While high-fidelity 6DOF aircraft models are available in the literature and open-source software, we limit ourselves here to the more simple point-mass model as proposed by . The state, control, and algebraic variables are defined as 4x:=(q,q˙,CL,ψ),u:=(C˙L,ψ˙),z:=λ, where q:=(q0,q1,q2)∈R9 are the positions of the centre point and the two wings, and CL∈R2 are the lift coefficients and ψ∈R2 the roll angles of the two wings. The algebraic variables λ∈R3 are the Lagrange multipliers associated with the holonomic constraints 5c(x,θ):=12q0⊤q0-lt2(q1-q0)⊤(q1-q0)-ls2(q2-q0)⊤(q2-q0)-ls2=0, which impose that the node positions are consistent with the tether lengths lt∈R and ls∈R of the main and secondary tethers, respectively. These parameters, together with the corresponding tether diameters dt∈R and ds∈R, can be optimized and are summarized in the parameter vector 6θ:=(lt,ls,dt,ds). Using the Lagrangian mechanics with index reduction approach proposed by and implemented by , we can derive the following equations of motion: 7M∇qc∇qc⊤0q¨λ=Fext with the mass matrix 8M:=Mt+2Ms12Ms12Ms12MsMs+mwI3012Ms0Ms+mwI3, where mw is the wing mass, and the terms Mt:=13μtltI3 and Ms:=13μslsI3 are used to describe the inertia of the tether. The density per unit length of the tether lengths is a function of the tether diameter, i.e. μt:=ρtπdt24 and μs:=ρtπds24, with ρt being the material density.

The external force vector is given as 9Fext:=F0-32g(Mt+2Ms)ezF1-g(mwI3+32Ms)ezF2-g(mwI3+32Ms)ez-∇q˙c˙⊤q˙, with F0, F1, and F2 being the aerodynamic forces. In the implementation, Baumgarte stabilization is added in the last element of this vector to ensure consistency of the solution in the context of periodic optimal control .

To compute the aerodynamic forces, we assume a uniform free-stream wind speed u∞∈R3 and constant density ρ∈R. Alternatively, realistic wind shear and density drop profiles can be incorporated as well in a straightforward fashion. Within the proposed framework of Sect. , the modelled wind shear would be taken into account automatically in the wake development. Within the scope of this paper, we limit ourselves to validating the model in uniform inflow conditions.

The wake state y(t,⋅) induces a velocity field uif(qi(t),y(t,⋅)) on top of the free-stream wind, resulting in a total wind field ut,i(qi(t),y(t,⋅)), which is evaluated at each of the wing positions qi(t) for i∈{1,2}: 10ut,i(qi(t),y(t,⋅)):=u∞+uif(qi(t),y(t,⋅)). For an explicit definition of the induced velocity functions uif, we refer to Sect. .

The apparent wind speed seen by each wing is then given by 11ua,i:=ut,i(qi(t),y(t,⋅))-q˙i(t),i∈{1,2}, where, for notational simplicity, we further omit the explicit dependence on time.

Then, assuming no side-slip and direct control of the roll angle ψi of the wings, the resulting lift and drag forces are given by FL,i:=12ρSwCL,i||ua,i||2(cos⁡(ψi)eL,i-sin⁡(ψi)eT,i)FD,i:=12ρSwCD,i||ua,i||ua,i, with the vectors eL,i and eT,i defined by 12eT,i:=ua,i×er,i||ua,i×er,i||,eL,i:=eT,i×ua,i||eT,i×ua,i||, where er,i is the tether direction 13er,i:=qi-q0||qi-q0||. The aerodynamic drag coefficient includes the induced-drag term of an elliptical wing in straight flight: 14CD,i:=CD,0+CL,i2πAR, with CD,0 being the parasitic drag and AR the aspect ratio. Note that the induced-drag term approximates the effect of the velocities induced by the part of the wake directly trailing the wing. This will be taken into account in the wake model in Sect.  so as to avoid double-counting, similar to .

