Method for heating a battery

Abstract

The method relates to a method for heating a battery (1) of a drive arrangement, wherein the drive arrangement has the battery (1), an inverter (3) electrically connected to the battery and an electric machine (2) connected to the inverter (3), said electric machine comprising a rotor and a stator with at least three phase windings. In the method, the inverter is operated such that a multiphase AC current with different current amplitude values flows between the inverter (3) and the at least three phase windings (U, V, W) of the electric machine (2).

Description

[0001] Description

[0002] title

[0003] Method for heating a battery

[0004] The present invention relates to a method for heating a battery as well as a computing unit and a computer program for carrying out the method.

[0005] Background of the invention

[0006] When a lithium (Li)-ion battery is very cold, it has a high internal resistance and therefore lacks performance. Charging performance, in particular, but also discharging performance, is severely limited at low temperatures, and charging at such low temperatures is detrimental to the battery's service life. This is especially true when charging with high charging currents. This can be particularly problematic in electric vehicles, as they can become very cold, especially in winter. The battery should therefore be warmed to a minimum temperature before a trip to ensure the electric vehicle has sufficient performance and recuperation capacity.

[0007] To heat the battery before driving or (fast) charging, a heating element (e.g. positive temperature coefficient, PTC) can be used in the battery cooling circuit. This heats the cooling water pumped through the battery, and thus the battery. However, a heating element is expensive because it is an additional component that must be connected to the on-board electrical system, the cooling water circuit, and the communications network. In addition, heating elements require a large surface area to the cooling water in order to quickly transfer the heat to the cooling water. Heating the battery via the cooling water, which is traditionally thermally coupled to the outside of the battery cell, also causes a significant temperature gradient within the battery. This leads to faster cell aging because the temperature gradient causes thermal stress.

[0008] Advantageously, the power loss for heating would be generated within the battery, resulting in a significantly more even battery heating. For example, DE10 2010 032 088 A1 describes an approach in which the generation of a high-frequency battery current ripple heats the battery through the resulting losses. To achieve this, the battery is divided into two sections, which are charged and discharged out of phase to achieve a constant battery voltage despite the battery current ripple. A load device generates a battery current ripple with frequencies above 25 kHz.

[0009] Disclosure of the invention

[0010] According to the invention, a method for heating a battery, as well as a computing unit and a computer program for implementing the method, are proposed, having the features of the independent patent claims. Advantageous embodiments are the subject of the dependent claims and the following description.

[0011] The invention is based on a drive arrangement, for example of a vehicle, which has a battery, an inverter or power converter electrically connected to the battery and an electric machine connected to the inverter and comprising a rotor and a stator with at least three phase windings. Charging or discharging the battery at low temperatures can sometimes significantly reduce the service life of the battery. The battery, which is in particular a lithium-ion battery, is in particular the traction battery, i.e. the battery which supplies the electric machine that drives the electric vehicle with energy. It is therefore provided to heat the battery using losses in the battery, e.g. before a charging process or ferry operation.Within the scope of the invention, the inverter is operated in such a way that a multiphase alternating current with different current amplitude values ​​flows between the inverter and the at least three phase windings of the electric machine. This advantageously results in an alternating current flowing between the battery and the inverter, which leads to the internal heating of the battery. Thus, the control or control of the inverter is used to generate electrical losses in the battery. These losses can be used to heat the battery from within.

[0012] When the battery cells are heated via their internal resistance, they heat up significantly more evenly and thus extend their service life more effectively than when heated via externally heated cooling water. Furthermore, the method advantageously allows for the greatest possible losses to be generated in the battery, while simultaneously minimizing losses in the electric drive or the vehicle electrical system, provided they are designed accordingly. The method according to the invention is therefore advantageously particularly (energy) efficient.

[0013] By heating the battery through internal losses, a heating element in the battery cooling circuit is no longer required. Furthermore, the method can be implemented using the standard hardware and much of the standard software of a conventional inverter. Only minor adjustments or extensions to the inverter software are necessary, allowing a conventional drive system to be used as a generator for current oscillations for battery heating with minimal hardware and software modifications. This allows existing components in electric vehicles to be used for battery heating, allowing the method according to the invention to be implemented cost-effectively.

