Power Limit of Crosswind Kites

Haocheng Li Aerospace Engineering Program, Worcester Polytechnic Institute

Abstract In this paper, a generalized power limit of the cross wind kite energy systems is proposed. Based on the passivity property of the aerodynamic force, the available power which can be harvested by a cross wind kite is derived. For the small side slip angle case, an analytic result is calculated. Furthermore, an algorithm to calculated the real time power limit of the cross wind kites is proposed. Keywords: Power Limit, Crosswind Kites, Available Power, Aerodynamics

1. Introduction 2. Available Power

Crosswind kite power is an emerging renewable energy In this section, the available power of the cross wind technology which uses kites or gliders to generate power kites will be derived. First, a classical aerodynamic model in high altitude wind flow. Compared to the conventional is presented and the passivity property of the aerodynamic wind turbine technology, the crosswind power systems can force is then proven. Physically, the passivity of the aero- achieve high-speed crosswind motion, which increase the dynamic force represents the power dissipated by the aero- power density significantly. Additionally, without the sup- dynamic force. Then available power of the crosswind kite port structure, the construction cost of such wind energy systems is then derived based on this property of the aero- system could be reduced compared to conventional tow- dynamic force. The derivation presented in this section ered wind turbines. This could eventually reduce the cost serves as a foundation of the power limit calculation of the of the wind energy dramatically. cross wind kite in next section. Based on the power generation modes, the crosswind power can be placed into two categories: the FlyGen sys- 2.1. Classical Model of Aerodynamics tems and the GroundGen systems. FlyGen systems, also As in classical aerodynamics, [3] the following two ref- referred to as drag mode systems, generate power through erence frames are used to describe the three-dimensional on-board devices, such as wind turbine. GroundGen sys- kite motion, tems, also referred to as lift mode systems, generate power through tether tension and power generator on the ground. • Cartesian Earth Frame C: iC jC kC Different industrial prototypes have been proposed for the airborne wind harvesting, • Body Frame B: iB jB kB Although the theoretical power limit of crosswind kites In crosswind kite systems, the Cartesian earth frame C is the most fundamental issue of such systems. In [1], is often assumed to center at anchor point of the tether. Miles Loyd first proposed a power limit for the crosswind The iC axis points to the upstream direction and jC points kite systems. Diehl provide a refined version of the power vertical downwards. The jC axis forms a right-hand coor- limit in [2] by considering the turbine power generation. dinate system with iC and kC axes. The body frame B, However, both Loyd and Diehl’s works are based on kite which centers at the gravitational center of the kite, follows motion in a two-dimensional wind field and no side force the North-East-Down conventions. has been taken into account in their analysis. In this pa- In this work, we will denote the rotational transfor- arXiv:1910.03807v1 [physics.class-ph] 9 Oct 2019 per, I propose the theoretical limit of the crosswind kite mation matrix from frame C to B as LBC , which can be systems in three-dimensional wind field, and the following represented either by Euler angles or quaternion. The the contributions are made. First, a general power loss of the kite and wind velocity measured in frame C are denoted crosswind kite system is derived based on the passivity as VC and W respectively, then the kite and wind velocity analysis of the classical aerodynamic model. Based on the observed in frame B are given by, power loss calculation, the available power of a crosswind

kite is derived, and a nonlinear optimization is presented VB = LBC VC , WB = LBC WC . (1) to determine the power limit.

Preprint submitted to Energy October 10, 2019 It is well known that the rotational transformation matri- CL ces are orthogonal, i.e., it inverse transformation, denoted as LCB, is its transpose, iB CD α 1 T LCB = L− = L . (2) BC BC V ∞

The kite apparent velocity, which is denoted as Va, is the kB key in determining the aerodynamic force acting on kites. Using the notation in equations (1) and (2), Figure 1: Lift and Drag Coeﬃcients

