1 Introduction

In aviation, the development of controllers for the automatic operation of flying vehicles continues to be a challenge. In the adjacent research field of airborne wind energy, the design of robust and safe control systems remains a key driver for the success of this emerging technology. As part of the EnerGlider project [1], the Institute of Flight System Dynamics analyzes the flight dynamics of flying wings and develops suitable flight controllers applicable in this field. Such airborne wind energy system is schematically shown in Fig. 1. In general, it consists of one or more controlled aircraft, a ground station, and one or more tethers that transmit the aerodynamic forces acting on the wings to the ground station [2]. As described by Erhard and Strauch [3], the design of these systems in terms of effectively generating power and enabling safe operation requires appropriate modeling. However, the tether forces encountered in tethered flight are often more than five times the aircraft weight [4]. Thus, the forces of the tether acting on the aircraft significantly impact the flight dynamics. Appropriate tether models have to extend the classical flight mechanics to model these airborne wind energy systems properly.

Fig. 1
figure 1

Components of an airborne wind energy System

Full size image

For this purpose, the tether is generally simplified as straight and as a mechanical spring. However, such a modeling approach does not consider a sagging of the tether due to external loads, such as gravity. Accounting for this tether sag is important because it causes a misalignment of the tether force at the airborne vehicle from the straight line connecting to the ground station. A more comprehensive introduction to the influence of the tether sag is given by Trevisi et al. [5]. To account for sag, the tether can also be modeled as a system of lumped masses discussed by Williams [6] and implemented by Eijkelhof and Schmehl within the open-source model MegAWES [7]. In these models, the tether is discretized by a finite number of lumped masses connected by tether segments and modeled as spring–damper systems. The induced stiffness and damping forces, as well as external loads, are then taken into account in the dynamics of the lumped masses. Besides this lumped masses approach, another approach for a dynamic model of the tether is based on the Lagrangian formulation, which is implemented by Sánchez-Arriaga et al. within the tool LAKSA [8].

These tether models play a critical role in the development phase of airborne wind energy systems. To test and design the flight controllers before field experiments, they need to be run considering the entire controlled system in a model-in-the-loop simulation environment. To facilitate an optimized design of the controller, such simulations are performed in real-time with a fixed size of the time step. In doing so, the simulated controller is clocked identical to the one integrated on the flight system. However, as the considered tether stiffness is high, such dynamic models can result in stiff mathematical problems. When an explicit solution method such as the Bogacki–Shampine is considered, the convergence performance depends on the problem's discretization in time. As dynamic tether models require a minimum discretization in space to map the tether sufficiently, they are bound to a specific time-step size according to the Courant–Friedrichs–Lewy condition, which links a discretization in space with a minimal discretization in time [9]. However, such increased discretization in time directly affects the computational effort and thus limits the general usability of these dynamic models, especially for such real-time simulation approaches.

To further reduce and simplify the problem, the tether force can be computed statically for every time step. In doing so, the time discretization of the simulation becomes independent of the tether model. Furthermore, complex high-frequency dynamic oscillations that can occur with dynamic lumped mass models are avoided by such a static tether model. This allows a first tether modeling focusing on the primary influences such as stiffness and deformation of the tether due to external loads. More complicated modeling parameters, such as the material-specific damping of the tether, can, therefore, be neglected for the time being. This motivated to extend the static spring model with a modeling approach introduced by Bigi et al. [10]. In contrast to the static spring model, such a model considers a sagging of the tether by analytically solving the catenary problem. As mentioned, however, it must be emphasized that this model does not represent the full dynamics of the tethers, as dynamic models, e.g., the lumped mass models, do. Nevertheless, the presented hybrid model may be a reasonable alternative to the simple spring model if a static modeling approach for the tether appears sufficient. For example, high-frequency dynamic oscillations can or should be neglected in modeling.

In the following, we present the spring model and the catenary model as the underlying tether models before proceeding with the development of a hybrid tether model. This is followed by an analysis and comparison with the lumped mass model considering various static configurations. Then, simulation results of the tethered flight are examined to assess and compare the hybrid tether model with the spring-based model. Finally, the results presented are summarized in a conclusion and a brief outlook is given.

