC = 200 V under standard test condition, S=1000 W/m2 and T=25 ◦ C, is shown in Fig. 5. Maximum power point (MPP) happens at D = [0.57 0.57]T . As is common in ES design (see for example, [11]), we make the following assumption about the system. Assumption 1: There exists D∗ ∈ Rn such that ∂P (D∗ ) = 0, (16) ∂D ∂2P (D∗ ) = H < 0, H = HT . (17) ∂D2 Fig. 6 shows the multivariable gradient-based ES design, where Kg is a positive diagonal matrix, and the perturbation signals are defined as T (18) Q(t) = a sin(ω1 t) · · · sin(ωn t) , T 2 sin(ω1 t) · · · sin(ωn t) , M (t) = (19) a
where ωj /ωk are rational for all j and k, and a is a real number, with the frequencies chosen such that ωj 6= ωk and ωj + ωk 6= ωm for all distinct j, k, and m. ˆ of the In particular, the design derives an estimate G gradient vector by adding a probing signal to the estimate ˆ = [D ˆ1 D ˆ2 · · · D ˆ n ]T of the pulse duration vector (of D all the DC/DC converters). With no additional information on the Hessian (and also for simplicity), we choose the amplitudes of the probing signals to all be the same value a. As before, smallness of the probing frequencies and the
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D1 D2 . . . Dn
D
DC/DC DC/DC . . . DC/DC
V1 V2 . . . Vn
PV 1 PV 2 . . . PVn
P1 P2 . . . Pn
Q(t) ˆ +D Fig. 6.
P =
Pn
M (t) ˆ G
Kg s
ωl s+ωl
×
j=1
Pj
D1 D2 . . . Dn
D
s s+ωh
DC/DC DC/DC . . . DC/DC
V1 V2 . . . Vn
PV 1 PV 2 . . . PVn
P1 P2 . . . Pn
Q(t)
z =P −η
ˆ +D
P =
Kn s
ˆ −ΓG
ˆ G
ωl s+ωl
×
ωj
= = =
Kg
=
j ∈ {1, 2, · · · , n}
ˆ Γ˙ = ωl Γ − ωl ΓHΓ (20) (21) (22) (23)
where ω and δ are small positive constants, and ωj′ , ωh′ , ωl′ and elements of Kg′ are O(1) positive real parameters. It can be shown that for sufficiently small ω, δ, and a, and with ˆ of the pulse duration vector and Kg > 0, the estimate D the output P converge to O(ω + δ + a)-neighborhoods of the optimal pulse duration D∗ = [D1∗ D2∗ · · · Dn∗ ]T and the peak power P ∗ respectively. The linearized update equation for the estimation error ˜ =D ˆ − D∗ is D ∂2P (D∗ ) (24) ∂D2 Pn where H is the Hessian of P = j=1 Pj with respect to pulse duration vector, D. Since P varies with irradiance, temperature, and degradation of the PV modules, so does H, and therefore a fixed adaptation gain Kg results in different (condition dependent) convergence rates for each converter. In order to alleviate the issue of unknown Hessian dependent convergence, we present in the next section a modified version of a multivariable Newton-based ES design that we developed in [6]. In comparison with the gradient-based multivariable extremum seeking presented in this section, the Newton-based algorithm makes the convergence rate of the parameter estimates user-assignable. In particular, all the parameters can be designed to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets. When applied to the MPPT problem in PV systems, the method offers the benefit of uniform convergence behavior under a wide range of working conditions, that include temperature and irradiance variations, and the non-symmetric power generation of the neighboring PV modules as a result of module degradation or mismatch. ˜˙ = Kg H D, ˜ D
Pj
s s+ωh
z =P −η W (t)
matrix gain Kg are ensured by selecting these as
ωh ωl
j=1
M (t)
Multivariable gradient-based ES for MPPT of a PV system.
ωωj′ , ωδωh′ ωδωl′ ωδKg′
Pn
H :=
IV. M ULTIVARIABLE N EWTON -BASED E XTREMUM S EEKING The multivariable Newton-based ES that we propose is shown schematically in Fig. 7. As is clear from the figure, the proposed scheme extends gradient-based ES with estimates
ˆ H
ωl s+ωl
×
Fig. 7. Multivariable Newton-based ES for MPPT of a PV system. The red dashed part is added to the gradient-based ES to estimate the Hessian.
