1 Introduction

With the rapid development of the internet of things, there is an increasing demand for wireless sensor networks (WSNs) (Gubbi et al. 2013). Since a WSN is composed of a large number of wireless sensor nodes to monitor a large area, the cost and installation of nodes should be as low as possible to make the WSN reasonable (Potdar et al. 2009). Therefore, WSNs with small-sized and low-cost nodes are becoming the mainstream (Botta et al. 2016; Jang et al. 2016). In most cases, the power module of the node occupies a considerable part of the whole system (Chou et al. 2016; Guan et al. 2017). Moreover, due to the limited size of nodes, power module is unable to provide too much energy, which causes a serious problem of power limitation, and researchers have to make WSNs more complex in either hardware or software (Gandelli et al. 2005; Razzaque and Dobson 2014). Thus, the optimization of the power modules is a practical approach to reduce both the size and cost of nodes.

Since photovoltaic (PV) cells have high power density, they have become one of the most commonly used energy harvesters in wireless sensor nodes. Among all the PV cell-based nodes, nearly every system uses the maximum power point tracking (MPPT) techniques to maximize the amount of power harvested and assist the load to collect more environmental data (Raghunathan et al. 2005; Simjee and Chou 2006; Brunelli et al. 2008, 2009; Hassanalieragh et al. 2014; Li et al. 2014;). One of the most well-known techniques is called the fractional open-circuit voltage (FOC) method, which limits the output voltage of PV module to about 75% of its open-circuit voltage. Unfortunately, the MPPT techniques usually require an additional circuit, making the system more complex, and the circuit also consumes energy to carry out its function (Han et al. 2008; Lu et al. 2010). The MPPT circuit in the wireless sensor nodes is contrary to the principle of being small and low-cost. However, the working efficiency of systems without the MPPT circuit is relatively low, and these systems have to sleep longer to collect energy, which is disadvantageous for monitoring (Minami et al. 2005). In conclusion, a method to remove the complex MPPT circuit without reducing working efficiency is lacking.

We propose a novel energy management strategy to solve the current problem. It optimizes the working efficiency of systems without an MPPT circuit so that the small-sized and low-cost system can replace it with a complex circuit. An energy model is established for systems without an MPPT circuit that shows the energy storage in different conditions. Based on the working mode durations analyzed by the model, an optimized strategy is proposed to maximize the load’s active time in one working period. Simulation of the load’s working process is conducted, and the total active time is calculated to be as long as that of the traditional system that includes the MPPT circuit. Experimental tests are conducted using a Solar Simulator, which verifies the strategy. Systems without an MPPT circuit have fewer electronic components, making them smaller and more affordable. The energy strategy optimizes their working efficiency to be comparable to that of traditional systems. Therefore, our work provides an effective solution to exclude the MPPT circuit for ultra-small-sized and low-cost wireless sensor nodes such as smart dusts (Kahn et al. 1999).

2 System modeling

Due to their long life cycle, relative high energy density and power density (Namisnyk and Zhu 2003), supercapacitors are becoming widely used working with PV cells to power wireless sensor nodes (Minami et al. 2005; Simjee and Chou 2006; Hassanalieragh et al. 2014). The hardware structure of nodes without an MPPT circuit is shown in Fig. 1a, while Fig. 1b depicts the most common structure. The simple system, a non-MPPT system, is composed of the following devices: PV cells for energy harvesting from the environment, a diode for damage protection of PV cells from current backflow, a supercapacitor for energy storage and output, a DCDC for provision of stable voltage, and a load for environmental monitoring. The traditional system, an MPPT system, includes an additional MPPT circuit that usually consists of DCDCs, comparators and diodes.

Fig. 1
figure 1

Hardware structure of wireless sensor nodes based on PV cells and supercapacitors: a the non-MPPT system; b the MPPT system

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In wireless sensor nodes, loads usually fluctuate between sleeping mode and active mode in the most typical way (Penella and Gasulla 2007). Here, T active /(T sleep  + T active ) in one working period is defined as R a , which describes the duty ratio of the active mode. The system’s total active time is longer over a long period of time with a higher R a , which means it can collect more environmental information. Therefore, the energy management strategy is designed to search the system parameters to maximize R a , and improve the maximum working efficiency of the non-MPPT system. A model for the non-MPPT system is established to analyze the energy storage and to calculate R a under different working conditions. A similar model for the MPPT system is also established to compare the R a values of the two different systems that help in evaluating the strategy.