2 Basic tether models

In a first approach, the tether can be simplified as a spring with a homogeneous relative stiffness distribution \({K}_{t}\). For this linear spring model, a tensile force \(F\) is induced due to a stretching of the tether from an unstretched length \({l}_{0}\) to a stretched one. Applied to airborne wind energy, the stretched length corresponds to the distance \(r\) between the aircraft and the ground station. Based on Hooke's law [11], this tension force can be derived as shown in Eq. (1).

$$F = K_{t} \frac{{r - l_{0} }}{{l_{0} }}.$$
(1)

In doing so, the relative tether stiffness is dependent on the tether cross section and the modulus of elasticity. However, this simple model does not consider any external loads or any deformation of the tether resulting from these loads and is only dependent on the stretching of the tether. A relevant parameter for further analysis is the sagging ratio \(k\) given in Eq. (2).

$$k = \frac{{l_{0} }}{r}.$$
(2)

It is defined as the ratio of the unwound tether length and the distance between the airborne vehicle and the ground station.

In the following, the influence of the tether length is analyzed for a configuration, where the airborne vehicle is kept at a constant position relative to the ground station. Therefore, only the sagging ratio \(k\) is varied. The resulting tether force at the airborne vehicle as a function of the sagging ratio is presented in Fig. 2. Here, a relative tether stiffness is chosen based on Fechner’s model [12], which is in order of \(6 \cdot {10}^{5}\, \mathrm{N}\). The figure shows the linear propagation for a sagging ratio smaller than one. As this corresponds to Eq. (1), this area is called the elastic stretching regime in the following. The figure also shows that the model does not consider any elastic forces for a sagging ratio larger than one as this spring model accounts for tensile forces only. However, since a tether would sag in this condition, this area can be defined as the sagging regime.

Fig. 2
figure 2

Tether force \(F\) derived from the spring model and plotted over the sagging ratio \(k\). Here, only the stretching of the tether is taken into account. Other influences such as gravity or aerodynamic drag are not considered in this model

Full size image

In contrast to this introduced spring model, the model based on the catenary and presented by Bigi et al. [10] considers also external loads such as the earth’s gravitational field to some extent. To approach this problem, an equilibrium of forces is established in a tether–load coordinate system shown in Fig. 3. This coordinate system is oriented in such a way that the vector \({z}_{t}\) acts in the direction of the external resultant load \({q}_{t}\). The coordinate \({x}_{t}\) is perpendicular to the load plane, which is spanned by \({q}_{t}\) and the distance vector \(OP\) from the ground station to the flight system.

Fig. 3
figure 3

Catenary-based tether model with uniform external load \({q}_{t}\)

Full size image

This forms the basis for the development of a differential equation describing the local curvature \(d{z}_{t}/d{y}_{t}\) of the tether as a function of the external tether line load \({\overrightarrow{q}}_{t}\) and the \({y}_{t}\)-directed local tether force \({F}_{yt}\) (compare Fig. 3). Bigi [10] has formulated this differential equation given in Eq. (3), which is also known as the catenary curve.

$$\left| {q_{t} } \right|\sqrt {1 + \left( {\frac{{dz_{t} }}{{dy_{t} }}} \right)^{2} } = F_{yt} \frac{{d^{2} z_{t} }}{{d y_{t}^{2} }}.$$
(3)

To keep this problem analytically solvable, the external tether line load has to be assumed constant. This assumption is necessary for this model, but it must also be viewed as very critical. In reality, the tether loads are rarely uniform as, for example, the wind speed varies with altitude. Nevertheless, if a constant line load is assumed, the tether curve and evolution of the tether force along the tether can be found by the integration of this differential equation (Eq. 3). As the differential equation is a function of the unknown tether force, it is possible to reformulate this problem to one only including the length of the tether \({l}_{0}\), the position of the airborne vehicle relative to the ground station, and the external tether line load \({q}_{t}\) [10]. Since this model is static, it can be calculated independently for each time step, taking into account the corresponding boundary conditions such as the position of the airborne vehicle as well as for example the wind direction and the body velocity considered in the external tether line load \({q}_{t}\).