The overall aerodynamic forces are given as 15F0=Ft,016Fi=FL,i+FD,i+Ft,i,i∈{1,2}, where Ft,i values are the tether drag forces, proportional to the tether lengths and diameters. For each tether, these forces are computed for the boundary nodes such that they produce resultant forces and moments equivalent to those obtained by numerically integrating the drag distribution along the tether. To simplify the exposition, we refer to Eq. 36 for the explicit expressions.

We propose to model each wake structure yi(t,⋅) as a continuous trail of infinitely small vortex rectangles made up of infinitely thin vortex filaments with variable circulation strength Γi(t,τ), as shown in Fig.  (left). The pair of chordwise vortex filaments with infinitesimal length ds model the rolled-up wing tip vortices. The wake is shed with the apparent wind speed; hence the infinitesimal length can also be written as a function of τ as ds:=||ua,i(t-τ)||dτ. We assume that these chordwise filaments are located at a distance of π4b from each other , symmetric relative to the rectangle centre position qv,i(t,τ). This distance equals the length of the pair of spanwise vortex filaments, which capture the shed vorticity caused by the variation in lift force over time. The orientation of the rectangles is determined by the normal unit vector en,i(t,τ) and chordwise unit vector ec,i(t,τ). The wake states are then defined as 20yi(t,τ):=qv,i(t,τ)Γi(t,τ)en,i(t,τ)ec,i(t,τ).

For the forcing term, we assume rigid downstream convection with a velocity uconv,i(y(t-τ,⋅))∈R3. This assumption implies that the shed wake elements are transported without deformation or dissipation: 21fi(y(t-τ,⋅)):=uconv,i(y(t-τ,⋅))07×1. In reality, the wake evolution results from complex vortex–vortex interactions and self-induced deformations, which are computationally prohibitive to resolve explicitly. To account for these effects in a tractable manner, Sect.  introduces and compares three heuristic strategies for selecting the convection velocity.

The shedding laws gi are defined such that the shed rectangle centre positions qv,i(t,0) coincide with the wings' aerodynamic centre, here assumed equal to the centre of mass positions qi(t). The trailing vortex strength Γi(t,0) follows from the Kutta–Joukowski theorem for elliptic wings, while the rectangle unit vectors en,i(t,0) and ec,i(t,0) are aligned with the lift force and apparent wind speed, respectively, which are orthogonal by definition: 22gix(t),y(t,⋅):=qi(t)2bπARCL,i(t)||ua,i(t)||FL,i(t)||FL,i(t)||ua,i(t)||ua,i(t)||.

Finally, the induced velocity at a position q^∈R3 by an infinitesimally small rectangle per unit width ds=||ua(t-τ)||dτ is given by 23u′(q^(t),yi(t,τ)):=duvl(q^,yi(t,τ),Δs)dΔsΔs=0||ua,i(t-τ)||, where the function uvl computes the velocity induced by a rectangular vortex loop of chordwise extent Δs, obtained from the Biot–Savart law applied to its four bounding vortex filaments, as written out in Appendix .

The symbolic complexity of Eq. () is relatively high, leading to expensive numerical derivatives and a high memory usage. This complexity can be reduced by applying a multi-pole expansion of the infinitesimal vortex loops and retaining only the leading non-vanishing term, which gives us a far-field approximation of the original induced velocity. This approximation can be interpreted as the induced velocity of a source dipole whose dipole moment is proportional to the enclosed area of the rectangular vortex loop.

Thus, we can also model the wake state as a continuous trail of vortex dipoles, visualized in Fig.  (right) as well, which can be conveniently written as a function of the vortex-loop state: 24yd(yi(t,τ)):=qv,i(t,τ)md,i′(t,τ), with the dipole moment per metre of trailing wake given by 25md,i′(t,τ):=Γi(t,τ)⋅πb4⋅en,i(t,τ).

The induced velocity function of the vortex dipole wake is given by the much simpler expression 26ud′(q^(t),yi(t,τ)):=14π⋅3r(r⋅md,i′)-md,i′||r||2||r||5⋅||ua,i(t-τ)||, with 27r:=q^(t)-qv,i(t,τ). Note that the vortex dipole induced velocity is written explicitly as a function of the vortex-loop state yi(t,τ) so that the same state representation can be used for this model as well.