2.1 Model of the non-MPPT system

The governing equations of the general system model (Eqs. 1–3) are derived from Kirchhoff’s laws and the properties of DCDC.

$$V_{pv} - V_{Diode} = V_{SC}$$
(1)
$$I_{pv} = I_{Diode}$$
(2)
$$V_{SC} \times \left( {I_{Diode} + I_{SC} } \right) \times \eta_{DCDC} = V_{load} \times I_{load}$$
(3)

V pv and I pv are the output voltage and output current of the PV cells. V SC is the output voltage of the supercapacitor, and I SC is the output current of the supercapacitor when it is positive and represents the charging current of the supercapacitor when it is negative. The instantaneous forward voltage of the diode is V Diode , and I Diode is its forward current. These parameters are variable and changeable under different circumstances. η DCDC and V load are the transfer efficiency and output voltage of DCDC, and I load is the load current. The former three parameters are determined by specific devices.

2.1.1 Model of PV cells

The single diode equivalent circuit, as shown in Fig. 2, is used to develop the current–voltage (I-V) characteristic equation (Eq. 4) of PV cells (Dzimano 2008).

Fig. 2
figure 2

Single diode circuit for the model of PV cells

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$$I_{pv} = I_{L} - I_{sat} \left[ {exp\left( {\frac{{V_{pv} + I_{pv} \times R_{s} }}{\alpha }} \right) - 1} \right] - \frac{{V_{pv} + I_{pv} \times R_{s} }}{{R_{sh} }}$$
(4)

There are five parameters in the model: I L is the light current, I sat is the reverse saturation current of the diode, α is the ideality factor, R s is the series resistance, and R sh is the shunt resistance.

These five parameters can be calculated according to environmental conditions, including solar radiance G, air temperature T and air mass AM, and their values under standard rating conditions are SRC with G* = 1000 W/m2, T* = 25 °C and AM* = 1.5 (Dzimano 2008).

2.1.2 Model of the diode

The current–voltage characteristic of diodes can be described by the Shockley function.

$$I_{Diode} = A \times (\exp \left( {B \times V_{Diode} } \right) - 1)$$
(5)

The two parameters, A and B, can be calculated based on the datasheet.

2.1.3 Model of the supercapacitor

Common double-layer supercapacitor characteristics can be described by the two-branch RC network (Zubieta and Bonert 2000) shown in Fig. 3.

Fig. 3
figure 3

Two-branch RC network for supercapacitor model

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Resistance R 3 reflects the leakage of the supercapacitor. The first branch, consisting of R 1, C 1 and C i , interprets the transient stage of charging or discharging. The second branch, consisting of R 2 and C 2, interprets the charge redistribution process during the retardant stage. Here, a switch K is involved that is open to disconnect the second branch when the supercapacitor works without a retardant stage. In the non-MPPT system, the supercapacitor is always in the quick charging or discharging state, meaning that K is always open. Equations 6–9 describe the transient stage of the supercapacitor. All the parameters can be determined from a constant current test.

$$I_{SC} \left( t \right) = I_{ 1} \left( {\text{t}} \right) - I_{ 3} \left( t \right)$$
(6)
$$V_{SC} \left( t \right) = I_{ 3} \left( t \right) \times R_{ 3}$$
(7)
$$V_{SC} \left( t \right) = V_{ 1} \left( t \right) - I_{ 1} \left( t \right) \times R_{ 1}$$
(8)
$$- I_{1} (t) = \frac{{dV_{1} (t)}}{dt}\left( {C_{1} + K_{V} \times V_{SC} (t)} \right)$$
(9)

2.1.4 Model of DCDC and load

η DCDC and V load are usually given by the datasheet of DCDC; the different load current I load in different working modes can be tested.

2.2 Model of MPPT system

The general system model of the MPPT systems is based on the energy conservation law instead of Kirchhoff’s laws. The governing equation for the MPPT system is as follows:

$$(P_{pv} \times \cdot \eta_{MPPT} + P_{SC} ) \times \eta_{DCDC} = P_{load}$$
(10)

P pv and P SC are the output voltage of the PV cells and supercapacitor, η MPPT is the conversion efficiency of the MPPT circuit, and P load is the power consumption of the load.