As already indicated and stated in the following equation (Eq. 4), external loads such as the aerodynamic lift and drag (\({q}_{l}, {q}_{d}\)) as well as inertia and gravitational loads (\({q}_{i}, {q}_{g}\)) act on the tether and can be superposed to the mentioned external tether line load \({q}_{t}\).

$$q_{t} \left( z \right) = q_{l} \left( z \right) + q_{d} \left( z \right) + q_{g} \left( z \right) + q_{i} \left( z \right).$$
(4)

Since these loads may vary with altitude \(z\) (compare Fig. 3), for instance considering a wind velocity gradient, the external tether load \({q}_{t}\) can be considered as a function of \(z\). However, as the differential problem of the catenary, Eq. (3) can only be solved for a constant external load \({q}_{t}\), these external loads have to be reduced to an equivalent constant load as given in Eq. (5).

$$q_{t, eq} = q_{l,eq} + q_{d,eq} + q_{g,eq} + q_{i,eq} .$$
(5)

If it is assumed that the aerodynamic velocity changes with the altitude, an equivalent inflow velocity can be found to calculate the equivalent aerodynamic lift and drag load (\(q_{l,eq} , q_{d,eq}\)). In this manner, it is also assumed that the tether sagging is small and thus a constant angle of attack between the tether and the inflow can be considered. According to Bigi et al. [10], this allows finding the equivalent aerodynamic velocity as:

$$\left| {V_{a,eq} } \right|_{ }^{ } = \sqrt {\frac{{\mathop \smallint \nolimits_{{O_{z} }}^{{P_{z} }} \left| {V_{a} \left( z \right)} \right|^{2} dz}}{{P_{z} }}} .$$
(6)

In doing so, the integration of the aerodynamic velocity is done from the height of the ground station \({O}_{z}\) to the one of the airborne vehicle \({P}_{z}\). Furthermore, the aerodynamic speed itself is composed of the wind and the body velocity. Here, the wind velocity is modeled with a velocity gradient according to the ITTC [13]. In doing so, the wind velocity at a fixed height \({z}_{w,ref}\), e.g., at the height of a wind velocity measuring station, is used with the corresponding wind velocity \({V}_{w,ref}\) and a friction coefficient \(n\) (compare Eq. 6). This allows deriving the wind velocity \({V}_{w}\) at the height \(z\). However, for the body velocity of the tether, a linear progression along the tether is assumed. Here, \({V}_{{b}_{AV}}\) refers to the body velocity of the airborne vehicle and \({P}_{z}\) to its position. In summary, the aerodynamic velocity is modeled as:

$$V_{a} \left( z \right) = V_{w,ref} \left( {\frac{z}{{z_{w,ref} }}} \right)^{n} - V_{{b_{AV} }} \frac{z}{{P_{z} }}.$$
(7)

By inserting Eq. (7) in (6), the equivalent aerodynamic velocity can be determined and used to compute the equivalent drag \({\overrightarrow{q}}_{d,eq}\) and lift \({\overrightarrow{q}}_{l,eq}\) loads according to Eqs. (8) and (9) (Bigi [10]).

$$q_{d,eq} = \frac{1}{2}\rho_{a} d_{t} \left[ {1.1\sin^{3} \left( {\alpha_{t} } \right) + 0.02} \right]\left| {V_{a,eq} } \right|_{ } V_{a,eq} ,$$
(8)
$$q_{l,eq} = \frac{1}{2}\rho_{a} d_{t} \left[ {1.1\sin^{2} \left( {\alpha_{t} } \right)\cos \left( {\alpha_{t} } \right)} \right] \left| {V_{a,eq} } \right|_{ } \frac{{V_{a,eq} \times \left[ {V_{a,eq} \times \left( {P - O} \right)} \right]}}{{\left| {V_{a,eq} \times \left( {P - O} \right)} \right|}}.$$
(9)

Here, \({\rho }_{a}\) is the density of air, \({d}_{t}\) the tether diameter, \({\alpha }_{t}\) the mean angle of attack, \({V}_{a}\) the aerodynamic velocity, \(P\) the position vector of the airborne vehicle and \(O\) the position of the ground station. Here, the lift and drag coefficients used in Eqs. (8) and (9) are taken from Bigi et al. [10].

This catenary model initially considers only static states and no dynamic loads. However, since the operating states of airborne wind energy systems are often accompanied by inertial loads, e.g., due to a constant turning rate, the model presented by Bigi is extended in the following. To do so, an inertia load \({q}_{i}\) is considered in the external tether line load \({q}_{t}\) that incorporates a linear progression of the body acceleration. Again, it is considered that the sagging of the tether is small to formulate the inertia loads \({q}_{i}\) as a function of \(z\). In doing so and for simplicity reason, the tether acceleration at point \(P\) equals the acceleration of the airborne vehicle. and equals zero at the ground station labeled P. Then, \({q}_{i}\) can be derived as:

$$q_{i} \left( z \right) = \frac{{\rho_{t} d_{t}^{2} }}{4}\left( {g + a_{{b_{AV} }} \frac{z}{{P_{z} }}} \right).$$
(10)

