Efficiency-Aware: Maximizing Energy Utilization for Sensor Nodes Using Photovoltaic-Supercapacitor Energy Systems

Abstract

Recently, photovoltaic-supercapacitor-based energy systems have become more and more popular in the design of energy harvesting wireless sensor networks (EH-WSNs) as an alternative to battery power. Existing research on this area mainly focuses on hardware design and the improvement of the charging efficiency. However, energy is wasted not only by the inefficient charging process, but also the inefficient discharging process and energy leakage. Therefore, to maximize node lifetime and energy utilization, all the previous energy loss should be considered. In this paper, we develop realistic hardware models of the complete photovoltaic-supercapacitor energy systems and propose the efficiency-aware, a systematic duty cycling framework to maximize energy utilization. We formalize the maximization problem as a nonlinear optimization problem and develop two efficient algorithms for its optimal solutions. The performance of our approaches is evaluated via extensive numeric simulations, and the results show that our efficiency-aware framework can, respectively, achieve 60% and 56% more active time (i.e. energy utilization) than the fixed duty cycle scheme and leakage-aware, a state-of-the-art scheme for photovoltaic-supercapacitor energy systems.

1. Introduction

Harvesting energy from the environment provides a promising solution to address the issues arising from energy scarcity in wireless sensor networks (WSNs) [1–4]. More and more researchers choose photovoltaic-supercapacitor hardware systems to design the sensor nodes that utilize energy-harvesting WSNs (EN-WSNs). The main reason is that solar energy is readily available and supercapacitors have larger charging-discharging efficiencies and much longer recharging cycles (thousands times or more) than batteries. More importantly, the residual energy of supercapacitors is easy to meter online with high accuracy, which can support power management algorithms and emergent energy-aware network protocols [5–9].

The key issue of sensor nodes with photovoltaic-supercapacitor energy systems is how to maximize the utility of the harvested energy. Due to the time-varying nature of solar energy and the highly nonlinear output versus voltage characteristics of supercapacitor, it is challenging to model the the whole energy system as well as to maximize its efficiency. There are several efforts [3, 10–14] aiming to improve the efficiency of energy systems. A quick-charge circuit to switch between parallel and series connection of photovoltaic (PV) cells is used in [3] to shorten the supercapacitor charging time. The well-known MPPT techniques [2, 3, 10–12] have obtained the maximum output power by adjusting the operating point of the solar panels dynamically. Reference [13] points out that the charging efficiency of MPPTs has a strong relationship with the mismatch between the maximum power point voltage of the solar panels ( $V_{\text{mpp}}$ ) and the supercapacitor voltage ( $V_{\text{cap}}$ ) and achieves high charging efficiency by selecting supercapacitors with appropriate capacities. However, they assume that the supercapacitor does not discharge in day time and their optimal solution is for a fixed solar irradiance model only. Leakage-aware [4] is the first indepth work that focuses on reducing the leakage of supercapacitors by dynamically adjusting the duty cycle. However, it does not consider the charging and discharging efficiency (charging efficiency, discharging efficiency, and charging-discharging efficiency are the energy conversion efficiencies of input regulator, output regulator, and energy storages (batteries or supercapacitors)) caused by the state of the supercapacitor, and it is quite difficult to find a suitable adjustment step length when the daily solar irradiation changes frequently. Besides, it needs to wake up the node frequently and can only assign duty cycles for a small duration in each calculation (tens of seconds, compared with tens of minutes of our efficiency-aware framework).

The energy transfer model of a representative photovoltaic-supercapacitor energy system can be simply illustrated by Figure 1. The power (the power harvested by a solar panel is highly depending on solar irradiance and the states of the solar panel. Here, it represents the harvesting power on the maximum power point of a solar panel.) ( $P_I$ ) harvested by the solar panel (PV Array) is transferred into the supercapacitor through the input regulator with the efficiency ${\eta }_{\text{in}}$ , and energy in the supercapacitor can be used by the node (e.g., Micaz) through the output regulator with the efficiency ${\eta }_{\text{out}}$ as well as being leaked away with the power $P_{\text{leakage}}$ . According to [4, 11–13], ${\eta }_{\text{in}}$ , ${\eta }_{\text{out}}$ , and $P_{\text{leakage}}$ highly depend on the voltage of the supercapacitor ( $V_{\text{cap}}$ ). Through a number of experiments, we also find that these variables can be expressed by functions of $V_{\text{cap}}$ through theoretical analysis or model fitting. As $P_I$ can be predicted by prediction algorithms such as EWMA [15] and WCMA-PDR [16], $V_{\text{cap}}$ can be controlled to the states beneficial to energy harvesting by adjusting the power ( $P_{\text{load}}$ ) consumed by the node, using duty cycle scheduling schemes.

Figure 1

Energy translation of a photovoltaic-supercapacitor energy system.