With these two modelling options on the table, it is intuitive to conceptualize a hybrid wake model, consisting on the one hand of a costly vortex-loop model that captures the induction of the closest wake segments in detail and, on the other hand, the numerically cheaper vortex dipole moment to include the influence of the wake segments further downstream, as illustrated by Fig. .

We devise the hybrid model by splitting the integrals in Eq. () at the location τ=TA for the self-induced velocity of each wing and at location τ=TB for the velocity induced by the wake of the other wing. For wing i=1, this gives 28u1f(q1(t),y(t,⋅)):=∫TTAu′(q1(t),y1(t,τ))dτ+∫TA∞ud′(q1(t),y1(t,τ))dτ+∫0TBu′(q1(t),y2(t,τ))dτ+∫TB∞ud′(q1(t),y2(t,τ))dτ, where the induced velocity for wing i=2 is simply obtained by switching the wing indices for the variables qi and yi. For each wing, we ignore the self-induced velocity for the first time interval with length T.

In this formulation, we can retrieve the full vortex-loop model (TA=TB=∞), the full vortex dipole model (TA=T,TB=0), or any variant in between.

Two near-elliptic wings are generated using a parametric mesh with 40 discrete spanwise sections of the NACA 4421 airfoil. The discretization introduces deviations from an ideal elliptical planform, leading to reduced aerodynamic efficiency. To account for this, a span efficiency factor e is incorporated into the induced-drag model (Eq. ): 29CD,i:=CD,0+CL,i2πARe, and the corresponding trailing vortex strength in Eq. () is scaled accordingly: 30Γi(t,0):=2bπAReCL,i(t)∥ua,i(t)∥. The efficiency factor, e≈0.75, is identified from a series of straight-flight DUST simulations across varying angles of attack. Inflow speeds are chosen to match the apparent wind conditions of the validation flight trajectory, ensuring consistency in Reynolds number. The factor e is then obtained by fitting Eq. () to the simulated drag polar.

To simulate the wake in DUST, we must prescribe a kinematic flight trajectory for the wings. To obtain a representative trajectory, we feed into DUST the optimal periodic wing position trajectories and orientations obtained by solving the optimization problem defined by Eq. () from Sect. 5. The rationale behind the dual-kite set-up is to generate an extremely high-load scenario with a small convection distance between the wings and the trailing vorticity structure. This constitutes a worst-case test for the ability of the vortex-loop model to accurately represent the induced velocity close to the wake structure. Lower-load scenarios (e.g. with a medium-scale single-wing system) are expected to be less demanding and to be representable with at least comparable, if not improved, accuracy.

One remaining problem is that the angle of attack is not explicitly modelled in the optimal orientation trajectory, since the optimization model assumes direct lift coefficient control. Therefore, in a preprocessing step, we apply a pitch correction to impose the angle of attack that would produce the same lift coefficient under the same conditions (including the induction effects predicted by the wake model), using the simulated 3D lift polar.

The simulation is initialized in a vortex-free state, with vortex particles being shed during the simulation. We run the DUST simulation for 6.5 periods of the optimal state trajectory to allow for the wake build-up to reach a steady state. Figure  visualizes the simulation result, showing a distinct helicoidal wake structure in the first part of the wake, followed by an increasing break-up of this structure due to the vortex-to-vortex interactions. In postprocessing, we can retrieve the vortex particle trajectories, the induced velocity field at all time steps, and the aerodynamic loads obtained from the lifting-line approach. The toolchain to generate the necessary DUST input files as well as the input data and more detailed simulation settings (e.g. particle size) are made publicly available.

To compare the DUST wake simulation with the proposed wake model, we evaluate the wake using the analytic solution defined by Eq. () and compute the induced velocities via Eq. () using a high-accuracy numerical integration scheme, running the integral up to 6.5 periods in the past. For the forcing term in Eq. (), describing the wake convection, we consider three different convection velocity variants. For each variant, we obtain a different wake solution and a different time-dependent induced velocity field, which can be used to correct the aerodynamic loads obtained in the prescribed flight trajectory.