of the Hessian. The perturbation matrix W (t) is defined as follows 16 1 2 Wjj = (25) sin (ω t) − j a2 2 4 sin(ωj t) sin(ωk t), j 6= k (26) Wjk = a2 where j, k ∈ {1, 2, · · · , n}. The product of W (t) and P generates an initial estimate of the Hessian, with the lowˆ pass filter reducing the high frequency oscillation in H. The inverse of the Hessian that is needed to implement the Newton algorithm, is estimated by a dynamic filter that has the form of a Riccati equation, and avoids the possible ˆ is singular. problem in inverting the Hessian if the estimate H It is shown in [6] that after a transient, the Riccati equation converges to the actual value of the inverse of Hessian matrix ˆ is a good estimate of H. if H Since we are integrating over a finite time period, and we set the phase delays of the periodic perturbation signals equal to zero, it is possible to exclude the condition ωj 6= ωk +ωm . The probing frequencies need to satisfy (see [6] for more details) n o 1 ′ ′ ′ ωj′ ∈ / ωk′ , (ωk′ + ωm ),ωk′ +2ωm , ωk′ +ωm ± ωp′ , (27) 2 for all distinct j, k, m, and p. Applying Taylor series expansion to P (D, t) at its maxi˜ + Q(t), we have mum point, and noting that D = D∗ + D T 1 ˜ ˜ ˜ D+Q(t) H D+Q(t) +R(D+Q(t)),(28) P =P ∗+ 2 ˜ + Q(t)) stands for higher where ∂P (D∗ )/∂D = 0 and R(D ˜ order terms in D+Q(t). Assuming the same amplitude for all probing signals, (28) shows that the averaged gradient vector, ˆ includes frequencies at ωj and ωj − ωk and the averaged G, ˆ includes frequencies at estimate of the Hessian matrix, H, 2ωj and ωj − ωk for all distinct j and k. The transient of the estimate of the gradient vector and the Hessian matrix contain frequencies that include harmonics of ωj − ωk . The bandwidth of the low-pass filter needs to be designed with respect to these values. ˜= Linearization of the update law for the error variable D ˆ − D∗ results in D ˜˙ = −Kn D, ˜ D (29)
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ˆ 1 (%) D
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Fig. 8. Generated power in a partial shading scenario. Newton-based ES governs the system to its MPP with a uniform transient less than 5 s.
which shows that the convergence rate of the parameter is independent of the shape of the cost function, and consequently, after a transient, when the Hessian is close enough to its actual value, the output power converges to the MPP with the same performance regardless of environmental or mismatch conditions. The analytical results in [6] guarantee that for the feedback system in Fig. 7, under Assumption ¯a 1, there exist δ, ¯ > 0 and for any |a| ∈ (0, a ¯) and ¯ δ ∈ (0, δ), there exists ω ¯ > 0 such that for any given a and δ and any ω ∈ (0, ω ¯ ) there exists a neighborhood ˆ G, ˆ Γ, H, ˆ η) = (D∗ , 0, H −1 , H, P (D∗ )) of the point (D, such that any solution of the Newton-based ES from the neighborhood exponentially converges to an O(ω + δ + |a|)– neighborhood of that point. Furthermore, P (D, t) converges to an O(ω + δ + |a|)–neighborhood of P (D∗ ). V. S IMULATION R ESULTS To show the effectiveness of the proposed Newton-based design in Fig. 7, and compare its performance with that of the gradient-based design in Fig. 6, we simulate a PV system with n = 2 cascade modules. The PV modules are model 215N from Sanyo, with datasheet parameters presented in Section II. For details on how to select the parameters of the gradient-based scheme, the reader is referred to [8]. The same criteria apply to the Newton-based design. As pointed out in Section IV in the paragraph following (27), care has to be taken that the cross-over frequency of the low-pass filters is chosen less than 5% of the least difference between probing frequencies. The cut-off frequency of the high-pass filter is simply chosen to be smaller than ωl . Based on the preceding remarks, the numerical values of the design parameters are: ω1 = 5000 rad/s, ω2 = 6000 rad/s, a = 0.01, ωl = 10 rad/s, ωh = 7 rad/s, H0 = −105 diag([1 1]), D0 = [0.5 0.5]T , and Kn = diag([1 1]). To make a fair comparison between gradient-based algorithm and Newton scheme Kg should be of the order of Kn Γ0 , where Γ0 = H0−1 . However, we select Kg = −30Kn Γ0 to make the comparison more strict for the Newton algorithm. The temperature T is assumed to be equal to 25 ◦ C for all modules throughout. The irradiance S is assumed to be equal to 1000 W/m2 initially, with a step change to 400 W/m2 for modules PV2 at t = 10 s and then back to 1000 W/m2 at
ˆ 2 (%) D
50 40 30 20
Gradient Newton
0
5
10
Fig. 9. Adaptation of the pulse duration. Newton-based ES shows a similar convergence rate for all parameters. The convergence rate of the gradientbased ES varies with power level and direction of its variation.