The PV cells and supercapacitor models of the MPPT system are based on the non-MPPT system models, and models of DCDC and load are the same as those of the non-MPPT system.

2.2.1 Model of PV cells

When the environmental parameters are input into the model of PV cells in chapter 2.1.1, the functional relationship between PV cell output power and output voltage can be determined; P pv in Eq. 14 is the maximum power value of the function. The diode that protects PV cells from current backflow damage is integrated into the MPPT circuit during the modeling of PV cells, and η MPPT is tested by a charging test (Enslin 1990).

2.2.2 Model of the supercapacitor

V 1, the voltage across the first branch’s capacitors, can be approximately regarded as V SC because the product of current and resistance in the first branch is usually much smaller than V 1 (shown in Eq. 8). In the models of the MPPT system, V 1 is used to replace V SC to simplify the computation.

3 Optimization and simulation

The models are developed to calculate R a . PV cells can charge the supercapacitor while providing enough energy for the load in the sleeping mode and can provide energy for the load together with the supercapacitor in active mode, which causes the supercapacitor’s voltage to increase during sleeping mode and the voltage to drop during active mode. In the stable working condition, the supercapacitor’s voltage changes between threshold voltages as shown in Fig. 4. Analysis for the impact factor of R a in the two systems will help the optimization process.

Fig. 4
figure 4

Curves of V SC and P load over time in a stable working condition

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3.1 Optimization for the non-MPPT system

T active and T sleep are affected by the consumption of load, operation voltage of the supercapacitor and output of PV cells. In the non-MPPT system, the consumption of load in different working modes is determined by the devices, and the operation voltage of the supercapacitor is determined by the threshold voltages shown in Fig. 4, i.e., V max and V min . When designing a monitoring system, the total area of PV cells is often fixed due to the restriction of the system’s size. Once the number of PV cells in series increases, the area of each cell decreases. Therefore, the output voltage is in proportion to the number of cells in series, while the output current is in inverse proportion. As a result, the number of PV cells in series, indicated as N, also affects T active and T sleep . Among all the parameters above N,\(V_{max}\) and V min are designed when optimizing the power module. Consequently, maximization of R a is achieved by changing N and threshold voltages; the calculation process is shown in Fig. 5.

Fig. 5
figure 5

Calculation process for R a in the non-MPPT system

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In Fig. 5, the minimum number of PV cells in series, designated N min , is determined by the DCDC’s lowest input voltage, dV is a voltage step in calculation, V oc is the open circuit voltage of PV cells, and N max is the maximum number of PV cells in series, which is determined by the supercapacitor’s rated voltage. The minimum V min is the start-up voltage of the DCDC, and the maximum V max is the PV cells’ V oc . With the calculation process shown in Fig. 5, R a can always be solved regardless of the working condition. In addition, T active should be longer than the threshold time according to practical requirements, so the situations in which T active is too short to transmit useful information should be excluded. Here, T a-min is defined as the minimum active time, and the situations in which T active  < T a-min are excluded during the calculation process.

A set of devices are selected for simulation. PV cells are made of single-crystalline silicon. An SDM10K45-7 (Schottky Barrier Diodes, produced by Diodes Inc.) is used to protect PV cells, a CLG05P030L17 (30 mF supercapacitor, produced by Cellergy) is used to store and output energy, and a TPS62234 and a CC2530 (both are produced by TI) are selected to be the DCDC and load. Each single-crystalline silicon cell’s open circuit is approximately 0.5–0.6 V, so N min is 5 since TPS62234 requires at least 2 V input voltage to start up, and N max is 9 since the rated voltage of CLG05P030L17 is 5.5 V. The minimum V min should be higher than the start-up voltage of TPS62234, which is set as 2.1 V, and T a-min is set as 1 s in the optimization process.

Optimization of R a in different environmental conditions is performed. For all conditions, the temperatures are 25 °C and air masses are 1.5, while the radiation intensity is 1000, 700, 400 and 100 W/m2, which correspond to the ultra-high, high, low and ultra-low light intensity, respectively, in the natural environment (Nfah et al. 2007). The results are shown in Fig. 6, and all the raw data are provided by Online Resource.

