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Modeling and Feedback Compensator Design for Power-Hardware-in-the-Loop System for Medium-Voltage Grid-Connected Power Converters
Hardware-in-the-loop (HIL) evaluation is a method in which the test subject is split into a physical part and a simulated part, and these parts are connected with interfaces to form a combined physical-numerical system. Power-hardware-in-the-loop (PHIL) systems, where actual hardware is connected to a real-time model, can exhibit issues of instability, inaccuracy and errors when operating in a closed-loop. The challenges of generating a PHIL setup are that, first, because of the limited dynamic response of the different parts of the system, the test results might be inaccurate, and, second, because of the high frequency noise introduced by the sensors to the closed-loop system, it can be difficult to design a compensator for the real-time emulator response, while stabilizing the closed-loop system at the same time. In this paper, different parts of a PHIL system are characterized, and the feedback compensator system design is proposed for the stable operation of the closed-loop PHIL system. The issues as observed in a PHIL system without any compensator are demonstrated using experimental results, and the effectiveness of a first order phase lead compensator is validated.
Thermionic converters and generators for space application. I - Thermionic converter performance and material problems.
Improving thermionic power conversion devices through various material design configurations
Converter-Interfaced CHP Plant for Improved Grid-Integration, Flexibility and Resiliency
GE Research and its partner GE Renewables have proposed the use of an interface converter solution to increase the penetration of small to medium-sized CHP (1MWe to 20MWe) into distribution grids and improve their flexibility and grid support capability. Indeed, the proposed interface converter solution thanks to presence of the grid-ready inverter, allows to streamline the compliance to grid codes requirements, reduce the interconnection delays and costs and ultimately one of the main barriers for CHP adoption by commercial and industrial facilities. An additional benefit provided by the interface converter is the use of the grid-ready inverter for reactive power which eliminates the need of sizing the generator for that capability. These two benefits highly favor the economic feasibility of converter-interfaced CHP. Five user cases, each in one of the leading U.S states for CHP potential reported by the DOE in its estimation of the U.S Technical Potential of CHP, were selected to compare the economic performances of converter-interfaced CHP as compared with directly-coupled. They include a college campus in California, a hospital in New York, a water reclamation plant in Texas, a hotel in Minnesota, and a large office building in Pennsylvania. Results showed that, the presence of the interface converter allows to increase the return on investment (ROI) by 0.5 to 2 percentage points in most of the cases (4 of 5). Indeed, the interface converter by shortening the interconnection process allows to accelerate revenues while reducing interconnection costs. Added to the reduced cost of the required generator these savings trade favorably the capital cost of the converter. The analysis also showed that the profitability of the converter-interfaced CHP is highly sensitive to the energy price, interconnection delay, and converter cost. However, it appears that if the interface converter can shorten the interconnection process by at least 6 months, adopting this solution will be more economically viable than directly-coupled configuration in almost all the +23,000 sites of the U.S Technical Potential CHP. The evaluation of the benefits of a converter-interfaced CHP also showed that it enables higher ROI when coupled with other distributed energy resources (DER) such as battery energy systems (BESS) or solar photovoltaic (PV). Indeed, in those scenario, the grid-ready inverter included in the interface converter eliminates the need of separate inverters if DC-coupling is used. On the technical performance, it has been verified that the presence of the interface converter allows to reduce by 70% to 80% the CHP short-circuit contribution to grid faults. This not only reduces the mechanical and thermal stresses exposed to the CHP electrical components but also increases the grid hosting capacity which ultimately enables higher penetrations CHP. Another key benefit of the interface converter validated with hardware-in-the-loop simulations and testing is its superior capability for reactive power support. Indeed, using a power hardware testbed with two +700kW inverters configured in back-to-back, a microgrid controller and actual facilities loads it was demonstrated that the presence of the interface converter can help maintain a power factor near ~1 or regulate the voltage to ~1.0pu at the point of common coupling. This benefit can be highly valuable if in the future, due to higher penetration of renewable distributed energy resources (DER), utilities start billing demand charge based on kVA instead of kW as currently. It was also validated that converter-interfaced CHP can dispatch heat and power commands and seamlessly switch between the two modes while consistently controlling the power factor or voltage at PCC. Indeed, the power hardware testing showed that grid-connected converter-interfaced CHP can follow either the power or heat demand while maintaining a unity power factor at converter output. This research proved that the adoption of an interface converter as the solution for interconnection of CHP system into the distribution grid can greatly improve the economic feasibility of small to medium-sized CHP as well as the plant power quality, flexibility and resiliency. Additionally, it allows increased penetrations of CHP into the distribution grid, extends their grid support capability, and facilitates the integration of BESS and PV DER by streamlining their collocation within the same facilities. This ultimately provides an opportunity for commercial and small industrial facilities in the U.S to accelerate their energy transition thanks to the high energy efficiency of CHP systems and its reliable, flexible, and resilient microgrid operation when interconnected with an interface converter.