20 s, so as to simulate partial shading. Central frequency of PWM is fc = 100 kHz. Fig. 8 shows the output power of the entire system, and ˆ and H ˆ of the pulse duration and Hessian the estimates D respectively, are shown in Fig. 9 and Fig. 10. It is clear from Fig. 8 that after the initial transient (roughly the first 5 s), the Newton-based algorithm performs a uniform and faster transient against step down (at 10 s) or step up (at 20 s) changes in the generated power. The transient of the gradient ES for all parameters is slightly faster than the Newton at the beginning as shown in Fig. 9, resulting in the faster transient performance initially1 . However, the transient behavior of the Newton-based ES is more uniform in response to sudden changes in power level, while the gradient-based ES has different convergence rates for every parameter which varies with the power level. Lastly, as mentioned in Section I, the Newton-based design moves the system in almost a straight line between extrema, in contrast to curved steepest descent trajectories of the gradient algorithm. This observation is demonstrated clearly in Fig. 11. VI. C ONCLUSIONS Using extremum seeking in a micro-converter configuration is a promising way to extract maximum power from a PV system. Conventionally used scalar gradientbased designs do so based on the generated power of each module. On the one hand, this requires two sensors per module, and on the other, the dependence on the level and 1 We remind the reader that at the beginning of the adaptation process, the estimate of Γ−1 is far from its actual value, which affects the transient convergence rate.
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proved overall performance. The scheme also only uses two sensors for the overall system, resulting in lower hardware cost. The dual advantages contribute towards reduced average cost/watt, enhancing the economic viability of solar.
x 10 0
Hessian
−2
R EFERENCES
−4 −6
(Γ−1 )11 (Γ−1 )22 (Γ−1 )12 = (Γ−1 )21
−8 −10
0
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Fig. 10.
Hessian matrix converges to its actual value.
30
60 55 50
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D2 (%)
45 40
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35 30
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25 20 45
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D1 (%) Fig. 11. Phase-portrait of the adaptation process. Newton-based algorithm moves the system along the straight line between MPP’s before and after changing irradiance.
direction of changes of the individual powers causes different transients in the parameter updates, particularly in response to sudden irradiance changes caused by partial shading. The multivariable gradient-based extremum seeking design removes some of these drawbacks. However, it still depends on the shape and curvature of the cost function. Since the Hessian of the entire system (and not individual modules) defines the performance of the parameter update, we can use the estimate of the Hessian to eliminate the dependency of the ES algorithm on environmental conditions that the Hessian depends upon. The Newton-based algorithm that we have presented does so, resulting in more uniform transients in response to irradiance and temperature changes, and im-
[1] K. Ariyur and M. Krsti´c, Real-Time Optimization by Extremum Seeking Feedback. Wiley-Interscience, 2003. [2] A. Bratcu, I. Munteanu, S. Bacha, D. Picault, and B. Raison, “Cascaded DC-DC converter photovoltaic systems: power optimization issues,” IEEE Transactions on Industrial Electronics, vol. 58, pp. 403– 411, 2011. [3] S. Brunton, C. Rowley, S. Kulkarni, and C. Clarkson, “Maximum power point tracking for photovoltaic optimization using ripple-based extremum seeking control,” IEEE Transactions on Power Electronics, vol. 25, pp. 2531–2540, 2010. [4] S. Dhople, J. Ehlmann, A. Davoudi, and P. Chapman, “Multiple-input boost converter to minimize power losses due to partial shading in photovoltaic modules,” in Proc. of IEEE Energy Conversion Congress and Exposition (ECCE), 2010. [5] T. Esram and P. Chapman, “Comparison of photovoltaic array maximum power point tracking techniques,” IEEE Transactions on Energy Conversion, vol. 22, pp. 439–449, 2007. [6] A. Ghaffari, M. Krsti´c, and D. Neˇsi´c, “Multivariable Newton–based extremum seeking,” Automatica, vol. 48, pp. 1759–1767, 2012. [7] A. Ghaffari, S. Seshagiri, and M. Krsti´c, “High-fidelity PV array modeling for advanced MPPT design,” in Proc. of IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), 2012. [8] ——, “Power optimization for photovoltaic micro-converters using multivariable gradient-based extremum-seeking,” in Proc. of American Control Conference, 2012. [9] D. P. Hohm and M. E. Ropp, “Comparative study of maximum power point tracking algorithms,” Progress in Photovoltaics: Research and Applications, vol. 11, pp. 47–62, 2003. [10] S. Jain and V. Agarwal, “Comparison of the performance of maximum power point tracking schemes applied to single-stage grid-connected photovoltaic systems,” IET Electric Power Applications, vol. 1, pp. 753–762, 2007. [11] M. Krsti´c and H.-H. Wang, “Stability of extremum seeking feedback for general nonlinear dynamic systems,” Automatica, vol. 36, pp. 595– 601, 2000. [12] P. Lei, Y. Li, Q. Chen, and J. Seem, “Extremum seeking control based integration of MPPT and degradation detection for photovoltaic arrays,” in Proc. of American Control Conference, 2010. [13] M. Miyatake, M. Veerachary, F. Toriumi, N. Fuji, and H. Ko, “Maximum power point tracking of multiple photovoltaic arrays: a PSO approach,” IEEE Transactions on Aerospace and Electronic Systems, vol. 47, pp. 367–380, 2011. [14] S. Moura and Y. Chang, “Asymptotic convergence through Lyapunovbased switching in extremum seeking with application to photovoltaic systems,” in Proc. of American Control Conference, 2010. [15] F.-S. Pai, R.-M. Chao, S. H. Ko, and T.-S. Lee, “Performance evaluation of parabolic prediction to maximum power point tracking for PV array,” IEEE Transactions on Sustainable Energy, vol. 2, pp. 60–68, 2011. [16] G. Petrone, G. Spagnuolo, and M. Vitelli, “A multivariable perturband-observe maximum power point tracking technique applied to a single-stage photovoltaic inverter,” IEEE Transactions on Industrial Electronics, vol. 58, pp. 76–84, 2011. [17] C. A. Ramos-Paja, G. Spagnuolo, G. Petrone, M. Vitelli, and J. Bastidas, “A multivariable MPPT algorithm for granular control of photovoltaic systems,” in Proc. of IEEE International Symposium on Industrial Electronics, 2010. [18] G. Vachtsevanos and K. Kalaitzakis, “A hybrid photovoltaic simulator for utility interactive studies,” IEEE Transactions on Energy Conversion, vol. EC-2, pp. 227–231, 1987.