V = L (V − W ) = u v w (3) a BC C C a a a , Therefore, the steady aerodynamic force acting on the kite, measured in frame B, can be calculated as for convenience, −Va may also be referred to as apparent wind velocity. Using kite apparent velocity, Va, the kite 1 A = ρSV 2C , (8) apparent attitudes, angle of attack α and sideslip angle β, B 2 a B are given by where S is the kite area and ρ is the air density. Cor- w v α = arctan a , β = arcsin a , (4) respondingly, the steady aerodynamic force acting on the ua Va kite, measured in frame C, is given by where Va = kVak is also refer as the kite apparent speed. 1 A = ρSV 2L C , (9) It is a common practice, [3], that the kite apparent C 2 a CB B velocity in i is assumed to be positive, which is clearly B To this end, all important notations that will be used in state as follows, the rest of the paper have been introduced. To achieve a

Assumption 1. The kite apparent velocity along iB di- more general power limit of crosswind kite, the passivity rection is positive, i.e. ua > 0. of the aerodynamic force should be ﬁrst discussed.

In conventional wind energy system, [4], Assumption 2.2. Power Loss 1 simply implies that the wind turbine is assumed always To establish the passivity property of the aerodynamic facing to the wind during the power harvesting. Under this model given in equations (8) and (9), some definitions in assumption, the kite apparent velocity Va can be repre- nonlinear system theory, [5], need to be reviewed first. sented by the apparent attitudes, α and β, in the following Consider a function, y = h(u), where y and u are input way, and output signal with compatible dimensions. Then, the T passivity of function h(u) is defined formally as follows, Va = kVak cos α cos β sin β sin α cos β . (5) Definition 1. A function y = h(u) is Passive if uT y ≥ 0. Equation (5) states that the kite apparent velocity is a The function is strictly input passive if there exist a func- function of kite apparent attitudes. In classical aerody- tion ϕ(u), with proper dimension, such that uT ϕ(u) > 0 namics, kite lift and drag coefficients, CL and CD, are also for all u 6= 0 and uT y ≥ uT ϕ(u). functions of apparent attitudes, The classical aerodynamics model introduced in the CL = CL(α),CD = CD(α). (6) previous section, i.e equations (3) - (8), can be represented using block diagram shown in Figure 2. The kite apparent By convention, lift and drag coefficients are defined based on the flow direction, i.e. direction of V shown in Figure Ct ∞ α 1. The aerodynamic coefficients along the body axes are C = C sin α − C cos α x L D Va α = arctan wa ua Cy = Cy 1 2 AB given by applying appropriate trigonometric transforma- − ρairSkVak β = arcsin va β 2 kVak tion, Cz = −CL cos α − CD sin α

C 1 0 sin α − cos α C x = L , Figure 2: Block Diagram of Aerodynamic Model Cz 0 −1 cos α sin α CD T T velocity, Va, can be treated as input to the model while CB = Cx Cy Cz − Ct 0 0 , the aerodynamic force in frame B, AB, is the output of where Ct is the turbine drag coefficient and Cy is the side the model. Then the following lemma can be proven, force coefficient. In a more compact form, the kite aero- Lemma 1. Under the Assumption 1, the aerodynamic dynamic coefficient in frame B is given by, force is strictly input passive with respect to apparent wind   velocity. CL sin α − CD cos α − Ct CB =  Cy  . (7) 1 − VT A = − ρkV k3SC > 0. (10) −CL cos α − CD sin α a B 2 a a 2 where Ca is given by, problem (16) is difficult to obtain. However, under certain simplified assumptions, the theoretical power limit of cross Ca = −CD cos β + Cy sin β − Ct cos α cos β. wind kite can be computed analytically. In this section, three different theoretical power limits, under three differ- Please refer to [6] for detailed proof of this lemma. The ent assumptions, will be derived. First, I will show that the passivity of the kite consists of two parts, the power dis- classical Loyd’s limit can be derived by assuming the side sipation due to kite structure and the power harvested by force acting on the kite is zero and the turbine drag and the on board turbine. Therefore, kite drag force are co-linear. Then, a power limit that only Remark 1. The pure power loss due to the kite structure assuming the side force is zero will be derived. Finally, the is given by, power limit, under small side slip angle assumption, will also be derived analytically. 1 VT A = ρkV k3S − C cos β + C sin β, (11) a k 2 a D y 3.1. Loyd’s Limit In classical aerodynamics, it is reasonable to assume where Ak = AB with Ct = 0. that the side force is negligible when the side slip angle is 2.3. Available Power of Crosswind Kites zero, that is Using the power extraction formula given in [2], the Assumption 2. If the side slip angle is zero then the side total power that can be extracted from the wind in the force coefficient is also zero, i.e. kite cross wind motion can be computed as follows, Cy = 0 if β = 0 (17) T T Pt = WC AC = WBAB. (12) To derive the Loyd’s limit, the following assumptions are The second equality can be derived by using equations also necessary, (1), (2) and (9). Combining with Remark 1, the available power in a cross wind motion is Assumption 3. The kite drag and turbine drag forces are co-linear. T T Pa = W AB + V Ak. (13) B a Under the Assumption 3, the aerodynamic coefficients given Denote the wind velocity measured in frame B as, in (7) can be simplified as follows,