Fig. 6
figure 6

R a in different environmental conditions: a relationship between Max_R a and N; b relationship between R a and V average when N = 9; c relationship between R a and \(\Delta V\) when N = 9 and V average  = 3.85 V

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Once N is determined, R a is a function of V max and V min , and it reaches the maximum value under certain threshold voltages. The maximum value when N is fixed is indicated as Max_R a . Figure 6a shows that there is a positive correlation between \(Max\_R_{a}\) and N in all conditions, and Max_R a reaches a maximum value when N = 9. The cause of this phenomenon is that the output current of PV cells is in inverse proportion to N, and the power consumption of the diode is in proportion to the current. Therefore, when N is higher, the power consumption of the diode decreases, and the power transmitted to the supercapacitor and load increases leading to a higher R a . Figure 6b, c show the threshold voltages to reach the maximum R a when N = 9. V average , defined as (V max  + V min )/2, significantly influences R a . Figure 6b shows that no matter how strong the radiation is, R a reaches a relatively high value when V average  = 3.85 V. This is because there is a maximum power point (MPP) for PV cells, and it does not change very much when the radiation changes. According to Eq. 1, the average operating voltage of a supercapacitor should be close to MPP to maximize the energy collection because V Diode is usually very low. Figure 6c shows that there is a negative correlation between R a and the range of operating voltage that is indicated as \(\Delta V\), and the suitable value for different conditions is 0.3 V. The goal of minimizing \(\Delta V\) is to make the operating voltage of the supercapacitor close to MPP. In conclusion, in order to maximize working efficiency, the non-MPPT system should set N = 9, V average  = 3.85 V and \(\Delta V\) = 0.3 V, i.e., V min  = 3.7 V and V max  = 4.0 V.

As a summary, the optimization process is achieved as follows:

Calculate R a in different environmental conditions, as shown in Fig. 5.

Plot the maximum value of R a as function of N, as shown in Fig. 6a. Based on the figure, determine N for the for the best system performance.

Plot R a as function of V average when N is determined, as shown in Fig. 6b. Based on the figure, determine V average for the for the best system performance.

Plot R a as function of \(\Delta V\) when V average and N are determined, as shown in Fig. 6c. Based on the figure, determine \(\Delta V\) for the for the best system performance.

The switch of the working modes can be achieved by using the analog-to-digital converter of the function module’s microcontroller to detect V SC periodically. The microcontroller turns the function module into active mode when V SC rises to V max and turns the function module into sleeping mode when V SC falls to V min .

3.2 Calculation for the MPPT system

In the MPPT system, the output power of the supercapacitor is determined by Eq. 11, which is the deformation of Eq. 10. In both the sleeping mode and active mode, P load , η DCDC , P pv , η MPPT are constant, which makes P SC constant in a working mode.

$$P_{SC} = \frac{{P_{load} }}{{\eta_{DCDC} }} - P_{pv} \times \eta_{MPPT}$$
(11)

The input energy to the supercapacitor in the sleeping mode is equal to the output energy of the supercapacitor in the active mode, which is described in Eq. 12.

$$|P_{{SC_{active} }} \times T_{active} | = \frac{1}{2}C_{1} (V_{max}^{2} - V_{min}^{2} ) + \frac{1}{3}K_{V} (V_{max}^{3} - V_{min}^{3} ) = |P_{{SC_{sleep} }} \times T_{sleep} |$$
(12)

Equations 11 and 12 indicate that R a is only affected by environmental conditions, and the function relationship is described in Eq. 13. T active is determined by Eq. 14.

