NOMENCLATURE

A _W

aspect ratio

C

battery capacity

C_D0

zero-lift drag coefficient

C_L

lift coefficient

D

drag coefficient

e

Oswald efficiency number

(f _p ) _battery

battery weight ratio

k

induced drag constant

κ _battery

battery specific energy

L

lift coefficient

n

Peukert constant

n _critical

critical n value

P _battery

output power

P _cruise

power required for level flight

q

dynamic pressure

R _t

battery hour rating

t

endurance

t ₁/ t ₂

endurance for each phase

t _total

total endurance

V

flying velocity

W

weight of the UAV

(W _P ) _battery

energy contained in the battery

η _bp

combined efficiency of battery power system

Subscripts

battery

battery condition

bp

battery power system

critical

critical value

cruise

cruise condition

1,2,3…

number of current phase

2.0 BATTERY WEIGHT AND ENDURANCE GOVERNING EQUATIONS

For a typical small electric-powered disposable UAV, it is usually assumed that the total weight remains unchanged throughout its flight. As this kind of UAV will maintain level flight for most of its flying time,^{(Reference Raymer25)} it is reasonable to consider that the UAV maintains level flying at sea level during the whole flight to simplify the calculation. Considering the efficiency of the electric power system, the required power and output power satisfy

(1) \begin{equation}{P_{battery}} * {\eta _{bp}} = {P_{cruise}}\end{equation}

If the battery is ideal, meaning that no current drain occurs during the discharge, the endurance t can be expressed as^{(Reference Feng, Min and Shou14)}

(2) \begin{equation}t = \frac{{{{\left( {{W_P}} \right)}_{battery}}{\eta _{bp}}}}{{{P_{cruise}}}} = \frac{{{{({m_P})}_{battery}}{\kappa _{battery}}{\eta _{bp}}}}{{{P_{cruise}}}}\end{equation}

where (W _P ) _battery is the energy contained in the UAV’s battery, P _cruise is the power required for level flight, η _bp is the combined efficiency of electric power system, (m _P ) _battery is the mass of the battery and κ _battery is the specific energy of the battery considering the impact of deep discharge.

However, the lithium-ion batteries generally used on disposable UAVs are not ideal and exhibit current drain. The battery’s discharge rate impacts its available capacity. The higher the current drawn, the lower the available capacity that can be expected from the battery. This phenomenon is known as the Peukert effect.^{(Reference Traub22,Reference Traub23)} For example, a 4000mAh lithium battery can supply power for exactly an hour at a current of 4A. However, if the required current is 2A, the battery will be capable of supplying the output for more than 2h. Meanwhile, for an 8A current, the discharge time may be less than half an hour.

The discharge time of lithium-ion batteries can be expressed as C/I ⁿ , where C represents the battery capacity and the parameter n is the Peukert constant, which depends on the type and development level of the battery. Unfortunately, this equation only applies when the discharge current is 1 A, but when applied in a disposable UAV, the battery is often required to supply a large current to provide sufficient power. Under this condition, this equation loses accuracy and must be corrected. A correction method proposed in reference (22) is shown below:

(3) \begin{equation}t = \frac{{{R_t}}}{{{I^n}}}{\left( {\frac{C}{{{R_t}}}} \right)^n},\end{equation}

where R _t refers to the battery hour rating (in hours), typically being 1 h for small rechargeable batteries; The Peukert constant n of a lead-acid battery is 1.3, while it is 1.0 for an ideal battery. As battery technology has advanced, the value of n for lithium-ion batters has dropped to about 1.2^{(Reference Shen, Wang and Lian26)} or even lower (~1.0). However, the Peukert constant may change as described above. Thus, in this article, four values of n (1.0, 1.1, 1.2 and 1.3) are considered to study the influence of the Peukert effect on the battery dumping strategy. In addition, research^{(Reference Shen, Wang and Lian26)} has also been carried out using other methods to correct the equation under high-rate discharging, such as C = I ⁿ t − It. However, equation (3) is used to correct it herein. Therefore, the output power P _battery of the battery pack can be expressed as

(4) \begin{equation}{p_{battery}} = UI = U\frac{C}{{{R_t}}}{\left( {\frac{{{R_t}}}{t}} \right)^{\frac{1}{n}}}\end{equation}