Radiation effects on DC-DC Converters
DC-DC switching converters are circuits that can be used to convert a DC voltage of one value to another by switching action. They are increasing being used in space systems. Most of the popular DC-DC switching converters utilize power MOSFETs. However power MOSFETs, when subjected to radiation, are susceptible to degradation of device characteristics or catastrophic failure. This work focuses on the effects of total ionizing dose on converter performance. Four fundamental switching converters (buck converter, buck-boost converter, cuk converter, and flyback converter) were built using Harris IRF250 power MOSFETs. These converters were designed for converting an input of 60 volts to an output of about 12 volts with a switching frequency of 100 kHz. The four converters were irradiated with a Co-60 gamma source at dose rate of 217 rad/min. The performances of the four converters were examined during the exposure to the radiation. The experimental results show that the output voltage of the converters increases as total dose increases. However, the increases of the output voltage were different for the four different converters, with the buck converter and cuk converter the highest and the flyback converter the lowest. We observed significant increases in output voltage for cuk converter at a total dose of 24 krad (si).
Radiation Effects on DC-DC Converters
In this work, several DC-DC converters were designed and built. The converters are Buck Buck-Boost, Cuk, Flyback, and full-bridge zero-voltage switched. The total ionizing dose radiation and single event effects on the converters were investigated. The experimental results for the TID effects tests show that the voltages of the Buck Buck-Boost, Cuk, and Flyback converters increase as total dose increased when using power MOSFET IRF250 as a switching transistor. The change in output voltage with total dose is highest for the Buck converter and the lowest for Flyback converter. The trend of increase in output voltages with total dose in the present work agrees with those of the literature. The trends of the experimental results also agree with those obtained from PSPICE simulation. For the full-bridge zero-voltage switch converter, it was observed that the dc-dc converter with IRF250 power MOSFET did not show a significant change of output voltage with total dose. In addition, for the dc-dc converter with FSF254R4 radiation-hardened power MOSFET, the output voltage did not change significantly with total dose. The experimental results were confirmed by PSPICE simulation that showed that FB-ZVS converter with IRF250 power MOSFET's was not affected with the increase in total ionizing dose. Single Event Effects (SEE) radiation tests were performed on FB-ZVS converters. It was observed that the FB-ZVS converter with the IRF250 power MOSFET, when the device was irradiated with Krypton ion with ion-energy of 150 MeV and LET of 41.3 MeV-square cm/mg, the output voltage increased with the increase in fluence. However, for Krypton with ion-energy of 600 MeV and LET of 33.65 MeV-square cm/mg, and two out of four transistors of the converter were permanently damaged. The dc-dc converter with FSF254R4 radiation hardened power MOSFET's did not show significant change at the output voltage with fluence while being irradiated by Krypton with ion energy of 1.20 GeV and LET of 25.97 MeV-square cm/mg. This might be due to fact that the device is radiation hardened.