The first variant (“free”) assumes no vortex-to-vortex interaction at all and prescribes convection with the free-stream wind velocity: 31uconv,i(y(t-τ,⋅)):=u∞. The second variant (“far”) assumes convection with reduced velocity due to far-wake induction at the time of shedding; i.e. 32uconv,i(y(t-τ,⋅)):=u∞+uif(qv,i(t-τ,0),y(t-τ,0)). The third variant (“near”) assumes convection with reduced velocity due to “near-wake” self-induction, as proposed by ; i.e. 33uconv,i(y(t-τ,⋅)):=u∞-CL,i∥ua,i∥πAReex.

Figure  shows the position in the xz plane of the DUST vortex particles for the upper part of the wake. We compare this result with the predicted vortex-loop wake trajectory and orientation as they cross this plane, for each of the three variants. Firstly, it is visible that in the first, most influential part of the wake, the wake orientation and the distance between the tip vortices match well with the DUST simulation, although the bottom DUST tip vortex is convected slightly downwards. Further downstream, the models start to diverge due to wake deformation and expansion.

We can now assess the convection velocity fit by comparing the locations of the tip vortices predicted by both models. The “free” variant strongly overestimates the convection velocity, as the consecutive DUST tip vortices pop up much sooner and closer to the wing trajectory. For the “near” variant, the opposite is true: it provides a strong underestimation. The “far” variant provides the best fit, although some overestimation remains. In the remainder of this work, we apply the latter variant and delegate further improvements to future work.

For the purpose of optimal control, we are primarily interested in the induced velocity evaluated at the wing position and its effect on the aerodynamic loads. However, the DUST model does not differentiate between the induced velocity by the “far”-wake structure and the local circulation around the wing that produces lift, as we do in the optimization model. Thus, to eliminate the local circulation effect around the wing, we probe the wind field by means of an imaginary pitot tube of length lp=0.9b pointing forward in the chord direction. For the vortex-loop model, we evaluate the induced velocity at the same location: 34u^if(t):=uif(qi(t)+lpec,i(t),y(t,⋅)), with ec,i being the pitch-corrected chordwise body frame vector.

Figure  (left) shows the component-wise comparison of the measured induced velocity, normalized with the free-stream wind speed. For the first half rotation, the DUST induced velocity is almost zero, as the wake has not developed yet. After that, the magnitude of the induced velocity increases rotation after rotation as the wake builds up until it reaches a steady state. When comparing the induced velocities at the end of the DUST simulation with those from the periodic vortex-loop simulation, we see very good agreement between the y and z components. Also for the x component, the vortex-loop model captures the unsteady nature of the wake very well. However, due to the overestimation in convection velocity, the vortex-loop model underestimates the induced velocity by up to 25 % in the dominant x direction.

Figure  (right) shows the effect of this mismatch on the different components of the aerodynamic loads F^i:=FL,i+FD,i. Again, we have very good agreement in the y and z directions. In the x direction, the induced velocity mismatch leads to an overestimation of the angle of attack and consequently an overestimation of the loads up to 5 %.

In summary, the vortex-loop model provides a strong match with the DUST benchmark. The y and z components of the induced velocities are captured with high accuracy. In the dominant x direction there is a moderate mismatch, mainly resulting from the limitations of the convection model. Nevertheless, it is important to note that the model reproduces the unsteady and non-axisymmetric nature of the wake, which conventional approaches fail to represent, very well.

For optimization purposes, it is important to have a good model for the induced velocity evaluated not only at the wing position, but also in its surrounding neighbourhood. Therefore, we also compare the induced velocity field of the vortex-loop model with that of DUST in the region around the optimal trajectory. Figure  (top row) shows a snapshot of the wake at the same time point t′ as in Fig. . It shows the induction factor ax in the x direction in a plane perpendicular to the average main tether direction, located at the average wings' position. The induction factor is computed taking into account the contribution of the entire wake. Figures  (middle row) and (bottom row) plot the induction factor in the x and z directions. In accordance with the findings in the previous section, we can discern a slight overall underestimation of the induced velocities in front of the wake. Nevertheless, the qualitative fit is very good in all directions.