For a disposable UAV maintaining level flight at sea level, power is consumed to maintain the thrust by overcoming the drag in the direction opposite to the velocity, which can be expressed as

(5) \begin{equation}{P_{cruise}} = D*V\end{equation}

The drag of a small electric-powered disposable UAV during level flight can be expressed as

(6) \begin{equation}D = qS\left( {{C_{D0}} + kC_L^2} \right) = \frac{1}{2}\rho {V^2}S\left( {{C_{D0}} + \frac{{C_L^2}}{{\pi {A_W}e}}} \right)\!,\end{equation}

where q is the dynamic pressure, C _D0 is the zero-lift drag coefficient of the UAV, k is the induced drag constant that can be expressed as k = 1/πeA _W , C _L is the lift coefficient, A _W is the aspect ratio and e is the Oswald efficiency number. Thus, the total drag on the disposable UAV can be expressed as the sum of the zero-lift drag and induced drag, while the required power can be expressed as

(7) \begin{equation}{P_{cruise}} = DV = \frac{1}{2}\rho {V^3}S{C_{D0}} + \frac{{2{W^2}k}}{{\rho {V^2}S}}\end{equation}

Then, applying equations (4) and (7) to equation (1), the endurance t can be derived as

(8) \begin{align} V\frac{C}{R_t}{\left( {\frac{{{R_t}}}{t}} \right)^{\frac{1}{n}}}{\eta _{bp}} & = \frac{1}{2}\rho {V^3}S{C_{D0}} + \frac{2{W^2}k}{\rho VS}\nonumber\\[5pt] t & = {R_t}^{1 - n}\left[ {\frac{{{\eta _{tot}}UC}}{1/2\rho {V^3}S{C_{D0}} + {{2{W^2}k}/{\rho VS}}}}\right]^{n} \end{align}

This equation is valid for any flight velocities inside the envelope. According to the aircraft performance estimation equations above, when the UAV reaches its maximum endurance, the following relationship should be satisfied:

(9) \begin{align}{C_{D0}} & = \frac{1}{3}kC_L^2\nonumber\\[9pt] V & = \sqrt {\frac{{2W}}{{\rho S}}\sqrt {\frac{k}{{3{C_{D0}}}}} } \end{align}

The maximum endurance can thus be expressed as

(10) \begin{align}{t_{\max }} = {R_t}^{1 - n}\left( \frac{{{\eta _{bp}}UC}}{\left(2/{\sqrt{\rho S}} \right)C_{D0}^{1/4}{{\left( {2W\sqrt {k/3} } \right)}^{3/2}}} \right)^n \end{align}

3.0 ANALYSIS OF BATTERY WEIGHT DISTRIBUTION STRATEGY

The battery weight ratio is one of the key factors affecting the endurance performance of an electric-powered disposable UAV. A higher battery weight ratio means there are more batteries inside the disposable UAV. This, however, does not always lead to longer endurance, because the greater weight of the batteries will increase the total weight, meaning that much more power will be required, which causes greater energy consumption. Generally, the endurance of a typical electric-powered disposable UAV is unimodal with increasing (f _p ) _battery . The maximum endurance occurs when (f _p ) _battery reaches 2/3.^{(Reference Chang and Yu17,Reference Traub21,Reference Traub22)} It is worth mentioning that this conclusion is applicable not only to disposable UAVs but also to other electric-powered UAVs, such as multi-rotor aircrafts.^{(Reference Jain and Mueller16)} However, when the total weight of the UAV is constant, the battery weight will be limited by the structure, airborne equipment and task load, which means it will be hard to reach a high level. Currently, the battery weight ratio (f _p ) _battery of an electric-powered disposable UAV may lie in the range of [0.2, 0.4], hence we take three values (0.2, 0.3 and 0.4) to study the effect of the battery dumping strategy on extending the endurance of a disposable UAV.