Assessment of Stirling Technology Has Provided Critical Data Leading Toward Flight Readiness of the Stirling Converter
The NASA Glenn Research Center is supporting the development of a Stirling converter with the Department of Energy (DOE, Germantown, Maryland) for an advanced Stirling Radioisotope Power System (SRPS) to provide spacecraft onboard electric power for NASA space science missions. A key technology assessment completed by Glenn and DOE has led to the SRPS being identified as a high-efficiency power source for such deep space missions as the Europa Orbiter and the Solar Probe. In addition, the Stirling system is now being considered for unmanned Mars rovers, especially where mission profiles may exclude the use of photovoltaic power systems, such as exploration at high Martian latitudes or for missions of long duration. The SRPS efficiency of over 20 percent will reduce the required amount of radioisotope by more than a factor of 3 in comparison to current radioisotope thermoelectric generators. This significantly reduces radioisotope cost, radiological inventory, and system cost, and it provides efficient use of scarce radioisotope resources. In support of this technology assessment, Glenn conducted a series of independent evaluations and tests to determine the technology readiness of a 55-We Stirling converter developed by Stirling Technology Company (Kennewick, Washington) and DOE. Key areas evaluated by Glenn included: 1) Radiation tolerance of materials; 2) Random vibration testing of the Stirling converter in Glenn's Structural Dynamics Lab to simulate operation in the launch environment; 3) Electromagnetic interference and compatibility (EMI/EMC) of the converter operating in Glenn's EMI lab; Independent failure modes, effects, and criticality analysis, and life and reliability 4. Independent failure modes, effects, and criticality analysis, and life and reliability assessment; and 5) SRPS cost estimate. The data from these evaluations were presented to NASA Headquarters and the Jet Propulsion Laboratory mission office by a joint industry/Government team consisting of DOE, Glenn, and Lockheed Martin Astronautics. This team concluded that there are no technical reasons that would rule out using the Stirling converter for deep space missions. As a direct result of the successful testing at Glenn, the DOE/Stirling Technology Company 55-We Stirling converter has been baselined for the SRPS. Glenn is now continuing an in-house project to assist in developing the Stirling converter for readiness for space qualification and mission implementation. As part of this effort, the Stirling converter will be further characterized under launch environment random vibration testing, methods to reduce converter EMI will be developed, and an independent performance verification will be completed. Converter life assessment and permanent magnet aging characterization tasks are also underway. Substitute organic materials for the linear alternator and piston bearing coatings for use in a high-radiation environment have been identified and have now been incorporated in Stirling converters built by Stirling Technology Company for Glenn. Electromagnetic and thermal finite element analyses for the alternator are also being conducted.
Distributed Embedded Energy Converter Technologies for Marine Renewable Energy (A Technical Report)
The domain of distributed embedded energy converter technologies (DEEC-Tec) is a nascent and underexplored paradigm for harvesting and converting marine renewable energy. The paradigm distinguishes itself through its use of many small distributed embedded energy converters (DEECs) that, ultimately, are assembled through the creation of "DEEC-Tec metamaterials" to create an overall larger marine renewable energy harvesting and converting structure. As an example, such a structure could be an ocean wave energy converter - a converter whose structure is made from various types of DEEC-Tec metamaterials that harvests ocean wave energy and converts that energy into something more useful such as electricity. To that end, DEEC-Tec can be viewed at three different technology levels: (1) individual distributed embedded energy converters, also known as DEECs; (2) DEEC-Tec metamaterials-essentially, pseudo-materials made from the interconnection of many DEECs; and (3) overall larger complete marine renewable energy harvesting-converting structures-these structures being made from DEEC-Tec metamaterials. Arising directly from the application of DEEC-Tec to harvest and convert ocean wave energy are several noteworthy benefits, some of which include: (1) the lack of load concentrations into singular components or subsystems, (2) broad-banded ocean wave energy frequency harvesting and conversion, and (3) inherent redundancy-failure of some individual DEECs does not represent a failure of an entire DEEC-Tec-based WEC. This report describes DEEC-Tec by way of descriptions of those three technology levels: individual DEECs, DEEC-Tec metamaterials, and DEEC-Tec-based WECs. Moreover, the report describes corresponding research approaches and methodologies for related concepts such as DEEC-Tec-based WEC topologies and morphologies in addition to manufacturing and fabrication techniques found suitable for the application of DEEC-Tec within the general domain of marine renewable energy-moving beyond only ocean wave energy conversion.