For all types of crosswind kite systems, the challenge is to compute periodic flight orbits that maximize a certain averaged performance index. In the AWE case, the performance index is typically the power output. For the atmospheric airborne actuator case, as well as for the ship-towing case, the tether-pulling force is to be maximized. For the airborne actuator case, this metric is used as a proxy for the amount of momentum that it can redirect into the wind farm: 35l(x(t),u(t),z(t),θ):=-Ft(t):=-λ0(t)lt while satisfying the following constraints: 36h(x(t),u(t),z(t),θ):=πdt2σmax-4Ft(t)(2.2b)2-(q2-q1)⊤(q2-q1)ν-νminνmax-ν≥0, which impose a maximum tether stress constraint, an anticollision constraint, and the bounds on the variables ν:=(x,u,z,θ,T).

Further, since the optimal state trajectory will be periodic, the wake state will be periodic in t as well. Therefore we can reformulate the wake transport equation as 37dydt(t,τ)+dydτ(t,τ)=f(y([t-τ],⋅)), with the modulo operator [t]:=tmodT so that the wake state only needs to be tracked for t∈0,T. Using this reformulation, we can formulate the following continuous and periodic PDE optimal control problem: minimizex(⋅),u(⋅),z(⋅)y(⋅,⋅),θ,T1T∫0Tl(x(t),u(t),z(t),θ)dt subject to 38F(x(t),x˙(t),u(t),z(t),y(t,⋅),θ)=0t∈[0,T]h(x(t),u(t),z(t),θ)≥0,t∈[0,T]dydt(t,τ)+dydτ(t,τ)-f(y([t-τ],⋅))=0t∈[0,T]τ∈[0,∞)y(t,0)-g(x(t),y(t,⋅))=0t∈[0,T]x(0)-x(T)=0. This problem is infinite-dimensional in both t and τ and numerically intractable. The first step is to transcribe the problem into a form that is only infinite-dimensional in t.

We propose discretizing the wake on a uniform grid of M points in the τ space: 390<τ1<…<τj<…<τM<T, and introduce the periodic wake state on this grid 40y(j)(t)≈y(t,t+τj), which approximately tracks the evolution of points j in the wake, originating at a time t=-τj=T-τj, which are convected away as time progresses. We can derive that the derivative of this wake state should approximate the forcing term in Eq. (): 41y˙(j)(t)≈f(y(-τj,⋅)), and we model its origination as an impulsive event along its periodic trajectory: 42y(j)(-τj+)≈g(x(-τj-),y(-τj-,⋅)). Note how in Eq. () the individual derivatives each depend on the entire wake history, which is counterproductive as it produces cross-couplings over different time instances. To arrive at a more classical state equation, we propose extending the wake states with these derivatives and collecting both in one overall wake state: 43Y(t):=y(1)(t)y˙(1)(t)⋮y(M)(t)y˙(M)(t). The state equation is trivially given by (assuming rigid convection) 44Y˙(t)=FY(Y(t)):=y˙(1)(t)0⋮y˙(M)(t)0, and the impulse effect can be written as 45Y(-τj+)=Gj(x(-τj-),Y(-τj-)), with the impulse function defined as 46Gj(x(t),Y(t)):=y(j)(t)-g̃(x(t),Y(t))y˙(j)(t)-f̃(Y(t)). Here, the functions f̃ and g̃ are defined by replacing the induced velocity expression in the functions f and g as follows: 47u1f(q1,y(t,⋅))≈ũ1f(q1,Y(t)), where the function ũ1f is obtained via a numerical integration scheme based on the discretized wake state Y(t) (see below).

To simplify notation, we further only consider the hybrid model where each wing experiences its own induction through the dipole model only (TA=T) and the induction caused by the other wing's wake only for one full rotation with the vortex-loop model (TB=2T). As discussed later in Sect. , this model choice is sufficiently accurate compared to the full vortex-loop model. Nevertheless, the following developments can be worked out for all different hybrid variants as well.