Similar to the aerodynamic line load, the equivalent inertia load can be found according to Eq. (11):

$$\left| {q_{i,eq} } \right|_{ }^{ } = \sqrt {\frac{{\mathop \smallint \nolimits_{0}^{{P_{z} }} \left| {q_{i} \left( z \right)} \right|^{2} dz}}{{P_{z} }}} .$$
(11)

As the gravitation field is constant along the tether, the equivalent gravitational load equals the local gravitational load (\({q}_{g}\left(z\right)={q}_{g,eq}\)). Finally, the different equivalent loads are summed up according to Eq. (5). This allows identifying the equivalent load vector \({q}_{t,eq}\) that acts constantly along the tether. Eventually, the catenary problem can be analytically solved and the emerging tether curvature, as well as the tether forces, can be calculated.

Regarding applications of airborne wind power systems, both models, the spring model, and the catenary-based model, require an unwound tether length and the position of the airborne vehicle to compute the occurring tensile tether forces. Moreover, the catenary-based model also needs an acting external constant load vector \({ q}_{t}\).

As for the spring model, the influence of the sagging ratio on the tether force at the airborne vehicle can be analyzed. Here, the same load scenario as shown for the spring model is considered. In contrast, also external gravitational, as well as aerodynamic loads, are considered. Thereby, it is assumed that the aerodynamic line load considers the aerodynamic model given in Eq. (6) with a reference wind velocity of \(5 \mathrm{m}/\mathrm{s}\). The following Fig. 4 shows the progression of the tether force at the airborne vehicle for this configuration and a varying sagging ratio \(k={l}_{0}/r\).

Fig. 4
figure 4

Tether force \(F\) derived from catenary tether model and plotted over the sagging ratio \(k\)

Full size image

In contrast to the spring model, a tether force acts at the airborne vehicle for a sagged tether with a sagging ratio \(k\) greater than one. Moreover, it is noticeable that for a decreasing tether length, the tether force increases nonlinear and significantly for sagging ratios close to one. For sagging ratios smaller one, it remains constant.

3 Design and implementation of the hybrid model

As shown in Fig. 4, a change of the tether force occurs only if the sagging ratio is greater than one, while for sagging ratios smaller than one, the tether force remains constant. This means that if the direct distance \({l}_{0}\) is greater than the unstretched tether length \(l\), no additional tether force from the stretching of the tether is generated. To gain a better representation of the tether in one model, the two models are combined in a hybrid model. Thereby, the catenary-based model is always active, while the spring model is only considered when the tether length is smaller than the distance.

For the formulation of the hybrid tether model, the computations of the tether forces from both models are superposed. Thus, as shown in Fig. 5, a changing tether force can also be determined for sagging ratios smaller than one. However, this form of a hybrid model does not take into account any influence of the tether stiffness for a sagging ratio \(k\) greater one. In this case, the hybrid model derives the tether force only with the catenary-based tether model, which is independent of the tether stiffness.

Fig. 5
figure 5

Tether force \(F\) derived from hybrid tether model and plotted over the sagging ratio \(k\)

Full size image

To implement the stiffness into the catenary-based model, an iterative approach shown in Fig. 6 is conducted. In doing so, the catenary-based model provides the tensile load along the tether, which is reduced to a mean tensile load. This can be done since the deformation within a tether section is neglected and since only a linear elastic stiffness is assumed here. Based on Hooke’s law given in Eq. (1), a resulting deformation of the tether can be derived. This deformation is then added to the unwound tether length \({l}_{0}\).

Fig. 6
figure 6

Iterative procedure to include the tether stiffness in the hybrid model

Full size image

As this new unwound tether length changes the boundary conditions of the catenary problem, the catenary-based model has to be computed once again for the new stretched tether length. This procedure is repeated until the change in the tether force becomes negligible.

For a relative tether stiffness in the order of \(6 \cdot {10}^{5}\,\mathrm{ N}\), the change in the tether force is less than 1% after three iterations and the problem converges. However, very elastic tethers deform stronger, leading to a larger number of iterations until the change in the tether force is less than 1% and thus negligible within the general model deviations.