Electrical performance characteristics of high power converters for space power applications
The first goal of this project was to investigate various converters that would be suitable for processing electric power derived from a nuclear reactor. The implementation is indicated of a 20 kHz system that includes a source converter, a ballast converter, and a fixed frequency converter for generating the 20 kHz output. This system can be converted to dc simply by removing the fixed frequency converter. This present study emphasized the design and testing of the source and ballast converters. A push-pull current-fed (PPCF) design was selected for the source converter, and a 2.7 kW version of this was implemented using three 900 watt modules in parallel. The characteristic equation for two converters in parallel was derived, but this analysis did not yield any experimental methods for measuring relative stability. The three source modules were first tested individually and then in parallel as a 2.7 kW system. All tests proved to be satisfactory; the system was stable; efficiency and regulation were acceptable; and the system was fault tolerant. The design of a ballast-load converter, which was operated as a shunt regulator, was investigated. The proposed power circuit is suitable for use with BJTs because proportional base drive is easily implemented. A control circuit which minimizes switching frequency ripple and automatically bypasses a faulty shunt section was developed. A nonlinear state-space-averaged model of the shunt regulator was developed and shown to produce an accurate incremental (small-signal) dynamic model, even though the usual state-space-averaging assumptions were not met. The nonlinear model was also shown to be useful for large-signal dynamic simulation using PSpice.
Cascaded resonant bridge converters
A converter for converting a low voltage direct current power source to a higher voltage, high frequency alternating current output for use in an electrical system where it is desired to use low weight cables and other circuit elements. The converter has a first stage series resonant (Schwarz) converter which converts the direct current power source to an alternating current by means of switching elements that are operated by a variable frequency voltage regulator, a transformer to step up the voltage of the alternating current, and a rectifier bridge to convert the alternating current to a direct current first stage output. The converter further has a second stage series resonant (Schwarz) converter which is connected in series to the first stage converter to receive its direct current output and convert it to a second stage high frequency alternating current output by means of switching elements that are operated by a fixed frequency oscillator. The voltage of the second stage output is controlled at a relatively constant value by controlling the first stage output voltage, which is accomplished by controlling the frequency of the first stage variable frequency voltage controller in response to second stage voltage. Fault tolerance in the event of a load short circuit is provided by making the operation of the first stage variable frequency voltage controller responsive to first and second stage current limiting devices. The second stage output is connected to a rectifier bridge whose output is connected to the input of the second stage to provide good regulation of output voltage wave form at low system loads.
Early Oscillation Detection Technique for Hybrid DC/DC Converters
Oscillation or instability is a situation that must be avoided for reliable hybrid DC/DC converters. A real-time electronics measurement technique was developed to detect catastrophic oscillations at early stages for hybrid DC/DC converters. It is capable of identifying low-level oscillation and determining the degree of the oscillation at a unique frequency for every individual model of the converters without disturbing their normal operations. This technique is specially developed for space-used hybrid DC/DC converters, but it is also suitable for most of commercial and military switching-mode power supplies. This is a weak-electronic-signal detection technique to detect hybrid DC/DC converter oscillation presented as a specific noise signal at power input pins. It is based on principles of feedback control loop oscillation and RF signal modulations, and is realized by using signal power spectral analysis. On the power spectrum, a channel power amplitude at characteristic frequency (CPcf) and a channel power amplitude at switching frequency (CPsw) are chosen as oscillation level indicators. If the converter is stable, the CPcf is a very small pulse and the CPsw is a larger, clear, single pulse. At early stage of oscillation, the CPcf increases to a certain level and the CPsw shows a small pair of sideband pulses around it. If the converter oscillates, the CPcf reaches to a higher level and the CPsw shows more high-level sideband pulses. A comprehensive stability index (CSI) is adopted as a quantitative measure to accurately assign a degree of stability to a specific DC/DC converter. The CSI is a ratio of normal and abnormal power spectral density, and can be calculated using specified and measured CPcf and CPsw data. The novel and unique feature of this technique is the use of power channel amplitudes at characteristic frequency and switching frequency to evaluate stability and identify oscillations at an early stage without interfering with a DC/DC converter s normal operation. This technique eliminates the probing problem of a gain/phase margin method by connecting the power input to a spectral analyzer. Therefore, it is able to evaluate stability for all kinds of hybrid DC/DC converters with or without remote sense pins, and is suitable for real-time and in-circuit testing. This frequency-domain technique is more sensitive to detect oscillation at early stage than the time-domain method using an oscilloscope.