[0014] In one embodiment, the electric machine has exactly three phase windings, and the inverter is operated in such a way that a first alternating current is generated in one of the three phase windings and a second alternating current is generated in each of the other three phase windings, the two second alternating currents being in total opposite to the first alternating current. This is achieved in particular by operating the inverter in such a way that the first alternating current and the second alternating current have a wave-like, in particular sinusoidal, form, the amplitude of the second alternating current corresponding to half the amplitude of the first alternating current, and the second alternating current being phase-shifted by an angle of 180° (or rt) relative to the first alternating current.

[0015] As a result, the first alternating current in one of the three phase windings and the second alternating currents in the other two phase windings add up to zero, so no rotating magnetic field is generated in the stator. This essentially keeps the rotor stationary. It can also be braked to prevent any oscillations. This allows the process to be carried out in a simple and energy-efficient manner.

[0016] In one embodiment, operating the inverter further comprises at least specifying at least one setpoint value that characterizes an amplitude of the first alternating current, and determining a drive signal for driving current switches (such as MOSFET, IGBT, etc.) of the inverter such that the first alternating current occurs in one of the three phase windings with an amplitude corresponding to the setpoint value, and the second alternating current occurs in the other phase windings with an amplitude corresponding to half the amplitude of the first alternating current and a phase shifted by 180° to the first alternating current. The current switches of the inverter are then controlled using the drive signal.

[0017] Determining the control signal can, in particular, comprise determining a control level or duty cycle for each affected current switch. A typical control of current switches in inverters can comprise a PWM-based method, wherein the control level characterizes the proportion of the ON state per PWM cycle. In one embodiment, the control signal is determined based on the control level. The control signal can be determined by a phase modulator, in particular by a space vector or pulse width modulator or space vector pulse width modulator. Space vector pulse width modulation (SVPWM) is a method for controlling rotating electrical machines based on pulse width modulation. The phase modulator determines the control signal output to the current switches of the inverter based on the supplied control signal as the input variable for the space vector pulse width modulation.Advantageously, the control signal can be generated using a phase modulator without the need for a controller, whereby the method can be carried out particularly cost-efficiently.

[0018] In one embodiment, a target value of a frequency of the first alternating current is further specified, wherein the frequency of the alternating current is in particular at least 50 Hz or at least 100 Hz or at least 150 Hz, and / or at most 150 Hz, or at most 500 Hz or at most 1000 Hz or at most 2000 Hz or at most 2400 Hz. A frequency in a range from 100 Hz to 150 Hz is particularly advantageous because the service life of the battery is not affected or only very slightly affected by the losses generated. In this case, determining a control signal comprises in particular determining a profile of the first alternating current based on at least the target value of the amplitude and the target value of the frequency or a corresponding dynamic target value signal. The control signal for controlling the current switches of the inverter is then determined as a function of the profile of the first alternating current.

[0019] If the alternating current used to drive the electric motor has a sufficiently high frequency – 100 Hz or higher for a common Li-ion battery system – it has little or no impact on the aging of the cell chemistry. The required frequency generally depends on the cell chemistry and the specific battery cell or battery. Furthermore, a frequency of no more than 2400 Hz is within the range that a conventional inverter with a PWM clock frequency of 20 kHz, for example, can generate. Furthermore, the losses generated in the rotor can be advantageously reduced, so that the specified frequency range allows the process to achieve the aforementioned advantages in a particularly service life-saving, energy-efficient, and cost-effective manner.

[0020] In one embodiment, determining the profile of the first alternating current further comprises determining the one phase winding of the at least three phase windings in which the first alternating current is to be generated, based on the position of the rotor relative to the stator. The method is carried out in particular on a stationary electrical machine, wherein, in particular, the rotor of the electrical machine is braked by a parking brake of the electric vehicle and, in particular, is held in a predetermined position by the locking positions of the parking brake of the electric vehicle.

[0021] To enable good and simple current control, the orientation of the rotor flux must be taken into particular consideration. Control is particularly simple when either the d- or q-direction of the rotor flux points towards the phase winding to be controlled in field-oriented coordinates. Depending on the machine design, it may be better to choose the q- or d-direction for power loss reasons. For example, in a permanent-magnet electrical machine, it can be particularly advantageous for the q-direction of the rotor flux to point towards the phase winding U in field-oriented coordinates. Various concept designs have shown that when a current is injected in the d-direction, large eddy current losses occur in the rotor magnets. In these studies, injecting a current in the q-direction resulted in the lowest rotor and machine losses. This is due to the fact that, unlike the rest of the rotor, the magnets are not laminated.Thus, the greatest losses occur when the magnetic flux is changed, for example by injecting a sinusoidal current into the stator through these magnets.