T  sin − ( + ) cos  WB = Wx Wy Wz . (14) CL α CD Ct α CB =  Cy  . (18) Then by substituting equations (5), (7), (8) and (11) into −CL cos α − (CD + Ct) sin α equation (13), the available power is given by Therefore, the available power of the kite given in equation 1 (15) can be approximated by P = ρkV k2SWT C + VT A a 2 a B B a k 1 1 2 = ρkV k2S W (C sin α − C cos α − C ) Pa = ρSkVak Wx(CL sin α − (CD + Ct) cos α) 2 a x L D t 2 + W C − W (C cos α + (C + C ) sin α) + WyCy − Wz(CL cos α + CD sin α) y y z L D t + kV k(−C cos β + C sin β) . (19) + kVak(−CD cos β + Cy sin β) . (15) a D y

Hence, it is clear that the theoretical power limit of a cross- Under the assumptions 2 and 3, the upper bound of the wind kite system can be computed from the following non- available power can be found as in the following theorem, linear optimization problem, Theorem 1. Assume that the sideslip angle β is zero and the kite drag is co-linear with the turbine drag, then the 1 2 max ρSkVak Wx(CL sin α − CD cos α − Ct) available power of the crosswind kite, with on-board tur- α,β,Va 2 bine, is bounded by + WyCy − Wz(CL cos α + CD sin α) p 2 ( C2 + (C + C )2)3 + kVak(−CD cos β + Cy sin β) . (16) p 2 2 3 L D t Pa ≤ ρS( Wx + Wz ) 2 . (20) 27 CD 3. Theoretical Power Limits Especially, without on-board turbine, the available power is bounded by It worth noting that the available power given in (15) p is a nonlinear function of the angle of attack α and side 2 p ( C2 + C2 )3 P ≤ ρS( W 2 + W 2)3 L D . (21) slip angle β, and the solution to the nonlinear optimization a x z 2 27 CD 3 Proof. Under the assumption 2, if β = 0 then Cy = 0, and It has been shown, in Theorem 1, that Loyd’s limit can the available power (19) can be simplified as follows, be derived using the proposed optimization problem under assumptions 2 and 3. However, if Assumption 3 is relaxed, 1 P = ρSkV k2 W (C sin α − (C + C ) cos α) a different power limit can be achieved. a 2 a x L D t Corollary 1. Assume the sideslip angle = 0, then the −Wz(CL cos α + (CD + Ct) sin α) − kVakCD . (22) β power limit of the crosswind kite system is given by Notice that p 2 ( (W 2 + W 2)(C2 + C2 ) − W C )3 ≤ x z L D x t (27) Wx(CL sin α − (CD + Ct) cos α) Pa ρS 2 . 27 CD 0 T 0 − Wz(CL cos α + (CD + Ct) sin α) = (W ) C , (23) B B Proof. If the side-slip angle β = 0, then the side force coef- 0 0 ficient = 0. The available power , given in equation where WB and CB are defined as Cy Pa (15), can be simplified as follows, 0 WB = Wx Wz , 1 2 0 CL sin α − (CD + Ct) cos α Pa = ρkVak S Wx(CL sin α − CD cos α − Ct) CB = . 2 −CL cos α − (CD + Ct) sin α − ( cos + sin ) − kV k (28) 0 0 Wz CL α CD α a CD . It is clear that the norm of CB and WB are given by