In this context, this paper proposes efficiency-aware: a realistic and complete way to maximize the energy utilization efficiency of photovoltaic-supercapacitor energy systems. Efficiency-aware is a three-layer design as shown in Figure 2. All the models in hardware layer are based on actual measurements or empirical formulas, and the duty cycle controller uses energy control algorithms to suggest optimal duty cycle to the adaptation layer. The major contributions of our work are as follows.

• (i)

Modeling all the power models of a commonly used photovoltaic-supercapacitor energy systems, describing the results, and using them to develop energy-harvesting-aware algorithms and systems.

(ii)

Proposing a framework that can be used in photovoltaic-supercapacitor energy systems with predictable solar energy to maximize energy utilization in such systems.

(iii)

Formulating the utility maximization problem as an optimization problem that produces duty cycle schedules and proposing algorithms to solve this optimization. We also make an extensive comparative performance analysis of leakage-aware, fixed duty cycle scheme, and our method.

Figure 2

Overview of efficiency-aware.

The rest of this paper is organized as follows. An overview of the efficiency-aware design architecture is presented in Section 2. In Section 3, we present power models of individual hardware components of the photovoltaic-supercapacitor energy systems. The formalization of the optimization problem and algorithms to solve the problem are presented in Section 4. Numeric results are presented in Section 5. Finally, we conclude the paper in Section 6.

2. Overview of Efficiency-Aware Framework

The efficiency-aware framework for the photovoltaic-supercapacitor energy systems is a three-layer architecture as shown in Figure 2.

The hardware layer provides offline hardware models and predicted solar energy to the control layer. The offline hardware models include a solar panel model, efficiency (i.e., charging and discharging) models, a residual energy model, and a leakage model.

Continuous time is divided into discrete slots in our design (as shown in Figure 3). At the beginning of every prediction interval, the control layer computes the optimal duty cycles for those slots based on the information provided by the hardware layer, predicted solar power, and QoS requirements.

Figure 3

Harvested energy prediction interval.

The adaptation layer changes its schedules (e.g., sensing and communication) according to the duty cycle determined by the control layer.

3. Power Models

In general, a photovoltaic-supercapacitor energy system includes a solar panel, an input regulator (except direct connection), a supercapacitor, and an output regulator. In order to provide a systematic understanding of the photovoltaic-supercapacitor energy system, it is necessary to know the characteristics of each individual component. To this end, we present the solar model, charging efficiency model for the input regulator, discharging efficiency model for the output regulator, the residual energy model, and the leakage model for the supercapacitor. In addition, a simple duty cycle model is given in this section.

3.1. Solar Panel Model

Previous work [12] presents an accurate simulation model of solar panels. By neglecting the shunt resistance and considering that the series resistor is large enough, the I-V characteristic of solar panels can be given by the following equation [12, 17]:

$$\begin{array}{cc}& I_{\text{panel}}=I_g-I_o\left(e^{q·V_{\text{panel}}/A·K·T}-\mathrm{1}\right),\end{array}$$

(1)

where $I_g$ is the generated current, q is the electron charge, $I_o$ is diode saturation current, A is an ideality factor, K is the Boltzmann's constant, and T is the solar panel temperature in degree Kelvin.

$I_o$ and $I_g$ depend on solar irradiance and temperature as [12, 17]

$$\begin{array}{cc}& I_o=I_{o,\text{STC}}{\left(\frac{T}{T_{\text{ref}}}\right)}^3·e^{\left(q·E_{\mathrm{gap}}/B·K·\left(\left(\mathrm{1}/T_{\text{ref}}\right)-\left(\mathrm{1}/T\right)\right)\right)},\end{array}$$

(2)

$$\begin{array}{cc}& I_g=\left\{I_{g,\text{STC}}+K_I·I_{g,\text{STC}}·\left(T-\mathrm{298.15}\right)\right\}·\frac{P_I}{\eta ·S·G_{\text{STC}}},\end{array}$$

(3)

where $I_{o,\text{STC}}$ and $I_{g,\text{STC}}$ are the diode saturation current and the generated current in Standard Test Condition (STC). $T_{\text{ref}}$ is the reference temperature. $E_{\mathrm{gap}}$ is the energy gap. B is the ideality constant. $K_I$ is the temperature coefficient. $G_{\text{STC}}$ is the solar irradiance in STC, typically 1000  $\text{W}/{\text{m}}^{\mathrm{2}}$ . η is the energy conversion efficiency of the solar panel, and S is the area of the solar panels. $I_{g,\text{STC}}$ can be considered equal to the short-circuit current in STC, and $I_{o,\text{STC}}$ can be presented as

$$\begin{array}{cc}& I_{o,\text{STC}}=I_{g,\text{STC}}·e^{-q·V_{\text{oc},\text{STC}}/A·K·T}.\end{array}$$

(4)

$V_{\text{oc,STC}}$ is the open-circuit voltage in STC. Given the solar irradiance $P_I$ and the solar panel temperature T, $I_g$ and $I_{g,\text{STC}}$ are determined by (3) and (4). Then $I_o$ can also be determined by (2). Finally the relationship between $V_{\text{panel}}$ and $I_{\text{panel}}$ can be explicitly determined according to (1).