Sizing and Location Selection of Medium‐Voltage Back‐to‐Back Converters for DER‐Dominated Distribution Systems
Medium‐voltage back‐to‐back (MVB2B) converters can connect two distribution systems and quantifiably transfer power between them. This function can enable the MVB2B converter to exchange distributed energy resource (DER)‐generated power between two systems and bring significant value to enhancing distribution system DER adoption. Our previous work analysed and demonstrated the value MVB2B converter can bring to DER integration. As continuous work, this paper presents a methodology that helps address the MVB2B converter sizing and location selection problem in distribution systems with high DER penetrations. The proposed methodology aims to address three critical problems for MVB2B converter implementation in the real world: (1) which distribution systems are better to be connected, (2) what converter size is appropriate for connecting the distribution systems, and (3) where the optimal connection points are in the systems for connecting the MVB2B converter. The proposed methodology has been demonstrated by case studies that include various scenarios involving distribution systems with different dominated load types and high photovoltaic penetrations. The results demonstrate that selecting the optimal converter size based on net revenue and time of return considerations leads to a balance between maximizing energy savings and minimizing financial payback periods. Furthermore, feeder pair selection based on load profile standard deviation effectively identifies systems that derive the greatest value from MVB2B integration. Finally, an optimized connection point selection approach using a voltage load sensitivity matrix ensures minimal system impact while facilitating efficient power exchange. These findings provide practical insights for the real‐world deployment of MVB2B converters to enhance DER hosting capacity and improve grid resilience.
Smart Adaptive Structures for an Ocean Wave Energy Converter
Ocean wave energy converters face significant challenges including cost-effectiveness, minimizing maintenance requirements, and withstanding extreme conditions. However, by utilizing smart materials, these converters could overcome these challenges. Such energy harvesters could use dielectric elastomer generators to convert ocean wave energy into electricity through their dynamic straining. Conversely, by applying electricity to these generators, they become actuators - dielectric elastomer actuators - thereby enabling them to alter their stiffness and adapt to the ever-changing ocean energy environments. Such active adaptation could enhance the converter's ability to: reach resonance with ocean waves and protect itself from dangerous waves. This study utilizes numerical analyses through the COMSOL software framework to evaluate the potential energy that could be harvested by a conceptual ocean wave energy converter based upon dielectric elastomer generators/actuators. The converter is composed of an external hull (that is a hollow cylinder), an inertial mass (that is a hollow cylinder and concentric with the hull), and 'spokes' - made of dielectric elastomer generators/actuators - that connect the hull to the inertial mass. Results of the numerical analyses include those outcomes arising from the conceptual converter being simulated via a sinusoidal motion analogous to ocean waves. That motion, therefore, causes relative motion between the converter's hull and inertial mass thereby dynamically stretching the corresponding connecting elastomers. The stretching of the elastomers enables them to 'gain elastic strain energy' and is, therefore, considered to be the theoretical limit of possible electrical energy conversion for the dielectric elastomer generator/actuator spokes. Additionally, the elastomer material properties of the spokes were altered to simulate the actuation of those same elastomers; with overall strain energy being subsequently investigated. Ultimately this is a preliminary study exploring the ability of such smart materials - electricity-generating and self-actuating elastomers - to actively adapt an ocean wave energy converter's structure to address and overcome the aforementioned challenges.
Angular-spectral filtering of recoil protons for optimization of fast neutron imaging employing proton converters
Fast neutron imaging is an important capability for diverse applications such as inertial confinement fusion diagnostics, cargo security, nuclear nonproliferation and arms control, and industrial inspection. Traditional phosphor image plates can be enhanced for fast neutron imaging using hydrogenous plastic converters which allow fast neutrons to scatter off hydrogen nuclei to produce energetic protons that can be recorded by the image plate. However, protons emitted by image plates are not constrained in their emission angle, which contributes to the blur of the resulting image. Here, we investigate two methods that can alter the spatial extent of converted protons that deposit energy in the image plate: reducing the converter thickness, and introducing a proton filter between the plastic converter and image plate to reduce the contribution of lower-energy, off-axis protons to the image. Here we determine the optimal plastic converter thickness for maximizing the signal intensity to be 2–3 mm through Monte Carlo simulations, and we benchmark this result against experimental measurements with a deuterium-tritium (DT) neutron generator. Next, we evaluate the image smearing and signal loss for various converters to show that solely reducing the converter thickness has the expected effect of reducing the blur from proton image smearing of the sharpness of an edge recorded on the image plate at the cost of reducing the signal intensity. The use of a proton filter is shown to achieve a similar improvement of edge sharpness as reducing the converter thickness while also sacrificing the signal intensity. We conclude that the use of proton energy filtering can improve the sharpness of fast neutron images in situations where the converter thickness cannot be reduced below some practical minimum. For more intense neutron sources, the signal intensity is of less concern, and optimizing the resolution of the image plate and therefore of the imaging system could have greater value. In these applications, proton filters may allow for improved fast neutron imaging measurements.