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Power Optimization for Photovoltaic Micro-Converters using Multivariable Newton-Based Extremum-Seeking Azad Ghaffari, Miroslav Krsti´c, and Sridhar Seshagiri Abstract— Extremum-seeking (ES) is a real-time optimization technique that has been applied to maximum power point tracking (MPPT) design for photovoltaic (PV) micro-converter systems, where each PV module is coupled with its own DCDC converter. However, most existing designs are scalar, i.e., employ one ES MPPT loop around each converter, and all current designs, whether scalar or mutivariable, are gradientbased. The convergence rate of gradient-based designs depends on the Hessian, which in turn is dependent on environmental conditions such as irradiance and temperature. Consequently, when applied to large PV arrays, the variability in environmental conditions and/or PV module degradation result in non-uniform transients in the convergence to the maximum power point (MPP). Using a multivariable gradient-based ES algorithm for the entire system instead of a scalar one for each PV module, while decreasing the sensitivity to the Hessian, does not eliminate this dependence. We present a recently developed Newton-based ES algorithm that simultaneously employs estimates of the gradient and Hessian in the peak power tracking. The convergence rate of such a design to the MPP is independent of the Hessian, with tunable transient performance that is independent of environmental conditions. We present simulation results that show the effectiveness of the proposed algorithm in comparison to multivariable gradientbased ES.
I. I NTRODUCTION a) Motivation: Maximum power point tracking (MPPT) algorithms for extracting the maximum achievable power from a photovoltaic (PV) system have been studied by several researchers [4], [13], [15], [16], [17], with detailed comparisons presented in [5], [9], [10]. Several recent works [2], [3], [12], [14] have focused on the application of gradient-based extremum-seeking (ES) [1] to MPPT design. In [8], we presented a multivariable gradient-based ES MPPT design for the micro-converter architecture, where each PV module is coupled with its own DC-DC converter. The design reduced the balance-of-system cost by reducing the number of required sensors (hardware reduction), and was shown to result in more uniform transients under sudden changes in solar irradiance and environmental temperature in comparison to a scalar gradient-based ES for each PV module. However, the convergence to maximum power point (MPP) is dependent on the unknown Hessian, which varies Azad Ghaffari is with Joint–Doctoral Programs (Aerospace and Mechanical) between San Diego State University and University of California at San Diego, La Jolla, CA 92093-0411, USA, [email protected]. Miroslav Krsti´c is with Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA, [email protected]. Sridhar Seshagiri is with Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA 92182-1309, USA, [email protected].
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
with irradiance, temperature, and module degradation and mismatch. In order to alleviate the issue of unknown Hessian dependent convergence of a gradient-based ES algorithm, we presented a multivariable Newton-based ES design for general nonlinear systems in [6]. In comparison with the standard gradient–based multivariable extremum seeking, the algorithm in [6] removes the dependence of the convergence rate on the unknown Hessian and makes the convergence rate of the parameter estimates user–assignable. In particular, all the parameters can be designed to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets. When applied to the MPPT problem in PV systems, the method offers the benefit of uniform convergence behavior under a wide range of working conditions, that includes temperature and irradiance variations, and the non-symmetric power generation of the neighboring PV modules as a result of module degradation or mismatch. b) Our contribution: We present a multivariable Newton-based ES scheme with the following features: (1) It is applied to micro-converter systems, and hence deals with the case of non-unimodal power characteristics, and deals specifically with the issue of module mismatch (for example, possibly different irradiance levels as a result of partially shaded conditions). (2) The use of the non-model-based ES technique makes the design robust to partial knowledge of the system parameters and operating conditions. (3) As opposed to gradient-based designs, our proposed Newtonbased design removes the dependence of the transient on the working condition and potential mismatch in PV modules. (4) Lastly, the multivariable design requires fewer sensors, which reduces the hardware cost compared to scalar designs. c) Organization: The rest of this paper is organized as follows: The mathematical model of a PV module, along with a discussion of the DC/DC converter power electronics, is briefly presented in Section II. This section is based on our previous work, and is reproduced here for the sake of completeness/clarity. Section III summarizes the multivariable gradient-based ES scheme in [8], and allows us to present key distinctions with the proposed design. Our proposed multivariable Newton-based ES is presented and discussed in Section IV. Some simulation results are presented in Section V, and our work is summarized and some concluding remarks made in Section VI.