3.2. Charging Efficiency Model

The charging efficiency ${\eta }_{\text{in}}$ is determined by $V_{\text{panel}}$ , $P_I$ , the solar panel temperature T, and $V_{\text{cap}}$ :

$$\begin{array}{cc}& {\eta }_{\text{in}}=f_{\text{in}}\left(V_{\text{panel}},P_I,V_{\text{cap}},T\right)\end{array}$$

(5)

Since the charging-discharging efficiency of supercapacitors is close to 100% in practice, we assume that it equals 100% in our model. In direct connection, $V_{\text{panel}}$ equals $V_{\text{cap}}$ and the charging current equals $I_{\text{panel}}$ . So ${\eta }_{\text{in}}$ is

$$\begin{array}{cc}& {\eta }_{\text{in}}=\frac{V_{\text{cap}}·I_{\text{panel}}}{P_I},\end{array}$$

(6)

where $I_{\text{panel}}$ can be determined by combining (1), (2), (3), and (4) as $P_I$ , T, and $V_{\text{panel}}$ have been determined.

3.3. Residual Energy Model

The residual energy of supercapacitors can be given by

$$\begin{array}{cc}& E_{\text{cap}}=\frac{\mathrm{1}}{\mathrm{2}}C{V}_{\text{cap}}^{\mathrm{2}}.\end{array}$$

(7)

3.4. Discharging Efficiency Model

As the voltage range of supercapacitors is wide and the operating voltage of sensor nodes may be very different from $V_{\text{cap}}$ (e.g., 0–2.7 V for supercapacitors, compared with 2.7–3.3 V for MicaZ), it is essential to adopt an output regulator between them. The DC-DC converter discharging efficiency ( ${\eta }_{\text{out}}$ ) is determined by input voltage, output voltage ( $V_{\text{out}}$ ), and output current ( $I_{\text{out}}$ ):

$$\begin{array}{cc}& {\eta }_{\text{out}}=f_{\text{out}}\left(V_{\text{cap}},V_{\text{out}},I_{\text{out}}\right).\end{array}$$

(8)

We assume that the node operates in two states only, active (MCU active and radio on for MicaZ) and sleep states. In sleep state, the current is tens of $\mu \text{A}$ and can be ignored. In the active state, the current as measured is about 22.3  $\text{mA}$ , and $V_{\text{out}}$ is a fixed voltage (e.g., 3 V). So ${\eta }_{\text{out}}$ is

$$\begin{array}{cc}& {\eta }_{\text{out}}=f_{\text{out}}\left(V_{\text{cap}}\right).\end{array}$$

(9)

3.5. Duty Cycle Model

Duty cycle (D) can be represented as

$$\begin{array}{cc}& D=\frac{t_{\text{active}}}{t_{\text{active}}+t_{\text{sleep}}},\end{array}$$

(10)

where $t_{\text{active}}$ is the node active duration and $t_{\text{sleep}}$ is the node sleep duration. Therefore, the average power consumed by the node ( $P_{\text{avg}}$ ) is

$$\begin{array}{cc}& P_{\text{avg}}=D·P_{\text{active}}+\left(\mathrm{1}-D\right)·P_{\text{sleep}},\end{array}$$

(11)

where $P_{\text{active}}$ and $P_{\text{sleep}}$ are the power consumptions of the active and sleep states, respectively. Since $P_{\text{sleep}}$ is small (about tens of $\mu \text{W}$ compared with tens of $\text{mW}$ of $P_{\text{active}}$ ), $P_{\text{avg}}$ can be approximately represented as

$$\begin{array}{cc}& P_{\text{avg}}=D·P_{\text{active}}.\end{array}$$

(12)

3.6. Leakage Model

The leakage power of a supercapacitor, $P_{\text{leakage}}$ , is mainly determined by $V_{\text{cap}}$ , and it can be approximately represented as

$$\begin{array}{cc}& P_{\text{leakage}}=f_l\left(V_{\text{cap}}\right).\end{array}$$

(13)

4. Duty Cycling Scheme Design

The duty cycle controller uses energy information (the models mentioned previously) and predicted solar energy to allocate duty cycles for several future slots, resulting in a maximum sum of duty cycles of those time slots.

