Nonlinear Equivalent Resistance-based Maximum Power Point Tracking (MPPT)

Nonlinear Equi v alent Resistance-based Maximum Po wer Point T racking (MPPT) Chaitanya Poolla ∗ , Abraham K. Ishihara † ECE, Carnegie Mellon Uni versity (SV) Mof fett Field, CA 94035 Email: ∗ cpoolla@alumni.cmu.edu, † abe.ishihara@west.cmu.edu Abstract —W e present a nonlinear equivalent resistance track- ing method to optimize the power output f or solar arrays. T racking an equivalent resistance r esults in nonlinear voltage step sizes in the gradient descent search loop. W e introduce a new model f or the combined solar module along with a DC-DC con verter which r esults in a highly nonlinear dynamical system due to the inherent non-linearity of the PV cell topology and the switched DC-DC conv erter system. T o guarantee stability over a range of possible operating r egimes, we utilize a feedback linearization control approach to exponentially conv erge to the setpoint. Simulations are presented to illustrate the performance and robustness of the proposed technique. Index T erms —Solar , Photovoltaic, Maximum P ower Point T racking, MPPT , Feedback Linearization, Buck-boost conv erter I . I N T RO D U C T I O N Recently , we have seen solar installations in the US more than double in ev ery market segment [1]. Despite this recent growth, improvements in reliability testing, and advancements in solar cell efficienc y , significant questions remain regarding the actual performance of PV modules in the field. Non- uniform changes in solar cell parameters may render modules more susceptible to hot-spot generation, especially under soil- ing and partial shading conditions. In order to optimize per- formance, per panel Maximum Power Point Tracking (MPPT) lev eraging intelligent control techniques has been shown to be a viable solution [2]. The problem of Maximum Power Point Tracking (MPPT) has been well-studied in the literature [3]. Common ap- proaches include, Hill-Climbing, Perturb and Observe (P&O), Incremental Conductance (IC), Fuzzy Logic (FL), Neural Networks (NN), and Ripple Correlation Control (RCC). The non-model based approaches such as Hill-Climbing and P&O seek to estimate the sign of the gradient at the operating point on the P-V curve. Assuming uniform conditions without faults the P-V curve is known to have a single maximum. Hence, knowledge about the sign of the gradient is sufficient to determine the direction of perturbation in the voltage space. Howe ver , inappropriate choice of step sizes often lead to oscillations at or near the MPP . Incremental conductance-based approaches [4] approximate the slope along the P-V curve as dP dV = I + V ∆ I ∆ V . Thus, measurements of instantaneous and incremental conductance are sufficient to determine the direc- tion of perturbation. While IC is capable of tracking changing weather conditions quickly , it can result in oscillations similar to P&O [5]. On the other hand, sophisticated approaches such as those based on Fuzzy Logic or Neural Networks require regular tuning for adaptability . The tracking performance of FL-based approaches depend on the choice of membership functions [6]. In case of NN-based approaches, the network needs to trained for a giv en PV array and tuned further to adapt to changing array characteristics [3]. Further , the Maximum Power Point (MPP) of the PV system depends on local weather conditions and hence accurate prediction of the en vironmental conditions [7] enables better MPP tracking [8]. In this work, we consider a PV plant integrated with a buck- boost conv erter supporting a load. The schematic is shown in Figure 2. By adjusting the duty cycle of the con verter , the operating point of the PV system can be driv en toward the MPP . Unlike sev eral Hill-Climbing or P&O approaches that seek to uniformly perturb the operating voltage, we propose to perturb the equi v alent resistance (or , conductance) at the operating point which results in nonlinear voltage changes. This perturbation sets up the tar get operating point for the inner loop tracker , which is implemented in the buck-boost con verter using feedback linearization. The main contribution of this paper is the combination of an equiv alent resistance tracking outer-loop based on a feedback linearization inner- loop control law that takes into account the highly nonlinear plant dynamics. The rest of the