In Fig. 7, the tether force at the airborne vehicle derived from the hybrid tether model is plotted over the sagging ratio \(k\). The considered position of the airborne vehicle and the ground station as well as the external loads are the same as presented in the previous configuration. The black solid line plotted in Fig. 7 shows the results, when not considering the stiffness for \(k>1\) corresponding to the results given in Fig. 5, whereas the dashed one shows the results when considering the stiffness for \(k>1\). Moreover, Fig. 7 shows the progression of the tether force for a more elastic tether. It can be seen, that although the tether is exposed to the same external load, the more elastic tether reveals a significant deviation to the stiffer one for sagging ratios very close to one. However, this deviation becomes negligible if \(l\) is greater than the direct distance \({l}_{0}\).

Fig. 7
figure 7

Tether force \(F\) derived from hybrid tether model with stiffness for \(k>1\) and plotted over the sagging ratio \(k\)

Full size image

4 Analysis of the hybrid tether model and comparison to a lumped mass model

A lumped mass model is used as a first validation of the developed hybrid model, for which three configurations are analyzed. The first of these three configurations represents a tethered flight, while the other two focus on outlining the characteristics of the hybrid model. It has to be mentioned that the results from the lumped mass model are compared after a decay phase to compare a quasi-static state with the hybrid model.

For the first configuration, the end of the tether connected to the airborne vehicle is positioned \(100\mathrm{ m}\) above the ground and \(100 \mathrm{m}\) in \(x\)- and \(y\)-direction. The tether is exposed to an inflow of \(20 \mathrm{m}/\mathrm{s}\) in \(x\)- and \(y\)-direction, while the sagging ratio is set to \(k=l/{l}_{0}=1.01\). The ground station is positioned in the origin. As shown in Fig. 8, the hybrid tether model shows a symmetric tether sagging, whereas the tether curve corresponding to the dynamic lumped mass model has a stronger sagging close to the airborne vehicle and a less distinctive one close to the ground station.

Fig. 8
figure 8

2D projection of the tether curves corresponding to the dynamic lumped mass and hybrid model (Exposed to:\(g=9.81 \mathrm{m}/{\mathrm{s}}^{2}\),\({\overrightarrow{V}}_{b,x}=20 \mathrm{m}/\mathrm{s}\),\({\overrightarrow{V}}_{b,y}=20 \mathrm{m}/\mathrm{s}\), \(k=l/{l}_{0}=1.01\))

Full size image

This deviation can be explained by the assumption made for the external load regarded in the hybrid model. Since the catenary problem can only be solved for a reduced and constant external load acting along the tether, the curvature is always symmetric. In contrast, the lumped mass model can take into account a varying external load, resulting in tether curvatures that do not necessarily have to be symmetric as shown for the given configuration.

Regarding the control of the airborne vehicle, the magnitude and direction of the emerging tether force at the airborne vehicle is of particular interest. For this first configuration, the force determined by the hybrid model deviates by \(7 \%\), to the force determined with the lumped mass model. However, the emerging angle between the force vector at the airborne vehicle determined by the lumped mass model and the one corresponding to the hybrid model is about \(8^\circ\). This deviation can be explained by the symmetric tether curve resulting in the hybrid model.

In a second configuration, the tether has the same geometric constraints but is subjected to an asymmetric external load. In contrast to the previous configuration, the gravitation is neglected and only an aerodynamic inflow in positive \(x\)-direction is considered. This is not a realistic operation condition, but it allows observing the influence of the aerodynamic loading independently. As a result, the deviation in the tether curvatures of the two models is more distinctive. Shown in Fig. 9, for the hybrid tether model, the curvature is only sagging in the \(x\)-\(z\)-plane.

Fig. 9
figure 9

2D projection of the tether curves corresponding to the lumped mass and hybrid model (Exposed to:\({\overrightarrow{V}}_{b,x}=20\, \mathrm{m}/\mathrm{s}\), \(k=l/{l}_{0}=1.01\))

Full size image

Since the hybrid model is only solved for a two-dimensional plane, spanned by the load vector and the relative position vector of the airborne vehicle and the ground station, the external load does not influence the deformation in the \(y\)-\(z\)-plane. As the lumped mass model considers a coupling between the two planes, the tether curvature shows a deformation also in the \(y\)-\(z\)-plane. Although the tether curvatures from the two models deviate more significantly, the force magnitude only differs by \(7 \%\), while the direction of the force vector acting at the airborne vehicle deviates by an offset angle of \(10^\circ\).