0 q The upper bound of Pa over all α can be found using 2 2 kCBk = CL + (CD + Ct) the following inequality, 0 p kW k = W 2 + W 2 B x z Wx(CL sin α − CD cos α) − Wz(CL cos α + CD sin α) Therefore the upper bound of the available power Pa is CL sin α − CD cos α = Wx Wy q −CL cos α − CD sin α 1 2 p 2 2 2 2 Pa ≤ ρSkVak Wx + Wz CL + (CD + Ct) q 2 ≤ ( 2 + 2)( 2 + 2 ) (29) Wx Wz CL CD . − kVakCD . (24) Substituting inequality (29) into (28) gives Diﬀerentiating the right hand side of (24) with respect to 1 q kVak gives, P ≤ ρkV k2S (W 2 + W 2)(C2 + C2 ) − W C a 2 a x z L D x t 0 q ∂ 2 2 2 kVak kWBk CL + (CD + Ct) − kVakCD − kVakCD . (30) ∂kVak 0 q 2 2 = kVak 2kWBk CL + (CD + Ct) − 3kVakCD . Taking the derivative of the right hand side of equation (30) with respect to kVak gives that The maximizing apparent speed can be solved by the fol- q lowing equation, 2 2 2 2 2 (Wx + Wz )(CL + CD) − WxCt − 3kVak∗CD = 0. 0 q kV k 2kW k 2 + ( + )2 − 3kV k = 0 a ∗ B CL CD Ct a ∗CD . That is

That is, p 2 2 2 2 2 (Wx + Wz )(CL + CD) − WxCt kVak∗ = . (31) p 2 2 3 C 2 0 CL + (CD + Ct) D kVak∗ = 0 or kVak∗ = kWBk . 3 CD Therefore, the upper bound is given by substituting equa-

If kVak∗ = 0, it is clear that the available power Pa is tion (31) into equation (30), zero, therefore the maximizing apparent speed is p 2 ( (W 2 + W 2)(C2 + C2 ) − W C )3 p x z L D x t 2 2 + ( + )2 Pa ≤ ρS 2 . p 2 2 CL CD Ct 27 C kVak∗ = Wx + Wz . (25) D 3 CD Substituting equation (25) into (24) gives, p 2 ( C2 + (C + C )2)3 3.2. Small Side Slip Case ≤ (p 2 + 2)3 L D t (26) Pa ρS Wx Wz 2 . Although the optimization problem (16) is difficult to 27 CD solve directly, for the case in which the side slip angle is This is exactly the power limit given by Diehl in [2]. The small, an analytic result can be found. Loyd’s limit, which is given in [1], can be obtained by simply setting Ct = 0 in inequality (26). 4 Theorem 2. If the side slip angle is small such that the By Cβ ≤ 0, we have γ2 ≥ 0. The inequality (37) then side force coefficient Cy can be approximated by the linear can be simplified using notation γ1 and γ2, function, 2 1 2 CD γ2 Pa ≤ ρkVak S(−kVakCD + γ1 + ). (38) Cy ≈ Cββ, (32) 2 3 kVak then, the power limit of the crosswind kite system is given Taking the derivative of the right hand side of (38) with by respect to kVak gives 2 + 2 2 2 + ( + ) 2 1 γ1 CDγ2 γ1γ3 γ1 γ3 ∂ 2 CD γ2 Pa ≤ ρS 2 , (33) kVak (−kVakCD + γ1 + ) 2 27CD ∂kVak 3 kVak q 2 2 2 2 2 2 CDγ2 γ1 = (Wx + Wz )(CL + CD) − WxCt, = − 3C kV k + 2γ kV k + . D a 1 a 3 r 3C γ = − β W 2, The optimizing apparent speed kV k satisfies the follow- 2 4 y a ∗ CD ing equation, q 2 2 2 γ3 = γ1 + CDγ2 . 2 2 CDγ2 −3CD(kVak∗) + 2γ1kVak∗ + = 0. (39) Proof. If the side slip angle is small such that sin β ≈ β, 3 cos β ≈ 1 and Cy ≈ Cββ, then from the second term of whose roots are given by inequality (11), p γ ± γ2 + C2 γ2 2 1 1 D 2 Cy sin β ≈ Cββ ≤ 0. kVak∗ = . 3CD This implies that the coefficient Cβ is not greater zero, i.e. Clearly, the positive root should be chosen, i.e.