[0022] The method can also be applied in cases where the d- or q-direction is not precisely aligned with a phase winding. In this case, a sinusoidal current in field-oriented control must be adjusted at exactly the right angle to the rotor position, for example, in the q-direction, in order to generate the lowest possible losses in the rotor. However, this is only practical at low frequencies.

[0023] Since torque oscillations occur in the electric motor at the frequency of the battery's alternating current when currents are applied in the q-direction, locking the electric motor in the parking brake's detent position has the additional advantage of significantly reducing noise. If the electric motor were not locked in one position, the transmission would oscillate at the frequency of the phase current. Fixing the electric motor to specific positions is not critical, as the electric motor would only have to move a few degrees between two positions, which corresponds to the current behavior of parking brakes.

[0024] By correctly positioning the rotor relative to the stator and braking the electric machine in this position, the process can be implemented cost-effectively and particularly efficiently, as the losses occurring in the rotor can be minimized.

[0025] In one embodiment, the electric machine further comprises a rotor cooling system, and the heat dissipated by the rotor to the rotor cooling system is additionally used to heat the battery. This allows the losses incurred in the electric drive to be recycled and the battery to be heated even faster and more efficiently, thereby shortening the time until the charging process begins and making the battery heating particularly (energy-)efficient.

[0026] In one embodiment, when a temperature of the battery reaches or exceeds a temperature threshold, the battery is charged. A charging current can, for example, be supplied externally, e.g. via a charging port of an electric vehicle in which the battery is installed. In this case, the operation of the inverter is stopped. Preferably, however, the charging current is generated within the vehicle by recuperation operation of the drive arrangement. The temperature can, in particular, also be determined before the inverter is operated, and if the battery is already above the temperature threshold in this case, the battery can be charged directly or the battery can not be heated. This makes it possible to heat the battery only when the temperature of the battery requires it, whereby the method can be carried out particularly efficiently.

[0027] In one embodiment, the temperature threshold is at least 0°C, or at least 10°C, or at least 20°C, or at least 30°C, and / or at most 10°C, or at most 20°C, or at most 30°C, or at most 40°C. These temperature values ​​depend heavily on the cell chemistry used and the desired charging power. For example, during fast charging, especially at a power of more than 50 kW, a conventional lithium-ion battery should be above 20°C.

[0028] The minimum temperature required to charge the battery in a way that preserves its lifespan depends primarily on the battery chemistry. Generally, the battery temperature should be above 0°C. A higher temperature can prevent battery damage, especially during fast charging.

[0029] The implementation of a method according to the invention in the form of a computer program or computer program product with program code for carrying out all method steps is also advantageous, since this entails particularly low costs, in particular if an executing control unit is also used for other tasks and is therefore already present. Finally, a machine-readable storage medium is provided with a computer program stored thereon, as described above. Suitable storage media or data carriers for providing the computer program are, in particular, magnetic, optical, and electrical memories, such as hard disks, flash memories, EEPROMs, DVDs, and others. Downloading a program via computer networks (Internet, intranet, etc.) is also possible. Such a download can be wired or cable-based or wireless (e.g., via a WLAN network, a 3G, 4G, 5G, or 6G connection, etc.).In one embodiment, the computing unit comprises a phase modulator, which is in particular a space vector pulse width modulator, which is configured to determine a control signal for the electric machine. Furthermore, the computing unit can comprise a phase current controller configured to regulate at least one phase current, and a control unit configured to determine a control level.

[0030] In one embodiment, additional switching elements are provided in the vehicle electrical system. These switching elements are designed to disconnect consumers supplied by the vehicle electrical system from the vehicle electrical system during the battery heating process. Since the vehicle electrical system's alternating current generated for heating affects not only the battery but also other capacitors connected to the vehicle electrical system, such as the inverters of the air conditioning compressor or other traction inverters, e.g., in all-wheel drive vehicles, unwanted losses can occur in these components. These losses can be prevented by the additional switching elements, and the capacitors do not need to be recharged.