4.1. Problem Formulation

As shown in Figure 3, $L·\mathrm{\Delta }T$ is the predicted interval consisting of L successive time slots with the duration $\mathrm{\Delta }T$ . We can choose a small $\mathrm{\Delta }T$ (e.g., 30 minutes), during which the solar irradiance changes slightly. Hence, we can assume that the solar power keeps constant during a slot and use $P_I^i$ to represent the solar power in the ith slot. Let $D^i$ be the duty cycle in slot i. Then the residual energy of the supercapacitor in slot $i+\mathrm{1}$ can be given by

$$\begin{array}{l}{E}_{\text{cap}}^{i+\mathrm{1}}=E_{\text{cap}}^i+{\int }_{i·\mathrm{\Delta }T}^{\left(i+\mathrm{1}\right)·\mathrm{\Delta }T}‍(\phantom{\frac{P_{\text{avg}}^i}{{\eta }_{\text{out}}}\left(t\right)}{P}_I\left(t\right)·{\eta }_{\text{in}}\left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{P_{\text{avg}}^i}{{\eta }_{\text{out}}\left(t\right)}-P_{\text{leakage}}\left(t\right))dt\end{array}$$

(14)

Since it is difficult to know ${\eta }_{\text{in}}(t)$ , ${\eta }_{\text{out}}(t)$ and $P_{\text{leakage}}(t)$ due to the changing $V_{\text{cap}}$ , we divide $\mathrm{\Delta }T$ into K smaller time slots (s-slot) with a duration $\mathrm{\Delta }t$ . $\mathrm{\Delta }t$ must be small enough (e.g., 30 seconds), so the supercapacitor voltage can be considered almost constant over this duration. Let k be the s-slot index, and let the tuple $(i,k)$ represent the beginning of the s-slot k in slot i. Then (14) can be replaced by (15).

$$\begin{array}{l}{E}_{\text{cap}}^{i,k+\mathrm{1}}=E_{\text{cap}}^{i,k}+P_I^i·{\eta }_{\text{in}}^{i,k}·\mathrm{\Delta }t\\ \hspace{1em}\hspace{1em}\hspace{1em}-P_{\text{avg}}^i·\frac{\mathrm{\Delta }t}{{\eta }_{\text{out}}^{i,k}}-P_{\text{leakage}}^{i,k}·\mathrm{\Delta }t,\\ \hspace{1em}\hspace{1em}\hspace{1em}{E}_{\text{cap}}^{i+\mathrm{1,1}}=E_{\text{cap}}^{i,K+\mathrm{1}}.\end{array}$$

(15)

According to (5), (7), (9), (12), (13), and (15), we have

$$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}C{\left(V_{\text{cap}}^{i,k+\mathrm{1}}\right)}^2=\frac{\mathrm{1}}{\mathrm{2}}C{\left(V_{\text{cap}}^{i,k}\right)}^2+P_I^i·f_{\text{in}}\left(P_I^i,V_{\text{cap}}^{i,k},T,V_{\text{panel}}\right)·\mathrm{\Delta }t\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}-D^i·P_{\text{active}}·\frac{\mathrm{\Delta }t}{f_{\text{out}}\left(V_{\text{cap}}^{i,k}\right)}-f_l\left(V_{\text{cap}}^{i,k}\right)·\mathrm{\Delta }t,\end{array}$$

(16)

$$\begin{array}{l}{V}_{\text{cap}}^{i+\mathrm{1,1}}=V_{\text{cap}}^{i,K+\mathrm{1}}.\end{array}$$

(17)

Reference [1] presents a performance model in terms of system utility to the user. To maximize the system performance and save energy, we have

$$\begin{array}{cc}& D_{\min }\le D^i\le D_{\max },\end{array}$$

(18)

where $D_{\min }$ is the application that defined minimal duty cycle requirement and $D_{\max }$ is the maximum duty cycle. Any assigned duty cycle that is larger than $D_{\max }$ cannot improve the system performance further and will lead to more energy consumption.

In order to ensure the survival time of the node (work at $D_{\min }$ until death), there should be some energy E left in the supercapacitor at the end of every prediction interval. Meanwhile, the supercapacitor voltage should be higher than the startup voltage ( $V_{\text{startup}}$ ) of the DC-DC converter and smaller than the maximum operating voltage of the supercapacitor ( $V_{\max }$ ). So we have

$$\begin{array}{cc}& V_{\text{cap}}^{L,K+\mathrm{1}}\ge V,\end{array}$$

(19)

$$\begin{array}{cc}& V_{\text{startup}}\le V_{\text{cap}}^{i,k}\le V_{\max }.\end{array}$$

(20)

Then, we formalize the cumulated duty cycle maximization problem as follows:

$$\begin{array}{cc}& \max z=E_{\text{cap}}^{L,K+\mathrm{1}}-E_{\text{cap}}^{\mathrm{1,1}}+{\sum }_{i=\mathrm{1}}^L‍D^i·P_{\text{active}}·\mathrm{\Delta }T\\ \text{s}.\text{t}.\mathrm{\hspace{0.17em}\hspace{0.17em}}\left(\text{17}\right)\text{–}\left(\text{21}\right),\phantom{\mathrm{2222222222222222222222}}\end{array}$$