Cβ ≤ 0. p 2 2 2 γ1 + γ1 + CDγ2 kVak∗ = . (40) Then, the available power equation (15) can then be sim- 3CD pliﬁed as follows, Substituting the optimizing apparent speed (40) into the

1 2 T T right hand side of inequality (35) gives, Pa = ρkVak SWBCB + Va Ak 2 2 2 2 1 1 γ1 + 2CDγ2 + γ1γ3 (γ1 + γ3) 2 Pa ≤ ρS , = ρkVak S Wx(CL sin α − CD cos α − Ct) + WyCy 2 2 2 27CD q 2 2 2 2 2 − Wz(CL cos α + CD sin α) + kVak(−CD + Cββ ) . γ1 = (Wx + Wz )(CL + CD) − WxCt, (34) r 3Cβ 2 γ2 = − Wy , Similar to equations (28) - (30), the available power (34) 4CD q is bounded above by, 2 2 2 γ3 = γ1 + CDγ2 . 1 q P ≤ ρkV k2S (W 2 + W 2)(C2 + C2 ) − W C a 2 a x z L D x t 2 + WyCββ + kVak(−CD + Cββ ) . (35) 4. Relations Between Power Limits Notice that the right hand side of (35) is concave with To this end, different power limits have been obtained respect to β since C is not positive. The maximizing side β from the optimization problem (16) using different set of slip angle is determined by assumptions. In this section, the relation between power 2kVakCββ∗ + WyCβ = 0, limits (20), (21), (27) and (33) will be demonstrated. For simplicity, let us take the following notations, which implies p 2 2 3 2 p 2 23 ( CL + CD) Wy P1 = ρS W + W , (41) β∗ = − . (36) 27 x z C2 2kVak D p 2 2 3 2 3 ( C + (C + C ) ) Substituting equation (36) into the right hand side of (35) p 2 2 L D t P2 = ρS Wx + Wz 2 , (42) yields, 27 CD p 2 2 2 2 3 1 q 2 ( (Wx + Wz )(CL + CD) − WxCt) 2 2 2 2 2 P3 = ρS , (43) Pa ≤ ρkVak S(−kVakCD + (Wx + Wz )(C + C ) 2 2 L D 27 CD 2 2 2 2 Wy 1 γ1 + 2CDγ2 + γ1γ3 (γ1 + γ3) − WxCt − Cβ ). (37) P4 = ρS 2 . (44) 4kVak 2 27CD 5 It will be first shown that if the turbine drag and side forces If Wx = 0, then P2 and P3 can be simplified as follows, are negligible then the power limits given in equations (41) p 2 2 3 - (44) are equivalent. Second, the order relations between 2 3 ( CL + (CD + Ct) ) P2|Wx=0 = ρS|Wz| 2 , (49) these limits will be discussed. 27 CD p 2 ( C2 + C2 )3 Remark 2. If Cβ = 0, then P3 = P4. Moreover, if Ct = 0 3 L D P3|Wx=0 = ρS|Wz| 2 . (50) and Cβ = 0, then P1 = P2 = P3 = P4. 27 CD