In a last observed configuration, an inertia load is superimposed to the previous configuration. Thereby, an acceleration of \(5 \mathrm{m}/{\mathrm{s}}^{2}\) in negative \(x\)-direction is set for the airborne vehicle. This configuration is intended to show the effect of such loading and does not necessarily represent actual flight operations. The tether sagging emerging for this case is similar to the second configuration, although the external loads are different. However, the emerging tether force is increased by \(7 \%\), while the deviation regarding the magnitude of the forces between the two models is \(6 \%\). In addition, the two tether force vectors at the airborne vehicle have a direction offset of \(9^\circ\) to one another.

Based on the comparison with the lumped mass model, it can be concluded that the hybrid model is generally suitable for mapping the physics of the tether. Nevertheless, it has been shown that there are deviations between the lumped mass and the hybrid model, especially concerning the angle between the force vector at the airborne vehicle and the position vector. Moreover, it has to be mentioned that the hybrid tether model is a static model with constant inertial loads. Consequently, this extended hybrid model cannot represent the full tether dynamics as a lumped mass model does. Nevertheless, this static model is independent of the time-step size and, therefore, suitable for computations with an explicit solution method and a limited resolution in time. In contrast to the spring model, this hybrid tether model is capable to take into account sagging due to external loads.

5 Analysis of the hybrid tether model for a tethered flight

In the following, a tethered circular flight above the ground station is considered. In this analysis, no wind field is assumed in order not to directly superimpose the effects from a tilted trajectory on the aerodynamic effects. To model the controlled system for such tethered flight, the tether force from the hybrid or spring tether model is also added to the gravitational and aerodynamic forces acting in the center of gravity of the airborne vehicle. Here, no further offset is assumed between the attachment point of the tether and the center of gravity of the airborne vehicle. Then, with the forces and moments acting in the center of gravity, the equation of motion can be formulated to derive the flight states for each time step. This dynamic system is controlled by a flight path and an attitude controller to command a circular flight path. A presentation of the attitude controller and further details about the considered airborne vehicle are given by Fuest [14].

For the tethered flight, the airborne vehicle is controlled on a prescribed circular flight path shown in Fig. 10. During this circular flight, the pitch angle is increased to a maximum of \(4^\circ\), while a constant mean thrust is provided. In the following, the results for this tethered flight with the spring model and the one corresponding to simulations with the developed hybrid tether model are presented. In doing so, the results for both simulations are analyzed and findings are compared against each other.

Fig. 10
figure 10

Flight path for tethered Flight with the spring model

Full size image

For the simulation with the spring model plotted in Fig. 11, it is noticeable that in the first few seconds the tether force remains zero, whereas the sagging ratio \(k=l/{l}_{0}\) remains greater one. However, since the spring model is considered here, this goes hand in hand with the presented results shown in Fig. 2 for \(k>1\). After a few seconds, however, the airborne vehicle experiences several strong peak forces from the tether, which can also be seen in the progression of \(k\) as downward directed peaks with \(k<1\). These peak forces are induced since the airborne vehicle is not capable of completely reaching the prescribed circular flight path until this moment. Here, these force peaks are still small enough. However, higher forces pose an increasing risk of tether rupture and potentially bring the aircraft into critical flight states.

Fig. 11
figure 11

Tethered flight with the spring model.  The figure gives the propagation of the tether force (a) and the sagging ratio (b).

Full size image

As shown schematically in Fig. 12, the first few force peaks pull the airborne vehicle repeatedly back into the center of the circular flight path until the magnitude of the tether force decreases and the system reaches a steady tethered flight state. In doing so, the tether force components \({F}_{AV, {y}_{b}},\) and \({F}_{AV, {z}_{b}}\) gradually increase until the prescribed maximum pitch angle is reached and a balanced state between the tether force and the aerodynamic lift force is generated.

Fig. 12
figure 12

Schematic representation of how the peak tether forces pull the airborne vehicle onto the circular path

Full size image

For this quasi-static state, the force component in \({x}_{b}\)-direction is negative. This can be traced back to the tether force acting in \({z}_{g}\)-direction. However, as the airborne vehicle is controlled to a positive pitch, the tether force must also include a comparatively small force component in negative \({x}_{b}\)-direction.

As shown in Fig. 13, unlike the simulation from the spring model, no peak forces or oscillation can be observed for simulations with the hybrid model. The transition from a comparatively untethered flight condition with tether forces below 50 N to a tethered flight condition with forces above 100 N is therefore smooth in this model. Furthermore, there are no significant force peaks during this transition with the same flight controller used.