[0031] Further advantages and embodiments of the invention will become apparent from the description and the accompanying drawings.

[0032] The invention is illustrated schematically in the drawing using exemplary embodiments and is described below with reference to the drawing.

[0033] Short description of the drawings

[0034] Figure 1 shows a block diagram of an electric vehicle which is designed to carry out an embodiment of the method,

[0035] Figure 2 shows a block diagram of an embodiment of a computing unit configured to carry out an embodiment of the method, and

[0036] Figure 3 shows a flowchart of an embodiment of the method. Figure 4 shows exemplary current profiles in the electrical machine during implementation of an embodiment of the method.

[0037] Embodiments of the invention

[0038] Figure 1 shows a block diagram of an electric vehicle 4 configured to implement an embodiment of the method. Figure 2 shows a block diagram of an embodiment of a computing unit 6 configured to implement an embodiment of the method. Figure 3 shows a flowchart of an embodiment of the method. Figure 4 shows exemplary current profiles occurring in the electric machine. All figures are described together below.

[0039] The electric vehicle 4 shown in Figure 1 has a battery 1 and an electric machine 2 with a rotor and a stator. The battery 1 is in particular a lithium-ion battery that supplies the electric machine 2, for example a permanent magnet electric machine, with energy. The electric machine 2 is in particular the machine used to drive the electric vehicle 4, wherein the battery 1 is the traction battery and, in the present example, has exactly three phase windings II, V and W. The electric vehicle 4 further has an inverter 3, which is controlled by the computing unit 6 and supplies the electric machine 2 with energy from the battery 1 based on the control by the computing unit 6. The computing unit 6 can in particular be the inverter control unit or a so-called motor control unit (MCU).Battery 1, inverter 3 and electric machine 2 are part of a drive arrangement of the vehicle.

[0040] The vehicle 4 can, for example, also provide additional switching elements in the vehicle electrical system, which are configured to disconnect all or only certain additional consumers, in particular those with inductive and / or capacitive resistance, such as inverters of the air conditioning compressor or other traction inverters, from the vehicle electrical system during a heating process of the battery 1 described below. Furthermore, the electric vehicle 4 has, for example, a charging port 5, via which the battery 1 can be charged by connecting a corresponding charging cable.

[0041] The vehicle 4 further comprises a parking brake 7, by means of which the rotor of the electric machine 2 can be held, in particular in a predetermined position.

[0042] Furthermore, the electric machine 2 has a rotor cooling system 8, in particular in the form of a liquid cooling system, with which the rotor can be cooled. The rotor cooling system 8 can be connected, in particular via lines, to a heating or cooling system of the battery 1, whereby the battery 1 can be heated with the heat emitted by the rotor.

[0043] The computing unit 6 is shown in detail as a block diagram in Figure 2 and has in particular a phase current controller 6a, a control unit 6b and a phase modulator 6c.

[0044] In one embodiment of the method, which is shown as a flow chart in Figure 3, the temperature of the battery 1 is first determined (S100) and, if the temperature of the battery 1 corresponds to at least a temperature threshold value, the battery 1 can be charged directly (S150), for example by recuperation or via the charging connection 6. The temperature of the battery 1 can in particular be determined not only before the further method steps, but also while the inverter 3 is operated according to the embodiment described below, in order to terminate the operation of the inverter 3 as soon as the temperature threshold value is reached or exceeded.

[0045] If, however, the temperature of the battery 1 is below the temperature threshold, the inverter 3 is operated (S110) such that a multi-phase alternating current with different current amplitude values ​​flows between the inverter 3 and the at least three phase windings U, V, W of the electric machine 2. The inverter 3 is operated in particular such that a first alternating current is generated in one of the three phase windings U, V, W, and a second alternating current I2 is generated in each of the other three phase windings I1, V, W, wherein the two second alternating currents I2 are in total opposite to the first alternating current. In the case shown, the first alternating current is generated in the phase winding U and the second alternating current I2 is generated in the phase windings V and W, respectively.

[0046] The first alternating current I ±and the second alternating current I2 can have a wave-like, in particular sinusoidal, shape, wherein an amplitude of the second alternating current I2 is half as large as the amplitude of the first alternating current and the second alternating current I2 is at an angle of 180° or n to the first alternating current I ± is out of phase. This results in a total current of zero, and no rotating magnetic field is generated in the stator, thus not driving the rotor.