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S is
+ vd −
ic
Rp
(a)
(+)
Current(A)
Rs
vc (-)
6
5
5
4
4
3
3
2 1 0 0
Fig. 1.
Equivalent circuit of a PV cell. Power(W)
200
II. P HOTOVOLTAIC M ODULES AND P OWER E XTRACTION Our design and analysis are based on the standard PV cell model described for example in [18], and shown schematically in Fig. 1. The PV cell is modeled as an ideal current source of value is in parallel with an ideal diode with voltage vd . Electrical losses and contactor resistance are accounted for by the inclusion of the parallel and series resistances Rs and Rp respectively. The amount of generated current is is dependent on the solar irradiance S and the temperature T through the following equation S , (1) is = (Is + ki (T − Tr )) 1000 where Is is a reference short-circuit current, Tr a reference temperature, and ki the short-circuit temperature coefficient. The diode models the effect of the semiconductor material and its I − V characteristics are given by vd id =i0 exp −1 , (2) N Vt 3 T kT Eg T (3) − 1 , Vt = exp i0 =I0 Tr N Vt Tr q where I0 , Eg and N are respectively the diode reference reverse saturation current, the semiconductor bandgap voltage (barrier height), and the emission coefficient, all three being cell material/construction dependent, Vt is the thermal cell voltage, and k = 1.38 × 10−23 J/K and q = 1.6 × 10−19 C are Boltzman’s constant and the charge on an electron respectively. The cell model described by the above equations along with KCL/KVL: ic = is −id −vd /Rp , vd = vc +ic Rs , is then scaled to represent a PV module by considering ns cells in series (each having thermal voltage Vt ), so that the terminal I − V relationship for the PV module is given by # v " v ( ns + iRs ) ns + iRs i = is−i0 exp −1 − . (4) N Vt Rp For the sake of model development and performing simulations, we pick the PV module 215N from Sanyo, with the following numerical values derived from the manufacturer’s datasheet [7]: Eg = 1.16 eV, N = 1.81, I0 = 1.13×10−6 A, Is = 5.61 A, ki = 1.96 mA/K, Tr = 298.15 K, Rs = 2.48 mΩ, Rp = 8.7 Ω, and the number of PV cells connected in series is ns = 72. The resulting I −V and P −V curves are shown in Fig. 2. As is clear from Fig. 2(b,d), the power-voltage (P–V) characteristic has a unique but (T, S) dependent peak (v ∗ , p∗ ). It is the job of the MPPT algorithm to automatically track this peak. In many grid-tied PV systems (including our current
(c)
6
2 1
1000 W/m2 80010 W/m220 600 W/m2 400 W/m2
30
40
(b)
50
0 0 250
0 ◦C 25 10◦ C 50 ◦ C 75 ◦ C
20
30
40
50
30
40
50
(d)
200
150 150 100
100
50 0 0
50
10
20
30
40
50
0 0
10
Voltage(V)
20
Voltage(V)
Fig. 2. Characteristic (a) I−V and (b) P−V for varying irradiance, T=25◦ C. Characteristic (c) I−V and (d) P−V for varying temperature, S=1000W/m2 .
work), this is done by means of a separate DC/DC power electronics stage that serves two functions: (i) regulating the output DC voltage at a (near) constant value, and (ii) extracting maximum power by forcing the PV module output v to equal v ∗ . Fig. 3 shows this setup for a DC/DC boost converter stage, whose output voltage is maintained constant as vdc . The ratio between the input voltage v and output voltage vdc can be controlled by changing the duty cycle of the transistor switch, which serves as the control input d. Under the assumption that the boost converter is working in Continuous Current Mode (CCM), and that the switching Pulse Width Modulation (PWM) frequency fc is significantly higher than the bandwidth of the control loop, the boost converter input-output voltage relationship is given by the following (averaged) relations: v
= =
idc
(1 − d)vdc (1 − d)i.
(5) (6)
From (4), (5) and Fig. 2(b,d), it follows that at the MPP def (v ∗ ,p∗ ), the power p = iv = f (v)v = J(v), satisfies g
=
h
=
∂J ∗ (v ) = 0 ∂v 2 ∂ J ∗ (v ) < 0. ∂v 2
(7) (8)
Also we have ∂v/∂d = −vdc then g¯ = ¯ h
=
∂J ∗ (d ) = −gvdc = 0 ∂d 2 ∂ J ∗ 2 (d ) = hvdc < 0. ∂d2 Diode
L
i + PV
v
Module T, S −
Fig. 3. bus.