Design and Demonstration of an 850 V dc to 13.8 kV ac 100 kW Three-phase Four-wire Power Conditioning System Converter Using 10 kV SiC MOSFETs
In this paper, an 850 V dc to 13.8 kV ac 100 kW modular multilevel three-phase four-wire dc/ac converter based on 10 kV SiC MOSFETs is designed and demonstrated. The design considerations of key components, including the dc-link, device cooling, gate driver, isolated gate driver power supply (GDPS), medium voltage (MV) ac filter inductor, MV and medium frequency transformer, and the mechanical design are discussed. Two converters are designed, and two prototypes are developed, to study the converter paralleling operation and scalability. Both converters are fully tested up to their voltage and power ratings. However, the two converters are not identical. Based on the design and test results of the first converter, the MV power stage, transformer design, GDPS, as well as the low voltage power stage in the second converter are improved for smaller size and/or higher efficiency. Compared to the version 1 converter, the version 2 converter achieves 49% volume reduction and 2 percentage point efficiency improvement, with a peak efficiency of 98.4% at the rated power.
Study of switching transients in high frequency converters
As the semiconductor technologies progress rapidly, the power densities and switching frequencies of many power devices are improved. With the existing technology, high frequency power systems become possible. Use of such a system is advantageous in many aspects. A high frequency ac source is used as the direct input to an ac/ac pulse-density-modulation (PDM) converter. This converter is a new concept which employs zero voltage switching techniques. However, the development of this converter is still in its infancy stage. There are problems associated with this converter such as a high on-voltage drop, switching transients, and zero-crossing detecting. Considering these problems, the switching speed and power handling capabilities of the MOS-Controlled Thyristor (MCT) makes the device the most promising candidate for this application. A complete insight of component considerations for building an ac/ac PDM converter for a high frequency power system is addressed. A power device review is first presented. The ac/ac PDM converter requires switches that can conduct bi-directional current and block bi-directional voltage. These bi-directional switches can be constructed using existing power devices. Different bi-directional switches for the converter are investigated. Detailed experimental studies of the characteristics of the MCT under hard switching and zero-voltage switching are also presented. One disadvantage of an ac/ac converter is that turn-on and turn-off of the switches has to be completed instantaneously when the ac source is at zero voltage. Otherwise shoot-through current or voltage spikes can occur which can be hazardous to the devices. In order for the devices to switch softly in the safe operating area even under non-ideal cases, a unique snubber circuit is used in each bi-directional switch. Detailed theory and experimental results for circuits using these snubbers are presented. A current regulated ac/ac PDM converter built using MCT's and IGBT's is evaluated.
Architecture for a High-to-Medium-Voltage Power Converter
A power converter now undergoing development is required to operate at a DC input potential ranging between 5.5 and 10 kV and a DC output potential of 400 V at a current up to 25 A. This power converter is also required to be sufficiently compact and reliable to fit and operate within the confines of a high-pressure case to be lowered to several miles (approx.5 km) below the surface of the ocean. The architecture chosen to satisfy these requirements calls for a series/ parallel arrangement of 48 high-frequency, pulse-width-modulation (PWM), transformer-isolation DC-to-DC power converter blocks. The input sides of the converter blocks would be connected in series so that the input potential would be divided among them, each of them being exposed to an input potential of no more than 10 kV/48 . 210 V. The series connection of inputs would also enforce a requirement that all the converter blocks operate at the same input current. The outputs of the converter blocks would be connected in a matrix comprising 6 parallel legs, each leg being a cascade of eight outputs wired in series (see figure). All the converter blocks would be identical within the tolerances of the values of their components. A single voltage feedback loop would regulate the output potential. All the converter blocks would be driven by the same PWM waveform generated by this feedback loop. The power transformer of each converter block would have a unity turns ratio and would be capable of withstanding as much as 10 kVDC between its primary and secondary windings. (Although, in general, the turns ratio could be different from unity, the simplest construction for minimizing leakage and maximizing breakdown voltage is attained at a turns ratio of unity.)