3.1. Double-pack battery strategy

In the previous section, an expression for the endurance t of a single-pack battery UAV was introduced. It is now assumed that the battery is divided into two packs having the same voltage. This means that the difference between the two battery packs lies in their capacity and mass, which are assumed to be linearly related. The two battery packs adopt the relay discharge method. Pack 1 is used first, and when its energy is exhausted, power will be supplied from pack 2. We assume that the weight ratio of pack 1 to the total mass of the battery is x and that the entire flight phase can be divided into two parts. After the first phase of the flight, pack 1 is dumped and the flying attitude is reset based on the updated flight weight to ensure the maximum endurance. Since the weight ratio of pack 1 to the total battery is x, the maximum endurance of the two stages t ₁ and t ₂ and the total endurance t _total can be expressed as

(11) \begin{align} {t_1} & = {R_t}^{1 - n}\left(\frac{{\eta _{bp}}UxC}{\left(2 /{\sqrt {\rho S} } \right)C_{D0}^{1/4} \left( {2W\sqrt {k/3} } \right)^{3/2}} \right)^n\nonumber\\[7pt] {t_2} & = {R_t}^{1 - n}\left( \frac{{\eta _{bp}}U\left( {1 - x} \right)C}{\left(2/{\sqrt {\rho S} } \right)C_{D0}^{1/4}\left(2\left( {1 - {{({f_p})}_{battery}}x} \right)W\sqrt {k/3}\right)^{3/2}} \right)^n\nonumber\\[7pt] {t_{total}} & = {t_1} + {t_2} = A\left({x^n} + \left(\frac{1 - x}{\left(1-({f_p})_{battery}x\right)^{3/2}}\right)^n\right)\end{align}

where here and below $A = {R_t}^{1 - n}\left[\frac{\eta _{bp}UC}{\left(2/{\sqrt{\rho S}}\right)C_{D0}^{1/4}\left(2W\sqrt{k/3}\right)^{3/2}}\right]^n$ .

According to equation (11), the factor A is a fixed value, so t _total is only affected by the variables in the second bracket, i.e. x, (f _p ) _battery and n. For the (f _p ) _battery values of 0.2, 0.3 and 0.4 and n values of 1.0, 1.1, 1.2 and 1.3, we calculate the endurance for x varying in the range of [0, 1], and solve for the maximum endurance. The maximum endurance is shown in Table 1, while the ratios of the two battery packs at the maximum endurance are shown in Table 2. The table only lists the parts where endurance benefits occur, expressed (here and below) as a multiple of the endurance with a single-pack battery. If the endurance provided by the double-pack battery is less than that of a single-pack battery at any pack ratio, the space is filled with “–”. The endurance for (f _p ) _battery of 0.2 and 0.4 is shown in Fig. 1.

Table 1 Maximum endurance for double-pack battery strategy

Table 2 The ratio of the two battery packs at the maximum endurance

Figure 1. Endurance for different values of the battery weight ratio (f _p ) _battery and Peukert constant n for the double-pack battery strategy. x is the weight ratio of pack 1, and the endurance is expressed as a multiple of that obtained for the conventional single-pack battery UAV.

From these tables and the figure, it can be concluded that all three main variables have a significant impact on the endurance extension. The main variations observed are as follows:

1. (1) When the Peukert constant is fixed, the endurance extension becomes more obvious with larger (f _p ) _battery , while if (f _p ) _battery is fixed, the endurance extension gets worse as n becomes larger. As the total capacity is fixed, the battery weight can also be considered to be fixed. Thus, larger (f _p ) _battery means a reduction in the total UAV weight, resulting in lower velocity, required power and also current drain. However, if (f _p ) _battery is fixed, a larger n leads to greater loss in the available capacity, that is, a greater loss in endurance;

2. (2) Define the maximum value of n at which the double-pack battery strategy yields an endurance benefit as the critical n value n _critical . As (f _p ) _battery declines, so does n _critical . Considering unit changes of 0.1 in the n value, the n _critical value at (f _p ) _battery of 0.3 is 1.2, while the n _critical value at (f _p ) _battery of 0.2 is only 1.1. This phenomenon can also be explained by the loss in available capacity mentioned in (1);

3. (3) When an endurance extension occurs, it generally shows a rising trend first but then declines with further increase of the pack 1 ratio x, or a slight drop appears at both ends. This probably occurs because, if the capacity of one pack is too small, it will suffer from serious current drain during its discharge, as the required current changes little. The x ratios that achieve the maximum endurance lie between 53% and 58%, very close to an equal split, and indeed some research has suggested that the equipartition law should be employed to split batteries in order to reduce the weight and design difficulties of BMDDs^{(Reference Chang and Yu17)};

4. (4) Ignoring the effects of temperature and fully considering the current state of battery technology development (n = 1.1, 1.2), it is estimated that the endurance extension obtained using the double-pack battery strategy lies between 2% and 14%. This result is not as attactive as the use of an ideal battery (9–20%) because of the Peukert effect.