The integrals in Eq. () extend over an infinite horizon in the τ domain, whereas the wake state Y(t) retains only a single period T of past history. However, using the analytic wake solution formula (Eq. ) and exploiting the wake periodicity, we can look further in the past by introducing (e.g. for a wing “i”, as seen by wing “1”) a finite number Nd of “duplicate” wake states y1i,d(j)(t), based on the tracked wake states yi(j)(t), which are convected downwind for d periods of duration T: 48y1i,d(j)(t):=yi(j)(t)+(d+p1in(t,τj))Ty˙i(j)(t). Note that we also introduced the “near-wake parameter” p1in(t,τj), which ensures that we only consider the duplicates of those wake states that are in the near-wake region at the current time point t. Formally, this parameter is defined as (for i∈{1,2}): 49piin(t,τj):=1ift≥-τj,0ift<-τj, and 50p12n(t,τj):=0ift≥-τj,1ift<-τj, and p21n=p12n.

Based on the introduced duplicate wake states, the induced velocity for wing 1, using the hybrid model, can be computed numerically as follows using the midpoint integration rule: 51ũ1f(q1(t),Y(t)):=∑i=12∑d=1Nd∑j=1Mud′(q1(t)),y1i,d(j)(t)Δτ+∑j=1Mu′(q1(t)),y1i,0(j)(t))Δτ. The induced velocity ũ2f(q2(t),Y(t)) for wing 2 is simply obtained by a change of index.

At every time instance t, the induced velocities defined by Eq. () depend on the entire wake state Y(t). This results in a dense sensitivity structure for the dynamics, which ultimately leads to a high memory usage, and more time spent in the linear solver backend of the optimization solver. Experience showed that it is crucial to exploit the fact that the influence of each wake element at any given time depends on how close it is to the position of the wing at that time. Therefore, the contribution of more remote elements can be neglected at a low loss of accuracy. We implement this by assigning a moving window of influence to each wing.

To this end, we divide the time grid in Nw equidistant intervals 520=t0<…<tk<…<tN=T, and as the wing passes through each interval, we only take into account the wake segments that originate from that particular interval and the neighbouring Nf intervals, as illustrated by Fig. .

At a given time point t, we determine the current interval k as 53k:=NtT, and the window that is taken into account in interval k is defined by the boundaries 54T¯f,k:=tk-NfTN,55T¯f,k:=tk+1+NfTN. We then introduce the “far-wake parameters” 56piif(t,τj):=1if-τj∈T¯f,k,T¯f,k,0otherwise, and 57p12f(t,τj):=1if{-τj,2T-τj}∩T¯f,k,T¯f,k≠∅0otherwise, with p21f=p12f. This parameter is then used to switch the contribution of the individual wake elements on and off, e.g. for wing i=1, 58ũ1f(q1(t),Y(t)):=∑i=12∑d=1Nd∑j=1Mp1if(t,τj)ud′(q1(t)),y1i,d(j)(t)Δτ+∑j=1Mp1if(t,τj)u′(q1(t)),y1i,0(j)(t))Δτ. The parameter Nf can then be tuned in numerical experiments to achieve the best trade-off between accuracy and computation time.

By transcribing the wake model as detailed above, we can reformulate the PDE optimal control problem from Eq. () into the more standard continuous-time formulation minimizex(⋅),u(⋅),z(⋅)Y(⋅),θ,T1T∫0Tl(x(t),u(t),z(t),θ)dt subject to 59F̃(x(t),x˙(t),u(t),z(t),Y(t),θ)=0t∈[0,T]h(x(t),u(t),z(t),θ)≥0,t∈[0,T]Y˙(t)-FY(Y(t))=0,t∈[0,T]Y(τj+)-Gj(x((τj-),Y(τj-))=0,j∈I1Mx(0)-x(T)=0Y(0)-Y(T)=0, where the dynamics F̃ are obtained by inserting the induced velocity expression from Eq. () into the dynamics F.