$$\left\{ {\begin{array}{l} {R_{a} = {\raise0.7ex\hbox{${T_{active} }$} \!\mathord{\left/ {\vphantom {{T_{active} } {(T_{sleep} + T_{active} )}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${(T_{sleep} + T_{active} )}$}} = {\raise0.7ex\hbox{${Q_{a} }$} \!\mathord{\left/ {\vphantom {{Q_{a} } {\left( {1 + Q_{a} } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {1 + Q_{a} } \right)}$}}} \\ {Q_{a} = {\raise0.7ex\hbox{${T_{active} }$} \!\mathord{\left/ {\vphantom {{T_{active} } {T_{sleep} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{sleep} }$}} = {\raise0.7ex\hbox{${P_{{SC_{sleep} }} }$} \!\mathord{\left/ {\vphantom {{P_{{SC_{sleep} }} } {P_{{SC_{active} }} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P_{{SC_{active} }} }$}}} \\ {P_{{SC_{sleep} }} = P_{pv} \times \eta_{MPPT} - \frac{{P_{{load_{sleep} }} }}{{\eta_{{DCDC_{sleep} }} }}} \\ {P_{{SC_{active} }} = \frac{{P_{{load_{active} }} }}{{\eta_{{DCDC_{active} }} }} - P_{pv} \times \eta_{MPPT} } \\ \end{array} } \right.$$
(13)
$$T_{active} = \frac{{\frac{1}{2}C_{1} (V_{max}^{2} - V_{min}^{2} ) + \frac{1}{3}K_{V} (V_{max}^{3} - V_{min}^{3} ) }}{{\frac{{P_{{load_{active} }} }}{{\eta_{{DCDC_{active} }} }} - P_{pv} \times \eta_{MPPT} }}$$
(14)

A similar set of devices are selected for calculation. The PV cells, supercapacitor, DCDC and load are the same devices as described in 3.1. MPPT is obtained by the use of FOC (Brunelli et al. 2009). The circuit includes a CPC1824 (reference PV cell, produced by IXYS Corporation), an LTC1440 (comparator, produced by Linear Technology), an LTC3401 (boost converter, produced by Linear Technology) and an SDM10K45-7, and its transfer efficiency is tested to be 85%.

According to Eq. 13, R a is calculated to be 88.3, 61.3, 33.8 and 6.66% in the condition of ultra-high, high, low and ultra-low light intensity. The results are also listed in Table 1.

Table 1 Analysis of the two systems’ simulation results
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3.3 Simulation analysis

High and low light intensity are most common under natural conditions (Nfah et al. 2007), and the two systems’ V SC -time curves under such conditions have been simulated. The non-MPPT system’s threshold voltages are set to be 3.7 and 4.0 V according to the optimization result, and the MPPT system’s threshold voltages are set to the same values for comparison. The monitoring time of the systems is set to be 2 min. The simulation results in 35 s are shown in Fig. 7. Voltage behavior in total 120 s are provided by Online Resource, and Table 1 lists the analysis results.

Fig. 7
figure 7

Simulation results of the two similar systems in different environmental conditions: a results of non-MPPT system under light intensity of 700 W/m2; b results of non-MPPT system under light intensity of 400 W/m2; c results of MPPT system under light intensity of 700 W/m2; d results of MPPT system under light intensity of 400 W/m2

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In each condition, the loads in the two systems start with the sleeping mode where the power consumption is extremely low. The PV cells in both systems charge the supercapacitors, and V SC rises at the same time. Once V SC rises to V max , the loads enter active mode, and the supercapacitors work together with PV cells to power the loads, which causes a decrease in V SC . The loads enter sleeping mode again when V SC drops to V min . When the radiation becomes stronger, the sleeping time decreases and the active time increases. The reason is that when the output of PV cells is higher, the charging current of supercapacitors in sleeping mode is higher and the discharge current in the active mode is lower. The simulation results show that the non-MPPT system has more working cycles during the monitoring period, and its total active time is longer than that of the MPPT system for all conditions. More importantly, its maximum R a is 5.8 and 3.7% higher than that of the MPPT system in the two conditions, so the conclusion that the non-MPPT system can be active longer is not only applicable to this specific monitoring time. When the monitoring time is long enough, the non-MPPT system total active time will always be longer. A longer active time means that the system has more time to sense and transmit data, so it can collect more data under the same radiation meaning higher working efficiency.

4 Experiment and discussion

Two systems have been developed for the experiment, and their block diagrams are shown in Fig. 8a, b. Circuits of two systems’ power module are shown in Fig. 8c, where the upper board is the circuit for the MPPT system and the lower board is the circuit for the non-MPPT system. The electronic components of the two boards are listed in Table 2. The board for the MPPT system has 24 components, while the other one has only 5 components, making the system smaller and simpler. A CC2530 module and a relay module are used to simulate the load. The relay module is used to detect V SC , and it connects the CC2530 and DCDC when V SC rises to V max to simulate the active mode power consumption of a wireless sensor node and disconnects the two parts when V SC falls to V min to simulate the sleeping mode power consumption. The experimental setup is shown in Fig. 8d, which includes a VeraSol (solar simulator, produced by Oriel Instruments), a myDAQ (data acquisition board, produced by NI) and a laptop.