Proof. Using definitions (41)-(43), it is clear that if Ct = 0, Therefore, it is clear that P3|Wx=0 ≤ P2|Wx=0 by equations then P1 = P2 = P3. On the other hand, by definition (33), (49) and (50). On the other hand, if Wz = 0 then P2 and P3 can be simplified as follows, γ2 = 0 if Cβ = 0. (45) p 2 2 3 2 3 ( CL + (CD + Ct) ) For a power generation operation, it is reasonable to as- P2|Wz =0 = ρS|Wx| 2 , (51) 27 CD sume that Wx ≤ 0 as shown in Figure 1. This indicates p 2 2 3 that 2 (|Wx| (CL + CD) − WxCt) P3|Wz =0 = ρS 2 . (52) 27 CD γ1 ≥ 0. (46) For a turbine generation operation, Wx ≤ 0, that is, Therefore, if γ2 = 0 then equation (52) can be further simplified as below,

γ3 = γ1. (47) p 2 2 3 2 3 ( (CL + CD) + Ct) P3|Wz =0 = ρS|Wx| 2 . (53) using equations (45) and (47), equation (33) can then be 27 CD simplified as follows, Calculate the following square difference, 2 2 2 1 γ1 + 2C γ2 + γ1γ3 (γ1 + γ3) q ρS D 2 2 2 2 2 2 ( (CL + CD) + Ct) − CL + (CD + Ct) 2 27CD 2 q 1 γ + γ1γ3 (γ1 + γ3) 2 2 1 =2Ct (CL + CD) − 2CDCt ≥ 0 = ρS 2 2 27CD 3 It is clear that P | ≤ P | . 2 ρSγ 2 Wz =0 3 Wz =0 = 1 27 2 CD P2 indicates that the power limit is only determined by the magnitude of the wind velocity if and are Using the definition of γ , it is clear that if C = 0, then CL CD 1 β fixed. on the other hand, P indicates that the wind di- P = P . Hence if C = 0 and C = 0, then P = P = 3 3 4 t β 1 2 rection also has influence on the power limit of the cross P3 = P4. wind kites. As shown in Remark 3, if Wx = 0, then the tur- The above remark shows that the power limits derived bine coefficient Ct has no contribution to the power limit in the previous section reduce to Loyd’s limit if the turbine of crosswind kites. For a cross wind kite with on-board drag and side force are negligible. turbine, one could argue that limit P3 is more reasonable than P2 from this observation, since wind turbine can not Remark 3. The order relations between the power limits harvest any power from a wind perpendicular to its direc- (41) to (44) is P1 ≤ P2, P1 ≤ P3 ≤ P4 Moreover, if Wx = tion. Moreover, P4 indicates that when the side force is 0, then P3 ≤ P2. On the other hand, if Wz = 0, then considered, the theoretical power limit of the cross wind P2 ≤ P3. kites is even higher. This is simply because nonzero side slip angle introduces nonzero side force which increases the Proof. It is clear that P1 ≤ P2 and P1 ≤ P3 by definition. 2 total aerodynamic force, hereby increases the total power It is also clear that γ3 is increasing with respect to γ2 . For a power generation operation, it is reasonable to assume as defined in equation (12). that Wx < 0, which implies that the power harvest by the turbine −WxCt > 0. This also implies that γ1 > 0. 5. Real Time Power Limit Using definition (44), P4 is a therefore a strictly increasing 2 function of γ2 . Hence, In the previous sections, the improved power limits of the cross wind kites have been proposed, and the rela-

min P4 = P4|γ2=0 = P3. (48) tions between the proposed limits and Loyd’s limit have γ2 been discussed. One of the fundamental observation in The second equality holds according to Remark 2. By the this paper is that the power limit of cross wind kites is above derivation, we have proven time varying with respect to wind velocity and kite aerodynamic states such as angle of attack and side slip angle. P1 ≤ P3 ≤ P4. The following simple but nontrivial case implies that under 6 certain situation, the theoretical power limit of the cross Again, due to the Assumption 3, there is no clear or- wind kites can be lower than the limits given in equation der relation between P0 and P2 as shown in the following (41), (43) and (44). corollary,

Corollary 2. If α = β = 0, then the power limit of the Corollary 4. cross wind kite is given by However, it still suﬃces to argue that if the angle of 2 (−W (C + C ) − W C )3 attack and side slip angle is known, better estimation on P = x D t z L (54) 0 ρS 2 . the cross wind kite power limit could be obtained. 27 CD