Because the system under consideration is not reeling, a solution consisting of a single periodic flight loop is of practical interest. Moreover, specifically for this dual-kite case, we can consider the relevant solution where both wings fly identical trajectories, phase shifted by half a period. In this case, while one wing flies, e.g. the downstroke part, the other one simultaneously flies the upstroke. After half a period, the roles are reversed, and the first wing flies the identical upstroke and the other one the identical downstroke. To avoid simulating the same upstroke and downstroke twice, we can instead integrate the dynamics over only half an orbit, and instead of enforcing periodicity on the wing states, we can implement the “role reversal” by matching the end state of each wing to the initial state of the other, as proposed by : 60x1(T)-x2(0)x2(T)-x1(0), with xi:=(qi,q˙i,CL,i,ψi), for i∈{1,2}. As a consequence, the variables N, Nw, and M can be reduced by a factor of 2 while still simulating the full flight loop and wake with identical accuracy. Only the number of duplicates should be doubled to retain the same wake history. The time variable T automatically becomes the desired half-rotation near-wake cut-off time period.

We then transcribe this problem into a nonlinear program (NLP) using a direct collocation approach. The OCP is discretized using N=8 collocation intervals. Each interval consists of four RadauIIa collocation points, and the transcription includes a time transformation approach to deal with the variable time grid, as detailed by . Additionally, the computational performance benefits from additional lifting of the induced velocities as algebraic variables to reduce nonlinearity and symbolic complexity. We implemented this problem in the open-source framework AWEbox , which relies on the symbolic framework CasADi for automatic differentiation and the interface to the NLP solver IPOPT , using MA57 as the linear solver backend. The problem is solved on an Intel Core i7 2.5 Ghz, 16 GB RAM.

To illustrate the proposed approach, we solve the OCP from Eq. () for a dual-kite system using the numerical parameters provided by Table  and with the variable bounds νmin and νmax summarized in Table . As mentioned in Sect. , the dual-kite set-up is chosen to generate an interesting scenario for the validation. The wing size is chosen to be in the large-scale range envisaged by e.g. to decrease the sensitivity of the solution to tether drag in favour of the induction effects. Other properties and bounds represent typical values from the AWE literature, e.g. for tether properties and aerodynamic properties and mass scaling . The induction effects are roughly invariant with respect to the chosen wind speed. However, since the wind speed is chosen relatively high (u∞=12ms-1⋅ex), the effect of the wing mass on the dynamic trajectory is rather small in this example.

To generate a first reference solution, the wake discretization parameters are chosen so that a good fit with the post-ex analytic wake solution is achieved. Figure  shows the obtained total aerodynamic forces compared to the analytic vortex-loop and DUST simulations using the same boundary condition. The fit is very good, which validates not only the reference transcription, but also the hypothesis of the hybrid model: that only the initial part of the wake needs to be modelled with vortex loops but that the rest of it can be modelled accurately with vortex dipoles. Table  summarizes the chosen parameters. Note that these parameters are valid for the half-periodic formulation of the OCP.

Figure  shows the resulting optimal flight trajectory in comparison to the case where no induction (NI) model is included. Note that the corresponding optimal wake state, induced velocities, and aerodynamic forces have already been discussed in Sect.  and shown in Figs.  to . Table  summarizes the main results of the optimization. The chosen induction model parameterization increases the computation time per iteration tCPU by a factor of 16 compared to the NI case.

The optimizer increases the secondary tether lengths from 2.5 in the NI case to almost 10 wing spans. While this leads to higher tether drag losses, it allows the system to fly longer time periods T, which creates a larger convection distance between the system and its wake. Hence, the induction losses are reduced up to the point where the tether drag becomes dominant.

Also, because the available wind speed is reduced, the wings fly more slowly and experience a 30 % lower airspeed compared to the NI case, and they produce 55 % less average tether force. As a result, the tether diameters can be dimensioned smaller, which helps to dampen the effect of the increased tether drag. In conclusion, both the optimal design and the performance of the considered system are highly sensitive to the inclusion of induction effects.