Fig. 13
figure 13

Tethered flight with the hybrid tether model. The figure gives the propagation of the tether force (a), the sagging ratio (b) and the tether deviation angle (c).

Full size image

Moreover, Fig. 13 depicts the propagation of the offset angle \({\Theta }_{\mathrm{offset}}\). This angle is defined between the position vector of the airborne vehicle relative to the ground station and the actual tether force vector \({\overrightarrow{F}}_{AV}\) acting at the airborne vehicle. This deviation angle \({\Theta }_{\mathrm{offset}}\) decreases simultaneously to the sagging ratio \(k\) but remains greater than \(10^\circ\) in this flight scenario. Eventually, this leads to a tether force smaller than the one from the simulation with the spring model. However, the magnitude of the negatively directed force component \({F}_{AV, {x}_{b}}\) is three times the one from the simulation with the spring model. The increased magnitude of this force component \({F}_{AV, {x}_{b}}\) can be traced back to the additionally considered external loads acting on the tether.

As this tether force component \({F}_{AV, {x}_{b}}\) counteracts the basic thrust of the airborne vehicle, which is the same in both simulations, the resulting aerodynamic velocity is \(20 \%\) smaller than in the simulation with the spring model.

This resulting deviation in the inflow velocity also explains the lower tether force component \({F}_{AV, {z}_{b}}\) for the simulation with the hybrid model. If the quasi-steady state is considered, the lift coefficient in both observed simulations is similar and compensated by the tether force component \({F}_{AV, {z}_{b}}\). In this case, the relation that is given in Eq. (12) can be concluded. According to this consideration, the tether force component \({F}_{AV, {z}_{b}}\) is proportional to the inflow velocity \(\left|\overrightarrow{V}\right|\).

$$F_{{AV,z_{b} }} \approx L \propto \left| {\vec{V}} \right|_{ } .$$
(12)

Considering this relation as well as the tether force component \({F}_{AV, {z}_{b}}\) from the simulation with the spring model and the inflow velocities from both simulations, the tether force \({F}_{AV, {z}_{b}}\) can be computed for the hybrid tether model. The resulting tether force component deviates only by \(10 \%\) to the one obtained from the simulation with the hybrid tether model. Thus, it can be concluded that the decreased inflow velocity in the hybrid model has a significant influence on the magnitude of the generated tether force, especially on the tether component \({F}_{AV, {z}_{b}}\).

The occurring tether force and curvature directly influence the bank angle. For the presented simulation, the reduced tether force and considered tether sagging for the simulation with the hybrid model lead to an increase of the bank angle.

As shown in Fig. 14, the centrifugal force, which is the product of the yaw rate \(r\), the body velocity \({\left|{\overrightarrow{V}}_{b}\right|}_{2}\) as well as the vehicle’s mass \(m\), acts opposite towards the tether force component \({F}_{AV, {y}_{b}}\) and a force component of the vehicle’s weight force. The corresponding equilibrium of forces in the \({y}_{b}\)-direction for this quasi-static state allows deriving a relation between the tether force component \({F}_{AV, {y}_{b}}\), the bank angle \(\phi\), and the body velocity \({\overrightarrow{V}}_{b}\) as:

$$F_{{AV,y_{b} }} = m\left( {\left| {\vec{V}_{b} } \right|_{2} r - \sin \left( \phi \right)g} \right).$$
(13)
Fig. 14
figure 14

Forces acting on the airborne vehicle’s center of gravity in the \({x}_{b}\)-\({z}_{b}\)-plane for a tethered curving flight

Full size image

For both, the spring tether model and the hybrid tether model, the tether force component \({F}_{AV, {y}_{b}}\) can be derived using this equation, whereby the results for both simulations differ less than \(2 \%\). Such small deviation is reasonable as both models consider no further tether loads acting in \({y}_{b}\)-direction.