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(9) (10)
idc +
C′
d
Q C iq
vdc
DC Bus
−
DC/DC boost converter for PV module supplying power to DC
Converter I1
(T1 , S1 )
70
IDC
Io1
350
350
′ C1
V1
C1
Vo1 380
PV1
65 P1
AC grid
Uinv
D2
440 5
∼
0
55 42
Converter (Tn , Sn )
Vn
0
′ Cn
Cn
50
IDC
Ion Von
400
In
0 38 0
35
PVn
45
0 30
Pn
350
P2
∼
0
0
380
PV2
35 30
430
VDC
D2 (%)
Vo2
38
0
C2
60
Igrid
40
′ C2
V2
420 430
Inverter Io2
44
Converter I2
(T2 , S2 )
400
D1
Dn
Fig. 4. A cascade PV system including n PV module. Each module has a separate DC/DC converter.
40 40
45
50
55
60
65
70
D1 (%)
Many MPPT techniques, including the classical perturb-andobserve (P&O) class of methods, and extremum-seeking (ES) techniques, are based on detecting the sign of the power gradient. The next section discusses gradient-based ES MPPT design for a cascade configuration of PV modules. III. M ULTIVARIABLE G RADIENT-BASED E XTREMUM S EEKING A cascade PV system is shown in Fig. 4. A DC/DC boost converter is assigned to each PV module to extract maximum power from the PV system. The output side of the converters are connected in series. The whole system is connected to the AC grid through a DC/AC inverter. It is assumed that the DC voltage at the input side of the inverter is held constant at VDC . Assume that the voltage and current ripple at the output side of converters are negligible. Applying electrical rules on the input side of the inverter gives n X
Voj
=
VDC
Ioj
=
IDC ,
(11)
j=1
∀j ∈ {1, 2, · · · , n}.
(12)
From (5), (6), and the I–V functional dependence Ij = fj (Vj ), the relation between the voltage V = [V1 V2 · · · Vn ]T of PV modules and the pulse duration D = [D1 D2 · · · Dn ]T is defined by n independent equations n X
Vj 1 − Dj
=
VDC
(13)
(1 − Dj )fj (Vj )
=
IDC ,
∀j ∈ {1, 2, · · · , n}. (14)
j=1
This means that for each set of pulse duration we have a unique set of voltages for PV modules. Assuming lossless converters results in Poj = Voj Ioj = Pj for all j ∈ {1, 2, · · · , n}. Using (12) we obtain Pj Voj = Pn
j=1
Pj
VDC ,
(15)
Fig. 5. Variation of the power of a cascade PV system including two Sanyo PV modules versus pulse duration, D = [D1 D2 ]T . Level sets show the power in Watt. S1 = S2 = 1000 W/m2 , T1 = T2 = 25◦ C, and VDC = 200 V.
which means that the share of the output voltage of each converter from DC bus voltage is defined by the generated power of its relevant PV module. The variation of the generated power of a PV system including two series module from Sanyo connected to a DC bus with VDC = 200 V under standard test condition, S=1000 W/m2 and T=25 ◦ C, is shown in Fig. 5. Maximum power point (MPP) happens at D = [0.57 0.57]T . As is common in ES design (see for example, [11]), we make the following assumption about the system. Assumption 1: There exists D∗ ∈ Rn such that ∂P (D∗ ) = 0, (16) ∂D ∂2P (D∗ ) = H < 0, H = HT . (17) ∂D2 Fig. 6 shows the multivariable gradient-based ES design, where Kg is a positive diagonal matrix, and the perturbation signals are defined as T (18) Q(t) = a sin(ω1 t) · · · sin(ωn t) , T 2 sin(ω1 t) · · · sin(ωn t) , M (t) = (19) a
where ωj /ωk are rational for all j and k, and a is a real number, with the frequencies chosen such that ωj 6= ωk and ωj + ωk 6= ωm for all distinct j, k, and m. ˆ of the In particular, the design derives an estimate G gradient vector by adding a probing signal to the estimate ˆ = [D ˆ1 D ˆ2 · · · D ˆ n ]T of the pulse duration vector (of D all the DC/DC converters). With no additional information on the Hessian (and also for simplicity), we choose the amplitudes of the probing signals to all be the same value a. As before, smallness of the probing frequencies and the
2423
D1 D2 . . . Dn
D
DC/DC DC/DC . . . DC/DC
V1 V2 . . . Vn
PV 1 PV 2 . . . PVn
P1 P2 . . . Pn
Q(t) ˆ +D Fig. 6.