According to the conclusions above, n has a significant effect on the endurance extension of the electric-power disposable UAVs. Using equation (11), it can be found that n mainly affects the endurance via the second multiplication term, which can be written using the function

(12) \begin{equation}f\left( x \right) = {x^n} + {\left( {\frac{{1 - x}}{{{{\left( {1 - {{({f_p})}_{battery}}x} \right)}^{3/2}}}}} \right)^n}\end{equation}

From a mathematical point of view, when n = 1.0, the first term in equation (12) corresponds to a straight line while the second term plots as a convex function curve. The sum of the two terms is always greater than 1 when x lies in the range of [0, 1], so whatever the value of (f _p ) _battery within the range considered here, a positive endurance extension is achieved. Meanwhile, when n is greater than 1.0, the first term becomes a concave curve, and its value gets smaller and smaller as n grows. Setting the battery weight ratio to 0.3, the second term in equation (12) is shown in Fig. 2 as a function of x at different n values. As the second term can be drawn as a convex curve and its value remains under 1.0, this term will show more linear growth with n, meaning that its value will become smaller for given x. Besides, the value of the second term decreases as (f _p ) _battery declines because of its position in the function. Therefore, when n is sufficiently large and (f _p ) _battery is sufficiently small, the value of this function will be below 1.0, indicating a decrease in endurance.

Figure 2. The value of the second term in equation (12) vs x for different values of n. As it can be drawn as a convex curve, this term becomes more linear as n increases, meaning that its value will be smaller for given x.

From a physical point of view, when n is greater than 1.0, the Peukert effect needs to be considered, and its influence increases with the growth of n. As the battery is divided into two packs, the capacity of each pack (in ampere-hours) will be lower than that of a single-pack battery. Thus, in the first phase, the capacity loss will be more serious, as the required current does not change with respect to the UAV with a single-pack battery. In the second phase, there will be a balance because the required current declines as the total weight decreases. When n is large enough, this loss will offset the benefits of the battery dumping strategy, and the endurance will decline.

In summary, the double-pack battery strategy will only result in a satisfactory endurance improvement when the battery weight ratio is large enough and the battery discharge performance is excellent (n is small).

3.2. Triple-pack battery strategy

Similarly, we now assume that the total battery is split into three packs, each having the same voltage and only differing in their capacities and masses. The packs discharge in order, with switching occurring when the previous one is exhausted. The variables a and b represent the proportions of the total weight represented by pack 1 and pack 2, respectively. Dividing the flight into three phases, the power supply will be switched to the next battery pack and the exhausted one will be dumped at the end of the first two phase. The flight velocity will then be re-matched for optimal endurance based on the new flight weight.

As the energy per unit weight of battery is assumed to be constant in this work, the energy proportions are the same as the weight proportions for all the battery packs. Therefore, the endurance for each phase, t ₁ , t ₂ , t ₃ and the summary t _total can be expressed as

(13) \begin{align} {t_1} & = {R_t}^{1 - n}\left(\frac{\eta_{bp}UaC}{\left(2/\sqrt{\rho S}\right)C_{D0}^{1/4}\left(2W\sqrt{k/3}\right)^{3/2}} \right)^n\nonumber\\[5pt] {t_2} & = {R_t}^{1 - n}\left(\frac{\eta_{bp}UbC}{\left(2/\sqrt{\rho S}\right)C_{D0}^{1/4}\left(2\left(1 - {({\kern1.3pt}f_p)_{battery}}a\right)W\sqrt{k/3}\right)^{3/2}}\right)^n\nonumber\\[5pt] {t_3} & = {R_t}^{1 - n}\left(\frac{\eta_{bp}U(1-a-b)C}{\left(2/\sqrt{\rho S}\right)C_{D0}^{1/4}\left(2\left(1 - {({\kern1.3pt}f_p)_{battery}}(a+b)\right)W\sqrt{k/3}\right)^{3/2}}\right)^n\nonumber\\[5pt] {t_{total}} & = {t_1} + {t_2} + {t_3} = A \left(a^n + \left(\frac{b}{\left(1 - ({f_p})_{battery}a \right)^{3/2}}\right)^n + \left(\frac{1-a-b}{\left(1 - ({f_p})_{battery}(a+b) \right)^{3/2}}\right)^n\right)\end{align}