paper is structured as follows: Section II provides an ov erview of the outer -loop iterative update along with the deriv ation of closed-form equations using the Lambert-W function in Section II-A. The model of the PV module integrated with a buck-boost conv erter along with cor- responding dynamics is described in Section III. The feedback linearization controller is derived in Section IV. Simulation results are presented in Section V along with conclusions in Section VI. I I . N O N L I N E A R E Q U I V A L E N T R E S I S TA N C E T R AC K I N G The effecti ve or equiv alent resistance seen by the solar module is given by R eq = V I (1) where V and I are the operating voltage and current of the module, respecti vely . Let us denote the operating point at time t k ∈ R + by the pair ( I k , V k ) ∈ R 2 . Gi ven an operating point ( I k , V k ) , the goal of the outer-loop MPPT algorithm is c  2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collectiv e works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. to determine a ne w operating point at time t k +1 giv en by ( I k +1 , V k +1 ) such that I k +1 · V k +1 > I k · V k . If the algorithm con verges, then the solar module will be operating at a local maximum of the power curv e. Under uniform conditions, there is only one global maximum and hence, gradient descent algorithms will conv erge to the maximum power point. A challenge is to determine change (∆ I k , ∆ V k ) d = ( I k +1 , V k +1 ) − ( I k , V k ) since the po wer v ersus v oltage (or current) landscape is highly nonlinear . It is common to use a constant ∆ V > 0 voltage increment or decrement depending on the slope of the power versus voltage curve, dP dV   ( I k ,V k ) . If dP dV   ( I k ,V k ) > 0 , V k +1 = V k + ∆ V . If dP dV   ( I k ,V k ) < 0 , V k +1 = V k − ∆ V . The problem with this approach is that a constant ∆ V that works in one region of the I-V curve may not work in another . For example, a small ∆ V increment that is suitable to the left of the MPP , may be too large of an increment for operating regimes to the right of the MPP due to the large negati ve slope of the I − V curve. An alternative approach is to consider a constant equiv alent resistance (or, conductance) change when the operating point is in region I (or , II) (Fig. 1). If the operating point is in region I, we increase the equiv alent resistance by a constant ∆ R eq . R eq k +1 = R eq k + ∆ R eq (2) where ∆ R eq > 0 is a constant for all t k . Using (1), (2) becomes V k +1 I k +1 = V k I k + ∆ R eq d = β k (3) When the operating point of the solar module is in region II (Fig. 1), we choose to increment by the inv erse of equiv alent resistance, or equiv alent conductance. That is: I k +1 V k +1 = I k V k + ∆ G eq d = 1 β k (4) It again follows that (compare to (3)) V k +1 = β k I k +1 (5) Giv en β k (Equations 3 or 4), we may use the one-diode model of the solar array and the Lambert-W function [9, 10] to compute the new (perturbed) operating point 1 as summarized below . A. Outer-Loop Update using Lambert-W function In the simplest representation, the PV cell is assumed to be a superposition of the dark and illuminated current-voltage characteristics. Along with the series and shunt resistances, the I-V relationship of the PV cell is described here. In what follows, I ph (A) denotes the photo-generated current, I 0 (A) denotes the dark saturation current, q (Coulombs) denotes the electric charge carried by a single photon, k denotes the Boltz- mann constant ( J · K − 1 ), and T denotes the cell temperature ( K ). Let us consider the following equations 2 to model the I-V characteristics of the PV cell: V = β I (6) 1 A two diode model can also be used [11] 2 For the sake of notational con venience, we omit the subscripts k from β k and k + 1 from V k +1 and I k +1 in Equation 5 Fig. 1: Regions I and II of I-V curve and I = I ph − I 01  e ( V + R s I ) V T 1 − 1  − V + I R s R sh (7) Substitution of (6) into (7) yields: V = c − 1 0 β  I ph + I 01 − I 01 e  β + R s β V T  V  (8) where c 0 = 1 + β + R s R sh Equation (8) is of the form y = d 0 − d 1 e α 1 y (9) where d 0 = c − 1 0 β ( I ph + I 01 ) d 1 = c − 1 0 β I 01 α 1 = β + R s β V T W e can transform (9) to W e W = x where W = α 1 d 0 − α 1 y x = α 1 d 1 e α 1 d 0 Using the Lambert-W function [10], we can obtain closed-form solutions with y = V as follows: y = 1 α 