(21)

where z is the total available energy in the prediction interval $L·\mathrm{\Delta }T$ . It equals the sum of the energy growth of the supercapacitor and energy consumption of the node. If $E_{\text{cap}}^{L,K+\mathrm{1}}$ and $E_{\text{cap}}^{\mathrm{1,1}}$ both equal E, then the system achieves energy neutral operation (ENO) [18] and (21) can be rewritten as follows:

$$\begin{array}{cc}& \max z={\sum }_{i=\mathrm{1}}^L‍D^i\phantom{20}\\ \text{s}.\text{t}.\mathrm{\hspace{0.17em}\hspace{0.17em}}\left(\text{17}\right)\text{–}\left(\text{21}\right).\end{array}$$

(22)

For a link, the previous optimization problems can be extended as data rate utility optimization problems. As shown in Figure 4, we assume that A→B→Sink is a fixed routing in a data collection sensor network. Node A and node B collect and transmit their data to the sink. Then, the data rate utility maximization problem can be formalized as follows:

$$\begin{array}{cc}& \max \sum _{i=\mathrm{1}}^L\left[U\left(r_{\text{A}}\left(i\right)\right)+U\left(r_{\text{B}}\left(i\right)\right)\right]\phantom{\mathrm{2021111111}}\end{array}$$

(23)

$$\begin{array}{cc}& \text{s}.\text{t}.\mathrm{\hspace{0.17em}\hspace{0.17em}}\left(e_{rx}+e_{tx}\right)·r_{\text{A}}\left(i\right)+e_{tx}·r_{\text{B}}\left(i\right)\le D_{\text{B}}^i·P_{\text{active}}\end{array}$$

(24)

$$\begin{array}{cc}& e_{tx}·r_{\text{A}}\left(i\right)\le D_{\text{A}}^i·P_{\text{active}}\phantom{2030202222212212}\\ \text{A},\text{B}:\text{constraints}\left(\text{17}\right)\text{–}\left(\text{21}\right),\phantom{2032012221222}\end{array}$$

(25)

where $r_{\text{A}}(i)$ and $r_{\text{B}}(i)$ are the data rates of node A and node B in slot i. $e_{rx}$ and $e_{tx}$ are the energetic costs to receive and to transmit one bit of data.

Figure 4

A fixed routing in a data collection sensor network.

4.2. Duty Cycle Assignment

According to (16) and (17), the determination of $D^i$ is an iterative process and $D^i$ cannot be expressed as linear combination of duty cycles under other slots. So the maximization problems previously mentioned can neither be solved by convex optimization nor standard linear programming. However, for quantized energy storage, the problem can be solved by a dynamic programming routine algorithm (see Algorithm 1).

Algorithm 1: Duty Cycle Allocation Algorithm.

$(1)$   $D_{\text{total}}(j,i)\leftarrow \mathrm{0},A(j,i)\leftarrow -\mathrm{\infty },B(j,i)\leftarrow \mathrm{0},D\leftarrow -\mathrm{\infty },\forall i<L,\forall j\le \left(V_{\max }-V_{\text{startup}}\right)/\mathrm{\Delta }V$ ;

$(2)$   for  $i\leftarrow \mathrm{1}$ to L do

$(3)$  for  $j_{\mathrm{1}}\leftarrow \mathrm{0}$ to $(V_{\max }-V_{\text{startup}}/\mathrm{\Delta }V)$  do

$(4)$    $V_{\text{cap}}^{i+\mathrm{1}}\leftarrow V_{\text{startup}}+j_{\mathrm{1}}·\mathrm{\Delta }V$ ;

$(5)$   for  $j_{\mathrm{2}}\leftarrow \mathrm{0}$ to $V_{\max }-V_{\text{startup}}/\mathrm{\Delta }V$  do

$(6)$     $V_{\text{cap}}^i\leftarrow V_{\text{startup}}+j_{\mathrm{2}}·\mathrm{\Delta }V$ ;

$(7)$     $D\leftarrow (\mathrm{1}/\mathrm{2}·C·(V_{\text{cap}}^{i+\mathrm{1}}{)}^{\mathrm{2}}-\mathrm{1}/\mathrm{2}·C·(V_{\text{cap}}^i{)}^{\mathrm{2}}+P_I^i·{\eta }_{\text{in}}·\mathrm{\Delta }T-P_{\text{leakage}}·\mathrm{\Delta }T)·{\eta }_{\text{out}}/P_{\text{active}}$ ;

$(8)$    if  $D_{\text{total}}(j_{\mathrm{2}},i)+D>D_{\text{total}}(j_{\mathrm{1}},i+\mathrm{1})$  then

$(9)$      $D_{\text{total}}(j_{\mathrm{1}},i+\mathrm{1})\leftarrow D_{\text{total}}(j_{\mathrm{2}},i)+D,A(j_{\mathrm{1}},i+\mathrm{1})\leftarrow j_{\mathrm{2}},B(j_{\mathrm{1}},i+\mathrm{1})\leftarrow D$ ;