According to these equations, the total endurance t _total varies with a, b, (f _p ) _battery and n. Setting (f _p ) _battery to 0.2, 0.3 and 0.4 and n to 1.0, 1.1, 1.2 and 1.3, a relationship between the endurance and a, b can be derived, and the results are listed in Table 3 and 4. These tables only list the benefits of endurance while “–” indicates no benefit in endurance or even a loss. From these data, it can be concluded that the triple-battery strategy is more advantageous when n equals 1.0, while both strategies offer similar benefits in terms of endurance enhancement when n is above 1.0. Besides, the endurance of the triple-battery strategy decreases more rapidly with increasing n.

Table 3 Maximum endurance with triple-pack battery strategy

Table 4 Ratios of three battery packs at maximum endurance

Figure 3 shows the variation of the endurance with n when the battery weight ratio is set at 0.3. When n = 1.0, indicating an ideal battery model, the proportion of each pack for the optimal endurance is 38%, 33% and 39% respectively, and the benefit reaches 1.19. When n = 1.2, the proportions at which the maximum endurance is achieved when using the triple-pack strategy are similar to those obtained when using the double-pack strategy, as the proportion of one of the packs is zero. In this case, the endurance benefit decreases to 1.02. If the value of n rises further, the benefit of the triple-pack battery becomes lower than that obtained using the double-pack battery. We thus define n _critical (the critical value of n) as the value at which the triple-pack strategy degenerates to the double-pack strategy. If n exceeds n _critical , few arrangements of the triple-pack battery strategy will result in an endurance benefit. Moreover, the endurance performance of the triple-pack strategy will be even worse than that with a single-pack battery, in accordance with the results obtained for the double-pack strategy.

Figure 3. The relationship between the endurance and the pack proportions for each n value when using the triple-pack battery strategy ((f _p ) _battery = 0.3).

Figure 4 shows the variation of the endurance with the proportions of pack 1 and pack 2 for two values of n and three values of (f _p ) _battery . As seen in each row of the figure, the battery dumping strategy offers remarkable endurance benefits when using high battery weight ratios, as analysed in subsection 3.1. In particular, when n = 1.2, this strategy causes a loss of endurance when (f _p ) _battery is low (below 0.3). This means a higher battery weight ratio will help to increase the endurance benefit of the battery dumping strategy.

Figure 4. The relationship between the endurance in the triple-pack battery strategy and the pack ratios (n = 1.0, 1.2).

In terms of n _critical , decreasing the battery weight ratio also leads to a reduction of n _critical . When (f _p ) _battery is 0.2, benefits can only be obtained when n is less than 1.1, which demands high-performance batteries. The reason for this is probably that the total weight of the UAV becomes too large and great power will be needed to keep it flying, leading to serious current drain. Therefore, similar to the case of the double-pack strategy, the battery dumping strategy will offer remarkable endurance extension benefits only when the battery weight ratio is large enough and n is very small. Besides, the triple-pack strategy appears to be more sensitive to n compared with the double-pack strategy. One possible reason for this is that the battery has been split into more packs, thus the battery rated capacity is much smaller, resulting in a more severe loss of the available capacity. Considering the variation in the ambient temperature, manufacturer, and age of different batteries described above, n may change by more than 0.1, indicating that the benefits obtained using the triple-pack strategy are less certain than those resulting from the double-pack strategy.