To verify the simulation results further, the tether force component acting in negative \({x}_{b}\)-direction can be observed in detail for both models. This force component can also be considered in the drag coefficient of the entire airborne wind energy system. According to Argatov, Rautakorpi, and Silvennoinen [15], the drag coefficient of the airborne system \({C}_{D,AWES}\) can be computed as given in Eq. (14). It is assumed that the tether is treated as inelastic and straight, the actual aircraft dynamics are not taken into account and the forces on the aircraft are modeled purely as lift and drag. This general empirical approach to derive the drag coefficient of the entire airborne wind energy system has also been numerically validated by them [16].

$$C_{D,AWES} = C_{D,AV} + \frac{1}{4}\frac{{d_{t} l_{0} }}{S}C_{D,t} .$$
(14)

Here, \({C}_{D,AV}\) considers the drag coefficient of the airborne vehicle without the tether, \({d}_{t}\) and \({l}_{0}\) consider the diameter and length of the tether, \(S\) is the wing area of the airborne vehicle and \({C}_{D,t }\) the drag coefficient of a cylinder exposed to a cross-flow. According to van der Vlugt, Bley, Noom, Schmehl [17] and Hoerner [18], \({C}_{D,t}\) can be considered to have a constant value of 1.1 for general airborne wind energy operations. For the simulated airborne wind energy system, the airborne vehicle has a drag coefficient of about 0.02 and measures an area of \(0.8 {\mathrm{m}}^{2}\). Based on Eq. (14), the drag coefficient of the entire system can be calculated to be \(0.244\). If a quasi-steady state is considered, this drag coefficient can also be derived as shown in Eq. (15). Here, the tether force component \({{F}_{AV,{x}_{b}}}\) is put in reference with the occurring aerodynamic velocity at the airborne vehicle, the air density, and the wing area of the airborne vehicle.

$$C_{D,AWES} = C_{D,AV} + \frac{{2 \left| {F_{{AV,x_{b} }} } \right|_{2} }}{{\rho_{a} \left| {V_{a} } \right|_{2}^{2} S}}.$$
(15)

Based on this formulation, the drag coefficient can be computed to \({C}_{D,\mathrm{AWES}, \mathrm{hybrid}}\) = 0.231. However, if one considers the quasi-steady state for the simulation with the spring model, the drag coefficient \({C}_{D,\mathrm{AWES},\mathrm{rod}}\) based on Eq. (15) is 0.007. Thus, the drag coefficient based on literature by Argatov, Rautakorpi, and Silvennoinen [15] correlates much better with the drag coefficient derived with Eq. (15) for the hybrid model. There, the relative deviation is about \(5 \%\). Thus, the hybrid tether model correlates better with existing tether model approaches from literature than the spring model.

6 Conclusion and future work

Within this paper, a spring tether model, as well as a catenary-based model, are presented to form the basis for the development of a hybrid tether model. This hybrid tether model considers an external load induced by wind and ground speed, a quasi-static acceleration as well as the earth’s gravitational field. Moreover, this hybrid model has a linear stiffness of the tether according to the one from a spring model.

Within the presentation of the hybrid tether model, the influence of tether stiffness on the tether forces is outlined. It can be concluded that for sagging tethers with low stiffness, the hybrid model represents the elongation of the tether more accurately than the basic catenary-based model. In addition, for different quasi-static configurations, the hybrid tether model has been compared to a lumped mass model. It has been shown that especially for asymmetric loadings along the tether, constant deviations between the two models exist. Moreover, it has been explained that these deviations are reasonable for the assumptions made. However, for the presented configuration the relative deviation in the relevant tether force at the airborne vehicle remains below \(10 \%\).

In addition, the hybrid tether model has been simulated within a model-in-the-loop environment for a tethered circular flight. The results have been compared with those for a tethered flight considering the spring model and thus neglecting external loads such as an aerodynamic tether drag. It has been outlined that in comparison to the simulation with the hybrid tether model, the occurring tether force for the spring model deviates not only in magnitude but also in the direction of the tether force vector acting at the airborne vehicle. Hence, it has been shown that the drag of the overall system, which is generated by the tether to a significant extent, can be taken into account much better by the hybrid model than by the spring model.

Overall, the developed hybrid tether model has been determined to model the physics of the system more precisely than the spring model. Since a tether force is always generated with the hybrid model, the airborne vehicle always experiences some sort of tether influence, for example, the tether's weight. Even if the tether is unstretched. Thus, the controlled system modeled with such a hybrid model comes closer to reality than with the spring model alone. In conclusion, the hybrid model can be considered as a suitable alternative, when the computational effort is limited and a quasi-static approach is tolerable. Besides online simulations for initial controller design, where such a static model may be sufficient in the beginning, this hybrid tether model could also be used in the framework of model-based control approaches. There, the computational effort for the models is limited but a proper mapping of the controlled system is important. However, further field tests with tethered flights have to be carried out, to validate this model experimentally.