We note that in this example, the resulting average airspeed of 190ms-1 in combination with the small turning radius leads to immense and practically infeasible centrifugal accelerations. A more realistic trajectory could be attained by lowering the design wind speed. Nevertheless, induction effects are roughly invariant with respect to the wind speed, and similar validation and design trade-off results are therefore expected for lower design wind speeds with lower flight speeds and therefore a more reasonable flight envelope.

Another improvement would be to add a mass-scaling law that takes into account the effect of the loading of the wing. In this very high loading case, this would lead to a significantly higher wing mass and therefore a different design trade-off possibly favouring even shorter tether lengths to minimize the upwards flight path length. However, in the dual-kite set-up considered here, where only one of the two wings flies upwards at a time, the effect of wing mass on the optimal trajectory is typically low , and we omit this extension for simplicity.

To assess the trade-off between computational effort and solution accuracy, we evaluate the impact of coarser wake parameterizations relative to the reference case. Each discretization parameter (M, Nd, and Nf) is independently reduced, while the others are held at their reference values. For each configuration, the optimal control problem is resolved, and the resulting computational cost and accuracy are compared against the reference solution. The error E is defined relative to the reference solution so as to capture both the accuracy of the cost and that of the optimal design: 61E:=F¯t∗-F¯t,ref∗F¯t,ref∗θ∗-θref∗θref∗. Figure  shows the trade-off between computational cost and accuracy across the tested wake parameterizations, including the no-induction (NI) case.

As the number of wake duplicates is reduced (Nd↓), the solution accuracy is significantly affected. In particular, whether to include the first duplicate (Nd=1) or not has a large influence. The computation time, on the other hand, is hardly affected, as the duplicates do not change the NLP dimensions or sparsity pattern. This suggests that the number of duplicates should be carefully selected but that they can be added at little additional computational cost.

Reducing the window of influence size (Nf↓) significantly decreases computation time, as the resulting NLP becomes increasingly sparse. However, when the window is chosen too small (in this case, Nf=0), solution accuracy deteriorates, and a spurious trajectory is obtained that performs even worse than the no-induced-wake (NI) case. These observations justify the introduction of a finite window of influence and highlight Nf as a key discretization parameter governing the balance between numerical efficiency and model fidelity.

Finally, reducing the number of wake elements (M↓) leads to a moderate decrease in both computation time and solution accuracy, indicating that relatively coarse discretizations can still yield meaningful results. However, for small M, oscillations appear in the time-resolved profile of the induced velocity, as illustrated by Fig. . This behaviour arises from the dependence of the induced velocity on the wing position: when the wing passes directly over a wake element, the induced velocity reaches a peak and subsequently drops to a minimum halfway to the next element. Consequently, the integration error in Eq. () is position dependent and sensitive to the choice of collocation grid in the time-continuous OCP (Eq. ). A potential remedy would be to implement a continuously moving window of influence that tracks the wing position, rather than the discrete interval-to-interval update scheme used here. While this would increase the computational complexity of the induced velocity evaluation, it could provide consistent integration accuracy, especially for small values of M. Such an extension is left for future work.

Figure  also shows the induced velocity from the analytic vortex-loop solution evaluated at the optimal solution for M=24. The induced velocity computed using the hybrid wake representation (comprising a single layer of vortex loops followed by dipole elements) shows good agreement with the one obtained from the full vortex-loop formulation, thereby justifying the use of dipoles for far-convected wake elements in this case. For applications involving larger convection distances, as are likely to occur for the case of single-kite AWE systems, the accuracy of the hybrid approximation is expected to improve further.

Based on the sensitivity analysis above, a new parameterization, denoted as “A”, is selected by choosing for each discretization parameter the lowest value that still yields a relative error E<5%. This configuration represents a Pareto-efficient trade-off between computational cost and model fidelity among the tested cases. At point A, it also holds that E<5%, while the average computation time per iteration is now only a factor of 3 more expensive than the NI case, which comes with a value of E=145%. In conclusion, when very high accuracy is not required – such as during design exploration or when other modelling errors are of comparable magnitude – there exist parameterizations that provide approximate solutions of the PDE OCP from Eq. () with only a moderate increase in computation time compared to the NI case.