4.0 CASE STUDY OF WEIGHT DECOMPOSITION FOR A DISPOSABLE UAV

Equation (10) describes the relationship between the endurance and flight velocity V when using a single-pack battery. After introducing the battery dumping strategy, equation (10) must be adapted to describe the sum of the various flight phases. The endurance in each phase should be calculated using its own capacity, flight velocity and weight. If it is assumed that the same lift coefficient can be used for the different flight phases, the endurance with the double-pack battery can be expressed as

(14) \begin{align} {t_1} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}UxC}{\dfrac{1}{2}\rho{V^3}S{C_{D0}} + {2{W^2}k}/{\rho VS}}\right)^n\nonumber\\[5pt] {t_2} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}U\left({1 - x}\right)C}{\dfrac{1}{2}\rho{V^3_2}S{C_{D0}} + {2{W^2_2}k}/{\rho V_{2}S}}\right)^n\nonumber\\[5pt] {t_{total}} & = {t_1} + {t_2} \end{align}

where ${W_2} = W\left(1 - x\left({\kern1.3pt}f_p\right)_{battery}\right)\!,{V_2} = V\sqrt{1 - x\left({\kern1.3pt}f_p\right)_{battery}} $ .

Similarly, the endurance when using the triple-pack battery can be expressed as

(15) \begin{align} {t_1} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}UaC}{\dfrac{1}{2}\rho{V^3}S{C_{D0}} + {2{W^2}k}/{\rho VS}}\right)^n\nonumber\\[5pt] {t_2} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}UbC}{\dfrac{1}{2}\rho{V^3_{22}}S{C_{D0}} + {2{W^2_{22}}k}/{\rho V_{22}S}}\right)^n\nonumber\\ {t_3} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}U(1-a-b)C}{\dfrac{1}{2}\rho{V^3_{3}}S{C_{D0}} + {2{W^2_{3}}k}/{\rho V_{3}S}}\right)^n\nonumber\\[5pt] {t_{total}} & = {t_1} + {t_2} + {t_3} \end{align}

where

\begin{align*} W_{22} & = W\left(1-a\left({\kern1.3pt}f_{p}\right)_{battery}\right)\!, V_{22} = V\sqrt{1-a\left({\kern1.3pt}f_{p}\right)_{battery}};\\[5pt] W_{3} & = W\left(1-\left(a+b\right)\left({\kern1.3pt}f_{p}\right)_{battery}\right)\!,V_{3} = V\sqrt{1-\left(a+b\right)\left({\kern1.3pt}f_{p} \right)_{battery} }\end{align*}

Setting (f _p ) _battery to 0.3, a case analysis of the battery dumping strategy for a disposable UAV can now be carried out. The proportions of the packs for which the UAV can reach its maximum endurance are applied in equations (14) and (15), respectively. Taking the velocity V as an independent variable, the effects of the velocity (and thus the required power) and the Peukert effect on the endurance were studied for different battery capacities (1, 2, 3, and 4Ah) and n values (1.0 and 1.2). The parameters of the disposable UAV are taken from reference (18). The take-off gross weight of the disposable UAV is W = 9.34N, its wing area is S = 0.32m², the ambient air density is ρ = 1.2kg/m³, the zero-lift drag coefficient is C_D0 = 0.015, the induced-drag constant is k = 0.13, the electric power system efficiency is η _bp = 0.5, the operating voltage is U = 11.1V and the battery hour rating is R _t = 1h. The resulting endurance is shown as Fig. 5:

Figure 5. Available endurance at different velocities (left: two packs; right: three packs).

The basic variation of the characteristics is similar to that mentioned for a single-pack battery in reference (Reference Peukert18), as follows:

1. (1) With increasing battery capacity, the endurance of the disposable UAVs increases. This is understandable because the battery weight is fixed, so more capacity means more supplementary energy. When the Peukert effect is considered at a capacity of 4Ah, the use of the double-pack battery strategy can reach a maximum endurance of 3.4h.

2. (2) With increasing velocity, the endurance first rises then decreases. At low velocity, the disposable UAV must fly at a high angle-of-attack, at which the lift-to-drag ratio is relatively low. Meanwhile, the thrust is also used directly to balance part of the weight. Therefore, high power is required to supply the thrust. As the velocity gradually increases, a situation with high lift-to-drag ratio is reached. The required power decreases, while the endurance increases. However, as the velocity increases further, the drag increases in a super-linear fashion, thus the required power increases again.