P =
Pn
M (t) ˆ G
Kg s
ωl s+ωl
×
j=1
Pj
D1 D2 . . . Dn
D
s s+ωh
DC/DC DC/DC . . . DC/DC
V1 V2 . . . Vn
PV 1 PV 2 . . . PVn
P1 P2 . . . Pn
Q(t)
z =P −η
ˆ +D
P =
Kn s
ˆ −ΓG
ˆ G
ωl s+ωl
×
ωj
= = =
Kg
=
j ∈ {1, 2, · · · , n}
ˆ Γ˙ = ωl Γ − ωl ΓHΓ (20) (21) (22) (23)
where ω and δ are small positive constants, and ωj′ , ωh′ , ωl′ and elements of Kg′ are O(1) positive real parameters. It can be shown that for sufficiently small ω, δ, and a, and with ˆ of the pulse duration vector and Kg > 0, the estimate D the output P converge to O(ω + δ + a)-neighborhoods of the optimal pulse duration D∗ = [D1∗ D2∗ · · · Dn∗ ]T and the peak power P ∗ respectively. The linearized update equation for the estimation error ˜ =D ˆ − D∗ is D ∂2P (D∗ ) (24) ∂D2 Pn where H is the Hessian of P = j=1 Pj with respect to pulse duration vector, D. Since P varies with irradiance, temperature, and degradation of the PV modules, so does H, and therefore a fixed adaptation gain Kg results in different (condition dependent) convergence rates for each converter. In order to alleviate the issue of unknown Hessian dependent convergence, we present in the next section a modified version of a multivariable Newton-based ES design that we developed in [6]. In comparison with the gradient-based multivariable extremum seeking presented in this section, the Newton-based algorithm makes the convergence rate of the parameter estimates user-assignable. In particular, all the parameters can be designed to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets. When applied to the MPPT problem in PV systems, the method offers the benefit of uniform convergence behavior under a wide range of working conditions, that include temperature and irradiance variations, and the non-symmetric power generation of the neighboring PV modules as a result of module degradation or mismatch. ˜˙ = Kg H D, ˜ D
Pj
s s+ωh
z =P −η W (t)
matrix gain Kg are ensured by selecting these as
ωh ωl
j=1
M (t)
Multivariable gradient-based ES for MPPT of a PV system.
ωωj′ , ωδωh′ ωδωl′ ωδKg′
Pn
H :=
IV. M ULTIVARIABLE N EWTON -BASED E XTREMUM S EEKING The multivariable Newton-based ES that we propose is shown schematically in Fig. 7. As is clear from the figure, the proposed scheme extends gradient-based ES with estimates
ˆ H
ωl s+ωl
×
Fig. 7. Multivariable Newton-based ES for MPPT of a PV system. The red dashed part is added to the gradient-based ES to estimate the Hessian.
of the Hessian. The perturbation matrix W (t) is defined as follows 16 1 2 Wjj = (25) sin (ω t) − j a2 2 4 sin(ωj t) sin(ωk t), j 6= k (26) Wjk = a2 where j, k ∈ {1, 2, · · · , n}. The product of W (t) and P generates an initial estimate of the Hessian, with the lowˆ pass filter reducing the high frequency oscillation in H. The inverse of the Hessian that is needed to implement the Newton algorithm, is estimated by a dynamic filter that has the form of a Riccati equation, and avoids the possible ˆ is singular. problem in inverting the Hessian if the estimate H It is shown in [6] that after a transient, the Riccati equation converges to the actual value of the inverse of Hessian matrix ˆ is a good estimate of H. if H Since we are integrating over a finite time period, and we set the phase delays of the periodic perturbation signals equal to zero, it is possible to exclude the condition ωj 6= ωk +ωm . The probing frequencies need to satisfy (see [6] for more details) n o 1 ′ ′ ′ ωj′ ∈ / ωk′ , (ωk′ + ωm ),ωk′ +2ωm , ωk′ +ωm ± ωp′ , (27) 2 for all distinct j, k, m, and p. Applying Taylor series expansion to P (D, t) at its maxi˜ + Q(t), we have mum point, and noting that D = D∗ + D T 1 ˜ ˜ ˜ D+Q(t) H D+Q(t) +R(D+Q(t)),(28) P =P ∗+ 2 ˜ + Q(t)) stands for higher where ∂P (D∗ )/∂D = 0 and R(D ˜ order terms in D+Q(t). Assuming the same amplitude for all probing signals, (28) shows that the averaged gradient vector, ˆ includes frequencies at ωj and ωj − ωk and the averaged G, ˆ includes frequencies at estimate of the Hessian matrix, H, 2ωj and ωj − ωk for all distinct j and k. The transient of the estimate of the gradient vector and the Hessian matrix contain frequencies that include harmonics of ωj − ωk . The bandwidth of the low-pass filter needs to be designed with respect to these values. ˜= Linearization of the update law for the error variable D ˆ − D∗ results in D ˜˙ = −Kn D, ˜ D (29)
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(a) 75 Gradient Newton
70 300 200 100
ˆ 1 (%) D
Power (W)
400
Gradient Newton 0
5
10
15 Time(s)
20
25
65 60 55
30
50
0
5
10
15 (b)
20
25
30
15 Time(s)
20
25
30
60
Fig. 8. Generated power in a partial shading scenario. Newton-based ES governs the system to its MPP with a uniform transient less than 5 s.