1 ( α 1 d 0 − W ( x )) . The implication is that, given β k and the I-V parameters, the new (perturbed) operating point can be ex- actly computed. This enables one to rapidly simulate electrical performance under the time-varying non-uniform conditions. The presented approach above is scalable, robust, and readily extends to arbitrary circuit topologies. Given N p strings of N s cells connected in series per string, the single cell parameters ( I ph , I 01 , R s , R sh , V T ) scale into the corresponding module parameters ( N p I ph , N p I 01 , N s N p R s , N s N p R sh , N s V T ) . I I I . M O D E L I N G S O L A R M O D U L E A N D D C - D C B U C K B O O S T C O N V E RT E R W I T H P A RA S I T I C L O S S E S W e consider a dynamical model of the PV -Buck-Boost system with parasitic resistances in the inductor, capacitor , and Fig. 2: Solar Module and DC-DC Buck Boost Conv erter with Parasitic losses. the on -state of the MOSFET . The circuit diagram is shown in Fig. 2. By representing the circuit elements as equations, we deriv e the dynamics of the system below . Switch-On Model: When the switch is in the on-state, the equations become: d dt   v pv v c i L   =    0 0 − 1 C pv 0 − 1 C ( R + R C ) 0 1 L 0 − ( R on + R L ) L      v pv v c i L   +   I pv C pv 0 0   = A 1 x + h 1 ( x ) (10) Switch-Off Model: When the switch is in the of f-state, the equations become: d dt   v pv v c i L   =    0 0 0 0 − 1 C ( R + R C ) R C ( R + R c ) 0 − R L ( R + R C ) − ( R L + R d + R c || R ) L      v pv v c i L   +   I pv C pv 0 − V D L   = A 2 x + h 2 ( x ) (11) A veraged Model: Using the averaged circuit model approach, the switched linear system can be approximated 3 by a single nonlinear system given by ˙ x = A ( x, d ) + dh 1 + (1 − d ) h 2 (12) A ( x, d ) d = dA 1 + (1 − d ) A 2 =    0 0 − d C pv 0 − 1 C ( R + R C ) (1 − d ) R C ( R + R C ) d L − (1 − d ) R L ( R + R C ) a 33    and a 33 = − 1 L { d ( R on + R L ) + (1 − d )( R L + R d + R C k R ) } . In the above, ( v pv , v c , i L ) are the state variables, d ∈ [0 , 1] is the control (duty-cycle) and the nonlinearities are due to the multiplicati ve control and state terms and the nonlinear function of the state variable: I ( v pv ) . 3 The degree to which the nonlinear system approximates the switched linear system can be measured by application of the Baker-Campbell-Hausdorff formula. I V . F E E D BA C K L I N E A R I Z A T I O N C O N T RO L For controller design we consider (12) but without parasitic resistances. Howe ver , in the simulation presented in Section V, the controller is found to be robust ev en in the presence of parasitic resistances. The dynamics without the parasitic resistances are given by:   ˙ v pv ˙ v c i L   =   0 0 − d C pv 0 − 1 C R 1 − d C d L − 1 − d L 0     v pv v c i L   +   I pv C pv 0 0   (13) In the following, we discuss only the mechanics of the feedback linearization controller design. W e do not discuss the stability nor robustness properties. Readers interested in the theory should consult [12]. Region I Controller: Consider an operating point in region I as shown in Fig. 1. W e assume a reference equi v alent resistance is generated via (2) denoted by R eq ref . In order to ensure tracking of R eq ref we define the output variable: y R eq ( x ) = R eq ref − R eq ( x ) Differentiating y R eq ( x ) with respect to time, we obtain ˙ y R eq ( x ) = − d dt ( R eq ( x )) = − I pv ˙ v pv − v pv ˙ I pv I 2 pv (14) Application of the chain rule yields: ˙ I pv = d dt I pv ( v pv ) = ∂ I pv ∂ v pv ˙ v pv (15) Consider N s cells in series with identical parameters. If we hav e N p strings of N s cells each, in parallel, then I pv = I ph − I 01 ( e v pv + R s I pv V T − 1) − v pv + R s I pv R sh (16) where, I ph = N p I ( c ) ph I 01 = N p I ( c ) 01 R x = N s N p R ( c ) s R sh = N s N p R ( c ) sh V T = N s V ( c ) T (17) Thus we have, ∂ I pv ∂ v pv = − I 01 V T e v pv + R s I pv V T (1 + R s ∂ I pv ∂ v pv ) − 1 R sh (1 + R s ∂ I pv ∂ v pv ) (18) = ⇒ ∂ I pv ∂ v pv (1 + I 01 R s V T e v pv + R s I pv V T + R s R sh ) = − I 01 V T e v pv + R s I pv V T − 1 R sh (19) Hence we get, ∂ I pv ∂ v pv ( I pv ,v pv )= − R sh I 01 e v pv + R s I pv V T + V T V T R sh + I 01 R s R sh e v pv + R s I pv V T + V T R s (20) W e may now ev aluate Equation 14 ˙ y = − 1 I 2 pv ( I pv − v pv ∂ I pv ∂ v pv ) ˙ v pv (21) = − 1 I 2 pv ( I pv − v pv ∂ I pv ∂ v pv )( − d C pv i L + I pv C pv ) (22) = − k y (23) Setting ˙ y = − k y where k > 0 and solving for d , we have − d C pv i L + I pv C pv = k yI 2 pv I pv − v pv ∂ I pv ∂ v pv di L C pv = I pv C pv − k yI 2 pv I pv − v pv ∂ I pv ∂ v pv = I pv ( I pv − v pv ∂ I pv ∂ v pv ) − k y I 2 pv C pv C pv ( I pv − v pv ∂ I pv ∂ v pv ) C pv i L = ⇒ d = I pv ( g ( v pv ) − k y I pv C pv ) i L g ( v pv ) (24) where, g ( v pv ) = I pv − v pv ∂ I pv ∂ v pv Region II Controller: Consider an operating point in region II as shown in Fig. 1. In this case, we assume a reference equiv alent conductance is generated via (4) denoted by G eq ref . In order to ensure tracking of G eq ref we define the output variable: y G eq ( x ) = G eq ref − G eq ( x ) Proceeding as abov e, we may solve for the feedback lineariz- ing control law that guarantees the output error con verges exponentially to the origin. Complete Control Law: W e combine the results of region I and II into the following globally v alid control law: d f lc =    1 i L  I pv − ky R eq I 2 pv C pv g ( v pv )  if Region I 1 i L  I pv + ky G eq v 2 pv C pv g ( v pv )  if Region II (25) V . S I M U L A T I O N R E S U LT S In this section we simulate the proposed feedback linearization-based control algorithm obtained in Equation 25 on the PV -b uckboost platform. The electrical parameters of a Kyocera PV module ( N s =36, N p =1) employed are provided here: R s = 0 . 01 (Ohms), R sh = 150 (Ohms), I 01 = 1 . 9795 · 10 − 10 (A), and I ph = 3 . 31 (A). For the DC-DC Buck-Boost con verter , the follo wing parameters are used: C = 220 e − 6 ( F ) , L = 3 e − 3( H ) , R = 10(Ω) , and C pv = 1 e − 3( F ) . W e take the sampling period of the buck- boost con verter to be T s = 1 / 100000( s ) . Under the standard test conditions, the module characteristics are shown in Figure 3. The MPP is achiev ed when the operating voltage is at Fig. 3: I-V and P-V characteristics of the PV module considered. Fig. 4: R eq and G eq profiles of the PV module in relation to P pv and V pv V mpp ≈ 18 . 1 V . Therefore, region 1 can be understood to span the interv al [0, V mpp ) and region 2 w ould span the interval [ V mpp , V oc ]. The equiv alent resistance for region 1 (and conductance for region 2) can be seen from Figure 4. The magnitude of R eq in region 1 and the magnitude of G eq in region 2 can be obtained from the portions of the graph that are left to the MPP . These magnitudes can provide cues to determine the step sizes associated with the iterative outer loop updates. Further , in order to adapt con vergence based on the proximity to the MPP , the perturbation step sizes ( ∆ R eq and ∆ G eq ) are chosen to be proportional to estimated slope ∂ P pv ∂ V pv . In this work, the PV -buckboost model was simulated in MA TLAB with the above parameters for different initial con- ditions. The feedback linearization-based control con ver ged to the correct Maximum Po wer Point by tracking the outer- loop set points iterati vely without prior kno wledge of the Fig. 5: Performance of Feedback Linearization Controller giv en in (25) for two initial conditions. MPP . In order to test for robustness, we first note that the feedback controller deri ved in 24 does not account for parasitic resistances ( R c , R L , R on , R d ) . Howe ver , for this simulation the values of ( R c , R L , R on , R d ) were set to (1 , 1 , 1 , 1000) . The simulated trajectories starting from two initial conditions ( V pv = 10 V , V pv = 20 V) are depicted in Figure 5. It can be noted that the system trajectory con ver ges to the MPP ev en in the presence of parasitic effects, demonstrating the robustness of the controller . Further , the choice of step sizes proportional to the estimated slope of the PV curve ensures that the system trajectory does not oscillate about the MPP . V I . C O N C L U S I O N This work presented a novel method for MPPT by a combination of an equiv alent resistance tracking mechanism achiev ed by feedback linearization of buck-boost conv erter dynamics. The analytical determination of the outer-loop set points using the Lambert-W function is discussed based on PV diode models. The buck-boost dynamics are derived for differ- ent switch positions and the av erage dynamics is formulated. A feedback linearization-based control law is derived to track the reference signal. 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