3. (3) The Peukert effect obviously affects the endurance of the disposable UAV. The required power is fixed, as is the battery weight, thus the current required is consistent for different capacities. Larger capacity means smaller current drain. For the double-pack battery, when the battery capacity is 1 or 2Ah, as the capacity is significantly lower than the current, the Peukert effect appears to reduce the endurance. However, when the capacity is increased to 3 or 4Ah, the capacity can exceed the current value under the condition of low required power (e.g. V = 10m/s). Therefore, considering the influence of the Peukert effect, the endurance will be longer than that with an ideal battery. The benefit of the Peukert effect is about 10% at 4Ah, where the endurance will be more than twice that at 2Ah. However, with rising or declining velocity, the current will gradually increase and the Peukert effect will once again weaken the performance characteristics.

4. (4) The Peukert effect on the triple-pack battery is basically similar to that on the double-pack battery. However, as the number of battery packs is greater, the ampere-hour rating of each pack will be lower than for the double-pack battery. Therefore, it is harder to gain benefits via the Peukert effect. The benefit can only be achieved at C = 4Ah, and it is quite small.

To observe the endurance extension in comparison with the single-pack battery at different velocities, we compare the endurance of the single-, double- and triple-pack batteries at the same operating condition in Fig. 6. The battery capacity is set to 4Ah, and the Peukert constant to 1.0 or 1.2. The results reveal that the battery dumping strategy has a positive effect on the endurance, while the benefit when using ideal batteries is also significantly higher than that for lithium-ion batteries with n = 1.2 (the maximum difference between the three red lines is greater than that of the blue lines). When n = 1.0 (red line), the endurance when using the battery dumping strategy shows obvious differences at each velocity, in the following order: triple-pack battery > double-pack battery > single-pack battery; while when n = 1.2 (blue line), the two battery splitting strategies show almost no difference and the benefits (10%) are less obvious than for the single battery (29%).^{(Reference Traub22)}

Figure 6. Comparison of endurance achievable when using the double- versus single-pack battery strategy at different velocities (C = 4Ah).

Figure 7. Comparison between the endurance in flight with constant velocity versus constant C_L with the double-pack battery strategy (C = 4Ah, (fp) _battery = 0.3).

In the derivation of equation (14), we assume the same lift coefficient for the several flight phases. This means that, if the flight weight changes by B times, the flight velocity will change correspondingly by $\sqrt B $ times. However, there is also another possible option, which is to maintain the flight velocity constant throughout all phases. The flight attitude of the aircraft, such as the angle-of-attack, must then be adjusted to maintain level flight. Taking the double-pack battery as an example, the expression for the endurance then becomes

(16) \begin{align} {t_1} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}UxC}{\dfrac{1}{2}\rho{V^3}S{C_{D0}} + {2{W^2}k}/{\rho VS}}\right)^n\nonumber\\[5pt] {t_2} & = {R_t}^{1 - n}\left(\frac{\eta_{tot}U(1-x)C}{\dfrac{1}{2}\rho{V^3}S{C_{D0}} + {2{W^2_{2}}k}/({\rho VS})}\right)^n\nonumber\\[5pt] {t_{total}} & = {t_1} + {t_2} \end{align}

where ${W_2} = W(1 - x{({\kern1.3pt}f_p)}_{battery})$ .

Assuming a capacity of 4Ah and a battery weight ratio of 0.3, the endurance when flying with constant velocity and constant lift coefficient are compared in Fig. 7. These results reveal that the disposable UAV can achieve greater endurance under the influence of the Peukert effect in the condition of constant lift coefficient (3.4h), being 10–13% greater than the other situation. When the battery discharge performance is desirable (n is lower), the difference between the two options is tiny. However, under the condition of fixed velocity, the UAV will reach its maximum endurance at lower velocity.

After the battery is dumped, a new balance with gravity can be achieved by changing the velocity or angle-of-attack (lift coefficient). Reducing either the velocity or the angle-of-attack will reduce the drag, as shown in equation (6). However, a reduction in velocity can lead to an extra decrease in the required power and current. The Peukert effect in this situation acts to improve the endurance, as shown in Fig. 5, which means that changing the velocity will lead to an extra improvement. Thus, longer endurance can be obtained by changing the velocity with a constant lift coefficient before or after the battery dumping than by keeping the same velocity. Nevertheless, for an ideal battery with n = 1.0, there is no obvious difference between these two strategies, as the Peukert effect has no influence in this situation.