Moreover, the power limit P0 satisﬁes the following rela- Lemma 2. If α and β is known at certain instance t, then tion, the power limit of cross wind kite at instance t is given by 2 W¯ 3 P0 ≤ P1. (55) P ≤ ρS (58) a 27 C¯2 Proof. If the angle of attack and side slip angle is zero, ¯ ¯ then available power of cross wind kites is given by where W and C are given by, ¯ 1 2 W =Wx(CL sin α − (CD + Ct) cos α) Pa(α = 0, β = 0) = ρkVak S Wx(−CD − Ct) − WzCL 2 + W C − W (C cos α + (C + C ) sin α), y y z L D t ¯ − kVakCD (56) C = − CD cos β + Cy sin β.

Taking the derivative of Pa with respect to kVak is given Lemma 2 can be proven similar to equation (24) - (26), by, which will not be repeated here. Notice that in steady aerodynamics, the lift and drag coeﬃcients are functions ∂Pa 2 of angle of attack α and the side force coeﬃcient is function = −2kVak(Wx(CD + Ct) + WzCL) − 3kVak CD ∂kVak of side slip angle β. Therefore, given α and β, W¯ and C¯ are also can be calculated. Therefore, in principle, Lemma Then the optimum apparent speed of the kite is given by 2 gives us an approach to calculate the real time power

2 Wx(CD + Ct) + WzCL limit of a cross wind kite using the following three steps, kVak∗ = − . (57) 3 CD 1. Measure the angle of attack and side slip angle of Substituting equation (57) into (56) gives that the cross wind kite. 2. Calculate W¯ and C¯ using definitions. P0 = max Pa(α = 0, β = 0) 3. Calculate the power limit using equation (58). Va k k 3 2 (−Wx(CD + Ct) − WzCL) = ρS 2 . 6. Conclusion 27 CD In this paper, the modification of the power limit formula of the crosswind kite energy systems is discussed. It is clear that, if α = β = 0, then the expression of First, a classical aerodynamic model was presented, and power limit of cross wind kite is simple. Moreover, it can its passivity was derived. Based on this passivity property, be shown that limit (54) is also lower than limits given in the power loss of the cross wind kite systems was obtained. (41), (43) and (44). A nonlinear optimization formulation, which determines the available power of the crosswind kites, was proposed. Corollary 3. If Ct = 0, then P0 ≤ P1. Otherwise, if Then the classical Loyd’s limit is derived using this for- Ct 6= 0, then P0 ≤ P3 ≤ P4. mulation. Moreover, a generalization of the power limit formula is also derived. Then the order relations between Proof. From Remark 3, P3 ≤ P4. For conciseness, we shall the proposed limits and classical Loyd’s limit are proven. first prove P0 ≤ P3. From Corollary 1, it is clear that Finally, a real time power limit calculation method is pro-

P0 = max Pa|α=0,β=0 ≤ max Pa|β=0 = P3 vided. Va Va ,α k k k k

Hence, if Ct = 0, from Remark 2, the following inequality References also holds, [1] M. L. Loyd, Crosswind kite power, Journal of Energy 4 (3) (1980) 106–111. doi:10.2514/3.48021. max Pa|α=0,β=0,Ct=0 ≤ max Pa|β=0,Ct=0 = P1. Va Va ,α [2] M. Diehl, Airborne Wind Energy Basic Concepts and Physical k k k k Foundations, Springer-Verlag, New York, 2013, Ch. 1, pp. 3–22. doi:10.1007/978-3-642-39965-7_1.

7 [3] J. D. Anderson, Fundamentals of Aerodynamics, McGraw-Hill, New York, 2011. [4] B. Etkin, L. D. Reid, Dynamics of Flight: Stability and Control, John Wiley & Sons, New York, 1996. [5] H. K. Khalil, Nonlinear Systems, 3rd Edition, Upper Saddle River, NJ, New York, 2001. [6] H. Li, D. J. Olinger, M. A. Demetriou, Apparent attitude track- ing of airborne wind energy system, Journal of Guidance, Con- trol, and Dynamics 42 (4) (2019) 958–962. doi:10.2514/1. G003527.