$(\mathrm{10})$      end if

$(\mathrm{11})$   end for

$(\mathrm{12})$  end for

$(\mathrm{13})$   end for

$(\mathrm{14})$   $j\leftarrow (V-V_{\text{startup}})/\mathrm{\Delta }V$ ;

$(\mathrm{15})$  for  $i\leftarrow L+\mathrm{1}$ to 1 do

$(\mathrm{16})$   $D^{i-\mathrm{1}}\leftarrow D(i,j)$ ;

$(\mathrm{17})$   $j\leftarrow A(i,j)$

$(\mathrm{18})$   end for

In Algorithm 1, the supercapacitor voltage is quantized from $V_{\text{startup}}$ to $V_{\max }$ with the step $\mathrm{\Delta }V$ . For every loop on line 2, the algorithm calculates the maximum cumulated duty cycle ( $D_{\text{total}}$ ) for every quantized supercapacitor voltage in slot $i+\mathrm{1}$ and stores the serial number $j_{\mathrm{2}}$ as well as the duty cycle D, corresponding to the maximum cumulated duty cycle, in two arrays A and B. Going “backwards”, Algorithm 1 determines a vector $\{D^{\mathrm{1}},D^{\mathrm{2}},\dots ,D^L\}$ that maximizes the cumulated duty cycle, and the vector is the optimal energy allocation. The running time of Algorithm 1 is $O(L·[(V_{\max }-V_{\text{startup}})/\mathrm{\Delta }V{]}^{\mathrm{2}})$ .

Since ${\eta }_{\text{in}}$ , ${\eta }_{\text{out}}$ , and $P_{\text{leakage}}$ (line 7 in Algorithm 1) vary with $V_{\text{cap}}$ , it is hard to calculate these values if the slot time ( $\mathrm{\Delta }T$ ) is very large (e.g., tens of minutes). In fact, $\mathrm{\Delta }T$ in most of the solar prediction algorithms is set to be tens of minutes. So we further divide every slot into K small slots. Then, D (in line 7) can be calculated as shown in Algorithm 2.

Algorithm 2: Duty cycle computation algorithm.

$(1)$   $D_{\mathrm{1}}\leftarrow D_{\max }$ ;

$(2)$  while  $D_{\mathrm{1}}>D_{\min }$  do

$(3)$  for  $k\leftarrow \mathrm{1}$ to K  do

$(4)$    Calculate $V_{\text{cap}}^{i,k+\mathrm{1}}$ using (16);

$(5)$  end for

$(6)$  if  $\left|V_{\text{cap}}^{i,K+\mathrm{1}}-V_{\text{cap}}^{i+\mathrm{1}}\right|<\delta$  then

$(7)$    $D\leftarrow D_{\mathrm{1}}$ ;

$(8)$   break;

$(9)$    $D_{\mathrm{1}}\leftarrow D_{\mathrm{1}}-\mathrm{\Delta }D$ ;

$(\mathrm{10})$  end if

$(\mathrm{11})$  end while

In Algorithm 2, $D_{\mathrm{1}}$ is quantized from $D_{\min }$ to $D_{\max }$ with the step $\mathrm{\Delta }D$ . The algorithm starts with $D_{\mathrm{1}}\leftarrow D_{\max }$ and calculates $V_{\text{cap}}^{i,K+\mathrm{1}}$ for every loop on line 1. If there are some duty cycles which can make the supercapacitor voltages at the end of the slot ( $V_{\text{cap}}^{i,K+\mathrm{1}}$ ) and $V_{\text{cap}}^{i+\mathrm{1}}$ equal, the algorithm will find the maximum duty cycle and assign it to D. δ is a small threshold compared to $V_{\text{cap}}^{i+\mathrm{1}}$ (e.g., 0.01 V).

In order to solve the cumulated duty cycle maximization problem, we developed Algorithms 1 and 2. With the supercapacitor voltage being quantized from $V_{\text{startup}}$ to $V_{\max }$ , the constraint (20) is satisfied. The constraint (18) can be obtained in Algorithm 2. We denote that the cumulated duty cycle is maximized when inequality (19) is equality. (Suppose that $\{D_{\mathrm{1}}^{\mathrm{1}},\dots ,D_{\mathrm{1}}^L\}$ is the optimum duty cycle vector of $V+j·\mathrm{\Delta }V$ and $\{D^{\mathrm{1}},\dots ,D^L\}$ is the optimum duty cycle vector of V; then $\{D_{\mathrm{1}}^{\mathrm{1}},\dots ,D_{\mathrm{1}}^{L-\mathrm{1}},D_{\mathrm{2}}^L\}$ can be a feasible vector of V. It is quite obvious that $D_{\mathrm{2}}^L$ is larger than $D_{\mathrm{1}}^L$ . So, ${\sum }_{i=\mathrm{1}}^L‍D^i\ge {\sum }_{i=\mathrm{1}}^{L-\mathrm{1}}‍D^i+D_{\mathrm{2}}^L\ge {\sum }_{i=\mathrm{1}}^L‍D_{\mathrm{1}}^i$ .) Then the cumulated duty cycle maximization problem can be solved by Algorithm 1 with its line 7 being replaced by Algorithm 2. Algorithm 2 runs in $O(K·(D_{\max }-D_{\min })/\mathrm{\Delta }D)$ . Thus, to solve the cumulated duty cycle maximization problem, it runs $O(K·L·(D_{\max }-D_{\min })·(V_{\max }-V_{\text{startup}}{)}^{\mathrm{2}}/(\mathrm{\Delta }D·\mathrm{\Delta }{V}^{\mathrm{2}}))$ time.