Fig. 8
figure 8

a Block diagram of the non-MPPT system; b block diagram of the MPPT system; c circuit board for two systems; d experimental setup

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Table 2 BOM (Bill of Material) list of the two power modules
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Experiments to prove the feasibility of the model are conducted with a light intensity of 725 W/m2. The threshold voltages of two systems are set to be 3.7 and 4.0 V, and the actual voltages has a little difference. Figure 9 shows the experimental results of the two systems in one working period, and they are compared to the simulation results. The experimental T sleep in the two systems are 0.937 and 1.003 s, and the simulation results are 0.98 and 1.06 s. The experimental T active values are 2.137 s and 1.93 s, and the simulation results are 2.26 and 1.74 s. The maximum relative error is less than 10%, which verifies the validity of the proposed model.

Fig. 9
figure 9

Simulation and experimental results of the two similar system: a non-MPPT system; b MPPT system

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Experimental tests in the most common conditions are conducted to explore the effectiveness of the optimization result. The actual light intensity is tested to be 725 and 420 W/m2, and the monitoring time is 2 min. The experimental results in 35 s are shown in Fig. 10. Voltage behavior in total 120 s are provided by Online Resource, and Table 3 lists the analysis results.

Fig. 10
figure 10

Experimental results of the two similar systems in different environmental conditions: a results of non-MPPT system under light intensity of 725 W/m2; b results of non-MPPT system under light intensity of 420 W/m2; c results of MPPT system under light intensity of 725 W/m2; d results of MPPT system under light intensity of 420 W/m2

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Table 3 Analysis of the two systems’ experimental results
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Experimental curves show a similar trend to the simulation curves shown in Fig. 7. Although there is a difference between the simulation and actual time duration, the experimental results of the working efficiency are very consistent with the simulation results. Table 3 indicates that the non-MPPT system can be active longer than the MPPT system during the monitoring period in the most common conditions. Its maximum R a is 6.3 and 1.3% higher than that of the MPPT system, which indicates that the non-MPPT system can be active a little longer when the monitoring time is long enough.

According to the simulation and experimental results, the non-MPPT system has a higher R a than the MPPT system in the most common conditions, which means its working efficiency is relatively higher. This is opposite to the common phenomenon and can be explained from the view point of energy transformation. The energy management strategy for the non-MPPT system can be regarded as a method to achieved MPPT without additional circuits, allowing PV cells to work around the maximum power point. Once the output power of PV cells is maximized, the energy transmitted to the supercapacitor and load would be close to that of the MPPT system. The conversion efficiency of the MPPT circuit is usually less than 100% that causes the harvested energy in the non-MPPT system to be slightly higher in some situations. In conclusion, no matter what devices are selected and how strong the radiation is, the working efficiency of the non-MPPT system can always be as high as that of the MPPT system with an appropriate energy strategy, and it can even be higher in some specific situations. Furthermore, our strategy is universally applicable, and it can simplify the hardware structure and reduce costs of systems with different application requirements without reducing their working efficiency.

5 Conclusion

A novel energy management strategy for wireless sensor nodes was proposed. By designing the number of PV cells in series and the working modes, the MPPT circuit for PV cells can be removed without reducing the working efficiency of a node. Models of the non-MPPT system and the MPPT system were developed for simulation. Based on real device parameters, simulations were implemented for most common conditions. The results show that the non-MPPT system’s maximum R a is slightly higher than that of the MPPT system. Experimental tests were also conducted, showing the same phenomenon that the non-MPPT system’s maximum R a is 6.3 and 1.3% higher than that of the MPPT system. All the results indicate that the working efficiency of the non-MPPT system is close to that of the MPPT system using the strategy. More importantly, the number of components in the non-MPPT system’s power module is only approximately 1/5 of that of the MPPT system. This energy strategy can simplify the hardware structure of PV cell based systems and will lead to the development of wireless sensor nodes with lower cost and smaller size.