[0047] In the illustrated embodiments, at least one setpoint is specified for this purpose (S120). The setpoint is an amplitude of the first alternating current I1. In the present case, the frequency of the first alternating current I ± Based on the specified setpoints, a control signal A for controlling current switches of inverter 3 is determined (S130) so that the first alternating current I ±in one of the at least three phase windings II, V, W, in this case in phase winding II, with an amplitude corresponding to the desired amplitude, and the second alternating current I2 is generated in the other two of the three phase windings, in this case V and W, with an amplitude corresponding to half the amplitude of the first alternating current. The second alternating current I2 is thereby at an angle of 180° or n to the first alternating current I ± phase-shifted. Exemplary current waveforms are shown schematically in Figure 4, where the current waveform shown with a solid line shows the first alternating current with amplitude A, which flows, for example, in the phase winding U.

[0048] The current waveform shown with a dotted line shows the current waveform of the second alternating current I2 with the amplitude A / 2, ie half the amplitude of the first alternating current I1, which is phase-shifted by the angle n or 180° to the first alternating current and flows, for example, in the phase windings V and W.

[0049] If, for example, only the amplitude of the first alternating current ^ is specified, this can be output directly to the phase current modulator 6c of the computing unit 6, which determines the control signal A based on the amplitude.

[0050] The specified frequency target value is, in particular, at least 50 Hz and a maximum of 2400 Hz. For battery currents with a frequency of 100 Hz and above, no significant lifetime losses occur, so a frequency of approximately 100-150 Hz, or generally at least 100 Hz, is particularly advantageous. A frequency higher than 2400 Hz cannot be generated by conventional inverters with a PWM frequency of 20 kHz. The frequency target value should therefore be within the specified range and can assume any value within this range, e.g., 200 Hz, 500 Hz, 1000 Hz, or 2100 Hz.

[0051] Based on at least the specified target values ​​for the amplitude and frequency, a curve of the first alternating current I ± determined (S131). The course of the first alternating current I ± can be determined in particular by equation I ±= sin(27T • ft), where A is the setpoint amplitude and f is the setpoint frequency. Furthermore, based on the position of the rotor relative to the stator, it can be determined in which of the phase windings II, V or W the first alternating current is to be generated (S131a). The position of the rotor can be held, for example, by braking the electric machine 2. For this purpose, the electric vehicle 4 and thus also the electric machine 2 are at a standstill. In particular, detent positions of the parking brake 7 can be used to hold the rotor in a predetermined position. The position is determined in particular based on the direction of the rotor flux in the field-oriented coordinate system. It is particularly advantageous if the q-direction of the rotor flux points in the same direction as the direction of a stator field excited by the phase windings II, V or W.For example, the q-direction of the rotor flux may coincide with the direction of the stator flux excited by phase winding II. In this case, the first alternating current is I. ± for the phase winding U.

[0052] The course of the first alternating current determined in this way and the phase winding in which the first alternating current I ± is to be generated, is fed to the control unit 6a of the computing unit 6, which calculates the first alternating current I ± (S132) by determining the phase voltage U1.

[0053] Then, using the regulated first alternating current I ± The modulation level is determined in the control unit 6b (S133). For this purpose, the control unit 6b receives not only the phase voltage U1 as an input variable, but also the phase voltages U2, U3 of the other two phase windings and an intermediate circuit voltage U ZK. From this, the control unit 6b determines the control level, in particular by dividing the maximum amplitude of the phase voltages U lt U2, U3 and the intermediate circuit voltage U ZK .

[0054] This control level is passed on to the phase modulator 6c, which is in particular a space vector pulse width modulator, which uses it to determine the control signal A (S134).

[0055] Subsequently, the electric machine 2 is controlled by the inverter 3 using the control signal A (S140).

[0056] The described method allows the battery 1 to be heated in an energy-efficient manner, so that the charging current used to charge the battery 1 has little or no impact on the service life of the battery 1, thus extending it. Since the method is implemented using components that are standardly installed in electric vehicles 4, it is particularly cost-effective.