For quantized energy values and duty cycles, the data rate utility maximization problem can be solved by an extension of Algorithms 1 and 2. Over all $\{r_{\text{A}}(i),r_{\text{B}}(i)\}$ that satisfy constraints of (24) and (25), the maximum utility can be determined as follows:

$$\begin{array}{l}h\left(V_{\text{A}}\left(i+\mathrm{1}\right),V_{\text{B}}\left(i+\mathrm{1}\right),i+\mathrm{1}\right)\\ \hspace{1em}=\underset{}{\max }\left[U\left(r_{\text{A}}\left(i\right)\right)+U\left(r_{\text{B}}\left(i\right)\right)+h\left(V_{\text{A}}\left(i\right),V_{\text{B}}\left(i\right),i\right)\right].\end{array}$$

(26)

Vectors $\{r_{\text{A}}(\mathrm{1}),\dots ,r_{\text{A}}(L)\}$ and $\{r_{\text{B}}(\mathrm{1}),\dots ,r_{\text{B}}(L)\}$ that maximize $h(V_{\text{A}}(L+\mathrm{1}),V_{\text{B}}(L+\mathrm{1}),L+\mathrm{1})$ are the optimal, and it has the complexity $O(K·L·(D_{\max }-D_{\min })·(V_{\max }-V_{\text{startup}}{)}^{\mathrm{4}}/(\mathrm{\Delta }D·\mathrm{\Delta }{V}^{\mathrm{4}}))$ to solve this problem. Solving this problem directly may be computationally expensive for nodes with limited capabilities. Instead, the nodes can determine their duty cycles (e.g., $D_{\text{A}}^i$ and $D_{\text{B}}^i$ ) independently using Algorithms 1 and 2, and then for each slot i, under constraints (24) and (25), the nodes can adjust their rates to obtain a solution to $\max [U(r_{\text{A}}(i))+U(r_{\text{B}}(i))]$ . For example, if we use $\log (·)$ as the function of data rate utility, the subproblem solution is $r_{\text{A}}=\min [D_{\text{A}}^i·P_{\text{active}}/e_{tx},D_{\text{B}}^i·P_{\text{active}}/(\mathrm{2}*(e_{tx}+e_{rx}))]$ and $r_{\text{B}}=D_{\text{B}}^i·P_{\text{active}}/e_{tx}-r_{\text{A}}$ .

5. Numerical Results

This section provides numerical results that demonstrate the high performance of efficiency-aware. Models based on actual measurements described in Section 3 are used as inputs to the algorithms described in Section 4.

Figure 5 shows the I-V measurements and simulations of a $\mathrm{110}\times \mathrm{70}\hspace{0.17em}\text{m}{\text{m}}^{\mathrm{2}}$ solar panel under different temperatures and solar irradiance. Figure 6 shows P-V measurements and simulations of the same solar panel under the same conditions. As shown, the error between numerical simulation and measurement results is less than 6% when $V_{\text{panel}}$ is smaller than the max power point voltage. Although the error will increase with the increase of temperature and solar irradiance when $V_{\text{panel}}$ is larger than the max power point voltage, this can be avoided by choosing a suitable solar panel. In fact, to achieve higher charging efficiency, it is better to choose a solar panel with its max power point voltage under sufficient solar radiation (e.g., at noon in a sunny day) being close to the maximum operating voltage of the supercapacitor, and therefore $V_{\text{panel}}$ can rarely be larger than the max power point voltage.

Figure 5

Measurements and simulations of solar panel output current.

Figure 6

Measurements and simulations of solar panel output power.

Previous works [12, 19] provide energy models of switching converters and MPPT circuits. For simplicity, we model the MPPT circuit in [12] using piecewise linear approximation based on real data measurements. However, it is worth noting that our efficiency-aware framework is not restricted to a specific model or a charging circle because our approach does not depend on the characteristics of the analysis model. The modeling results are shown in Figure 7. As shown, the charging efficiency of the MPPT circuit increases with the increasing of the supercapacitor voltage when the supercapacitor voltage is smaller than 1.8 V and decreases slowly as the voltage goes larger than 1.8 V. The variation trend of the efficiency is similar to the results of [11]. Moreover, our model is also quite accurate (the relative error is no more than 5% when the irradiance is low).

Figure 7

Measurements and a simulation of a MPPT circuit charging efficiency.

