Power Limit of Crosswind Kites

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 '(u), with proper dimension, such that uT '(u) > 0 namics, kite lift and drag coefficients, CL and CD, are also for all u 6= 0 and uT y ≥ uT '(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 1 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 0 1 velocity. CL sin α − CD cos α − Ct CB = @ Cy A : (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.

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