which shows that the convergence rate of the parameter is independent of the shape of the cost function, and consequently, after a transient, when the Hessian is close enough to its actual value, the output power converges to the MPP with the same performance regardless of environmental or mismatch conditions. The analytical results in [6] guarantee that for the feedback system in Fig. 7, under Assumption ¯a 1, there exist δ, ¯ > 0 and for any |a| ∈ (0, a ¯) and ¯ δ ∈ (0, δ), there exists ω ¯ > 0 such that for any given a and δ and any ω ∈ (0, ω ¯ ) there exists a neighborhood ˆ G, ˆ Γ, H, ˆ η) = (D∗ , 0, H −1 , H, P (D∗ )) of the point (D, such that any solution of the Newton-based ES from the neighborhood exponentially converges to an O(ω + δ + |a|)– neighborhood of that point. Furthermore, P (D, t) converges to an O(ω + δ + |a|)–neighborhood of P (D∗ ). V. S IMULATION R ESULTS To show the effectiveness of the proposed Newton-based design in Fig. 7, and compare its performance with that of the gradient-based design in Fig. 6, we simulate a PV system with n = 2 cascade modules. The PV modules are model 215N from Sanyo, with datasheet parameters presented in Section II. For details on how to select the parameters of the gradient-based scheme, the reader is referred to [8]. The same criteria apply to the Newton-based design. As pointed out in Section IV in the paragraph following (27), care has to be taken that the cross-over frequency of the low-pass filters is chosen less than 5% of the least difference between probing frequencies. The cut-off frequency of the high-pass filter is simply chosen to be smaller than ωl . Based on the preceding remarks, the numerical values of the design parameters are: ω1 = 5000 rad/s, ω2 = 6000 rad/s, a = 0.01, ωl = 10 rad/s, ωh = 7 rad/s, H0 = −105 diag([1 1]), D0 = [0.5 0.5]T , and Kn = diag([1 1]). To make a fair comparison between gradient-based algorithm and Newton scheme Kg should be of the order of Kn Γ0 , where Γ0 = H0−1 . However, we select Kg = −30Kn Γ0 to make the comparison more strict for the Newton algorithm. The temperature T is assumed to be equal to 25 ◦ C for all modules throughout. The irradiance S is assumed to be equal to 1000 W/m2 initially, with a step change to 400 W/m2 for modules PV2 at t = 10 s and then back to 1000 W/m2 at
ˆ 2 (%) D
50 40 30 20
Gradient Newton
0
5
10
Fig. 9. Adaptation of the pulse duration. Newton-based ES shows a similar convergence rate for all parameters. The convergence rate of the gradientbased ES varies with power level and direction of its variation.
20 s, so as to simulate partial shading. Central frequency of PWM is fc = 100 kHz. Fig. 8 shows the output power of the entire system, and ˆ and H ˆ of the pulse duration and Hessian the estimates D respectively, are shown in Fig. 9 and Fig. 10. It is clear from Fig. 8 that after the initial transient (roughly the first 5 s), the Newton-based algorithm performs a uniform and faster transient against step down (at 10 s) or step up (at 20 s) changes in the generated power. The transient of the gradient ES for all parameters is slightly faster than the Newton at the beginning as shown in Fig. 9, resulting in the faster transient performance initially1 . However, the transient behavior of the Newton-based ES is more uniform in response to sudden changes in power level, while the gradient-based ES has different convergence rates for every parameter which varies with the power level. Lastly, as mentioned in Section I, the Newton-based design moves the system in almost a straight line between extrema, in contrast to curved steepest descent trajectories of the gradient algorithm. This observation is demonstrated clearly in Fig. 11. VI. C ONCLUSIONS Using extremum seeking in a micro-converter configuration is a promising way to extract maximum power from a PV system. Conventionally used scalar gradientbased designs do so based on the generated power of each module. On the one hand, this requires two sensors per module, and on the other, the dependence on the level and 1 We remind the reader that at the beginning of the adaptation process, the estimate of Γ−1 is far from its actual value, which affects the transient convergence rate.
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4
proved overall performance. The scheme also only uses two sensors for the overall system, resulting in lower hardware cost. The dual advantages contribute towards reduced average cost/watt, enhancing the economic viability of solar.
x 10 0
Hessian
−2
R EFERENCES
−4 −6
(Γ−1 )11 (Γ−1 )22 (Γ−1 )12 = (Γ−1 )21
−8 −10
0
5
10
15 Time(s)
20
25
Fig. 10.
Hessian matrix converges to its actual value.
30
60 55 50
D0
D2 (%)
45 40
Step Up
35 30
Step Down
25 20 45
Gradient Newton 50
55
60
65
70
75
D1 (%) Fig. 11. Phase-portrait of the adaptation process. Newton-based algorithm moves the system along the straight line between MPP’s before and after changing irradiance.
direction of changes of the individual powers causes different transients in the parameter updates, particularly in response to sudden irradiance changes caused by partial shading. The multivariable gradient-based extremum seeking design removes some of these drawbacks. However, it still depends on the shape and curvature of the cost function. Since the Hessian of the entire system (and not individual modules) defines the performance of the parameter update, we can use the estimate of the Hessian to eliminate the dependency of the ES algorithm on environmental conditions that the Hessian depends upon. The Newton-based algorithm that we have presented does so, resulting in more uniform transients in response to irradiance and temperature changes, and im-
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