Figure 8 shows discharging efficiencies of two commonly used DC-DC converter chips (NCP1402 [22] and LTC3401 [21]), and Figure 9 shows the leakage of a 10 F and a 100 F supercapacitors. We also model the efficiencies of those two chips and the leakage of a 100 F supercapacitor based on the measurements. As shown, the discharging efficiencies and the leakage both increase with the increasing of the supercapacitor voltage, and the models are also of high accuracies (the relative error is no more than 2%).

Figure 8

Discharging efficiency of NCP1402 and LTC3401.

Figure 9

Leakage of a 100 F and a 10 F supercapacitor.

Figure 10 shows the solar profile of three common winter days in Ashland [20]. The solar radiation is collected once every 5 minutes and has been converted into the power that can be harvested by the solar panel.

Figure 10

Solar profile of 3 days.

We evaluated two sets of data under four conditions. MPPT, NCP1402, and a 100 F supercapacitor were used in condition 1; direct connection, NCP1402, and a 100 F supercapacitor were used in condition 2; MPPT, LTC3401, and a 100 F supercapacitor were used in condition 3; and direct connection, LTC3401, and a 100 F supercapacitor were used in condition 4. In all the four conditions, $V_{\text{startup}}=1$  V, $D_{\min }=\mathrm{1}\%$ , $D_{\max }=\mathrm{100}\%$ , and we ignored the influence of temperature and the prediction error of the solar radiation. $\mathrm{\Delta }V$ , $\mathrm{\Delta }D$ , and δ are 0.02 V, 1%, and 0.01 V. The prediction interval was set to be one day. $\mathrm{\Delta }T$ and $\mathrm{\Delta }t$ were 30 minutes and 30 seconds. Figure 11 shows cumulated active time of the fixed duty cycle scheme under these four conditions. As shown, the cumulated active time increases with increasing duty cycle and decreases after some duty cycle. The reason is that energy surplus is large when the duty cycle is low and high duty cycle leads to low working voltage and inefficiency. Figures 12, 13, 14, and 15 show the supercapacitor voltage and cumulated active time of efficiency-aware, leakage-aware, and the fixed duty cycle scheme over 72 hours under these four conditions. The fixed duty cycles were set to 0.76, 0.45, 0.8, and 0.51 where the fixed duty cycle scheme can achieve the max cumulated active time. Table 1 shows the numerical results. The results reveal that our algorithms can achieve 56% more cumulated time than leakage-aware and 60% more cumulated time than fixed duty cycle scheme. In Figures 12(a), 13(a), 14(a), and 15(a), the supercapacitor voltages of our algorithms at 24th, 48th, and 72nd hour are 1.5 V, compared with 1.25 V of leakage-aware and 1 V with fixed duty cycle scheme. It illustrates that our algorithms perform better in terms of sustainable operation (e.g., nodes never run out of energy in their supercapacitors). It worth noting that leakage-aware uses leakage as the only factor in determining nodes' life, while a practical one should consider the minimum requirements of the application as well as leakage.

Table 1

Cumulated active time of 3 days.

Scheme	Condition
	1	2	3	4
Efficiency-aware	18.5 hr	16.7 hr	17.7 hr	17.6 hr
Leakage-aware	9.0 hr	10.7 hr	10.1 hr	10.4 hr
Fixed duty cycle	10.1 hr	6.3 hr	11.1 hr	7.1 hr

Figure 11

Cumulated active time of the fixed duty cycle scheme.

Figure 12

Simulation results under condition 1.

Figure 13

Simulation results under condition 2.

Figure 14

Simulation results under condition 3.

Figure 15

Simulation results under condition 4.

Figure 16 shows the results for the data rate determination problems under condition 1. The solar profile of the first day was used as the energy input for both nodes A and B, and $\log (·)$ was used as the utility function of data rate ( $U(·)$ ). We assumed that both $e_{tx}$ and $e_{rx}$ are 50 nJ/bit.

Figure 16

Date rates of node A and node B.

6. Conclusion

In this paper, we study how to optimize the energy efficiency of photovoltaic-supercapacitor systems for sustainable node operations in solar-powered wireless sensor networks. Based on realistic power model of every hardware component, we present energy-aware, the first duty cycling framework that maximizes the energy utilization efficiency of the whole photovoltaic-supercapacitor system. We consider all possible energy losses in the energy conversion process (charging, discharging, and leakage) and model all the power models of a commonly used photovoltaic-supercapacitor energy systems. We have proposed a general duty cycling optimization problem and associated efficient algorithms to solve the problem. Numerical results show that our approach performs better than fixed duty cycling schemes and leakage-aware [4], a state-of-art duty cycling scheme for photovoltaic-supercapacitor systems. As efficiency-aware is model-independent, it can be used in most of photovoltaic-supercapacitor energy systems with predictable harvesting energy for local power management or to meet other QoS requirements in wireless sensor networks.