Claims

Claims 1. A method for heating a battery (1) of a drive arrangement, wherein the drive arrangement comprises the battery (1), an inverter (3) electrically connected to the battery and an electric machine (2) connected to the inverter (3) comprising a rotor and a stator with at least three phase windings, the method comprising: Operating (S110) the inverter (3) in such a way that a multi-phase alternating current with different current amplitude values ​​flows between the inverter (3) and the at least three phase windings (II, V, W) of the electrical machine (2).

2. The method according to claim 1, wherein the electrical machine has exactly three phase windings and the operation (S110) of the inverter (3) is carried out in such a way that a first alternating current ( ) is generated in one of the three phase windings (II, V, W), and a second alternating current ( / 2) is generated in each of the other three phase windings (U, V, W), the two second alternating currents ( / 2) being in total opposite to the first alternating current ( / .

3. The method according to claim 2, wherein the operation of the inverter (3) is carried out in such a way that the first alternating current ( ) and the second alternating current ( / 2) have a wave-like, in particular sinusoidal, shape, wherein an amplitude of the second alternating current ( / 2) corresponds to half the amplitude of the first alternating current ( ) and the second alternating current ( / 2) is phase-shifted by an angle of 180° to the first alternating current ( ).

4. The method according to any one of claims 2 or 3, wherein operating (S110) the inverter (3) comprises: Specifying (S120) at least one setpoint value which characterizes an amplitude of the first alternating current ( / , Determining (S130) a control signal (A) for controlling current switches of the inverter (3) such that the first alternating current ( / in one of the three phase windings (II, V, W) with an amplitude corresponding to the setpoint, and the second alternating current ( / 2) in each of the other three phase windings (II, V, W) with an amplitude corresponding to half the setpoint and with a phase shifted by 180° to the first alternating current ( / , Controlling (S140) the current switches of the inverter (3) using the control signal (A).

5. The method according to the preceding claim, further comprising: specifying (S120) a target value of a frequency of the first alternating current ( ), wherein the frequency of the first alternating current ( / in particular at least 50 Hz or at least 100 Hz or at least 150 Hz, and / or at most 150 Hz, or at most 500 Hz or at most 1000 Hz or at most 2000 Hz or at most 2400 Hz.

6. The method according to the preceding claim, wherein determining (S130) a control signal further comprises the following steps: Determining (S131) ​​a profile of the first alternating current ( / based on at least the setpoint value of the amplitude and the setpoint value of the frequency, determining (S134) the control signal (A) for controlling the current switches of the inverter (3) as a function of the profile of the first alternating current ( / .

7. The method according to the preceding claim, wherein determining (S131) ​​a profile of the first alternating current ( / further comprises: Determining (S131a) the one phase winding of the at least three phase windings (U, V, W) in which the first alternating current ( ) is to be generated, based on the position of the rotor relative to the stator.

8. Method according to one of the preceding claims, wherein the method is carried out on a stationary electrical machine (2), wherein in particular the rotor of the electrical machine (2) is braked via a parking brake (7) of a vehicle (4).

9. Method according to the preceding claim, wherein the rotor of the electric machine (2) is held in a predetermined position, in particular by locking positions of the parking brake (7) of the vehicle (4).

10. Method according to one of the preceding claims, wherein the electric machine (2) further comprises a rotor cooling system (8) and the heat given off by the rotor to the rotor cooling system is additionally used to heat the battery (1).

11. A method according to any one of the preceding claims, wherein the method further comprises: Determining (S100) a temperature of the battery (1) and, if the temperature of the battery (1) reaches or exceeds a temperature threshold, charging (S150) the battery (1).

12. The method according to claim 11, wherein the temperature threshold value is at least 0°C or at least 10°C or at least 20°C or at least 30°C and / or at most 10°C or at most 20°C or at most 30°C, depending on the battery type used.

13. Computing unit (6) configured to carry out the method according to one of the preceding claims.

14. A computing unit (6) according to claim 13, wherein the computing unit (6) comprises a phase modulator (6c), which is in particular a space vector pulse width modulator, which is configured to determine a control signal (A) for current switches of an inverter (3).

15. A computing unit (6) according to the preceding claim, wherein the computing unit (4) further comprises a phase current regulator (6a) which is configured to regulate the first alternating current ( ) of the one phase winding.

16. A computer program that causes a computing unit (6) to perform all method steps of a method according to one of claims 1 to 12 when executed on the computing unit.

17. A machine-readable storage medium having a computer program according to claim 16 stored thereon.

Images