Analysis of the Residential, Commercial and Industrial Electricity Consumption


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1 1 Analysis of the Residential, Commercial and Industrial Electricity Consumption John D. Hobby and Gabriel H. Tucci Abstract This work explains how to analyze the aggregate electricity consumption of many consumers, and extract key components such as heating, ventilation and air conditioning (HVAC), residential lighting, and street lighting consumption from the total consumption. To avoid explicit modeling of dependencies on time of day and on working versus nonworking days, leastsquares fitting for outside temperature and natural illumination dependency proceeds independently for each hour of the day. Cubic polynomials model dependencies on Steadman apparent temperature and on logscale illumination, but spline surfaces are best when considering these variables jointly. The primary focus is on residential consumption, but the same techniques can be used for studying street lighting, commercial and industrial consumption. Index Terms Smart Grid; Electricity Consumption; Smart Metering; RECS Survey; Time of Use Survey I. INTRODUCTION AND OVERVIEW An important property of the electricity grid is that production must be carefully matched to consumption in order to keep voltage and frequency stable and avoid damaging expensive infrastructure. On the other hand, customer activities, needs, and desires, as well as weather, shape the patterns of electricity use, which vary seasonally and hourly. These patterns typically result in high concentrations of electricity use in peak periods. The larger the peak demand, the greater the amount of electrical resources (distribution, transmission, and generation assets) that are needed to meet it. Hence, it is of fundamental importance to have an accurate model of the demand of electricity consumption. The unpredictable variations in consumption force suppliers to operate some power plants inefficiently at less than full capacity, e.g., as spinning reserve. The more accurately demand can be predicted, the more efficiently the plant can be operated. Furthermore, it helps to predict demand a day or more in advance because some power plants take that long to start up, and there are other aspects of power production and distribution that can benefit from such advance notice. For instance, coal plants require significantly more start up time than natural gas plants. Upcoming smart grid technology should allow producers to shape demand, but they will still need to know in advance what changes are desirable. Smart grid technology uses computers and advanced telecommunications for twoway communication between power consuming appliances and the electrical John D. Hobby and Gabriel H. Tucci are with Bell Labs, Alcatel Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, and grid. Devices can respond intelligently to price variations and alter their consumption accordingly. It may be difficult to get consumers (particularly residential consumers) to accept realtime electricity pricing. One alternative is dayahead pricing whereby suppliers announce prices about 24 hours in advance, so human consumers and smart grid devices have time to react. Producers clearly need to be able to predict future demand in order to make effective use of such a system. The 24 hour time frame (or perhaps a few days) is convenient because electricity consumption is known to be weather dependent, and future demand predictions require weather forecasts. Suppose we have historical consumption data for a mediumsized city or a suitable metropolitan area, and we want to analyze this in conjunction with historical weather data in order to understand total consumption and use weather forecasts to make future predictions. The historical consumption data presumably has separate categories for residential, commercial and industrial consumers, so their weather dependencies can be analyzed separately. The U.S. government s Residential Energy Consumption Survey [21] shows that heating and cooling are the largest components of residential electricity consumption and that large parts of the country have substantial consumption for electric heat. Hence, Section II begins by extracting the weatherrelated component from the total residential consumption. More specifically, given the hourly electricity consumption in a mediumsized American city, we obtain the hourly temperature and relative humidity of this city for each hour of the day. With these data, we compute the hourly electricity consumption due to HVAC (Heating, Ventilation and Air Conditioning). Since residential consumption for lighting depends on natural illumination, we also extract the outside illumination and compute the hourly electricity consumption due lighting. It is often useful to extract these components jointly via least squares surface fitting. In Section III we apply these techniques to analyze the street lighting consumption, and Section IV considers commercial and industrial consumption. Remaining components of consumption can be understood in terms of a collection of electrical appliances, but we reserve that subject for a longer version of this paper. Finally, Section V presents prediction results. The numerical results we show are based on a single case. We have the residential electricity consumption data for a US city with a population of approximately 100,000 people for 2007, 2008 and We use the consumption of 2007 as past consumption data, take the weather data for
2 as it is (we assume perfect weather data here), and predict the hourly consumption for the entire year of We do the same with 2008 to predict II. WEATHER RELATED AND ILLUMINATION ELECTRICITY CONSUMPTION COMPONENTS A. Weather Related Component In this Section, we present a method for separating the weather related electricity consumption from the total electricity consumption. More specifically, and as mentioned in the introduction, we have the hourly residential, commercial and industrial consumption data from a mediumsized American city for the years 2007, 2008 and We also have the hourly temperature and relative humidity for the same city. We would like to extract from this information, a function of time that tells how much of this electricity is going to cooling and heating. We shall refer to this function as the weatherrelated electricity consumption. The difference between it and the total consumption is the nonweatherrelated electricity consumption. Our analysis depends on apparent temperature, i.e., how hot or cold it feels to the human body. The apparent temperature combines the air temperature and the relative humidity in an attempt to determine the humanperceived equivalent temperature. The human body uses evaporative cooling, which is less effective when the relative humidity is high. After studying this and other issues, Steadman [1] developed a formula for apparent temperature that depends on the partial pressure of water vapor. For this partial pressure, it is convenient to use an approximation due to [2]: e = ( T T T 3) ( 5450 ) exp H, T where T is the temperature in Celsius and H the relative humidity (as a fraction 1). Then Steadman s formula for apparent temperature in degrees Celsius [1] can be defined as at := T e. (1) This formula generalizes the popular heat index which we do not use because it is only properly defined when the actual temperature is above 27 Celsius. Here is how we extract the weather related component: First localize the data for each hour of the day (also separating working from non working days), and consider the electricity consumption versus apparent temperature, using the Steadman equation as we described above. In other words, for each hour of the day, we can plot of the electricity consumption as a function of the apparent temperature. We interpolate these data with a cubic polynomial and shift it down so the minimum is zero. The shifted curve is our model for the weatherrelated electricity consumption for this particular hour. Figure 1 gives this plot for 20h. A few remarks are in order here. First, the appendix explains why shifting this curve down is a reasonable thing to do. Fig. 1. The weather related model for 20 hrs (8 p.m.) Fig. 2. The nonweather related electricity consumption versus hour of the day in the month of January (blue) and July (red). Heuristically, this is based on the assumption that there exists a comfortable temperature where no air conditioning or heating is necessary. Second, the reason for using a cubic polynomial is to make the model as simple as possible while capturing the essence of the problem. We feel that higher degree polynomials, splines, or more complicated curves are unnecessary. Thus, for each hour of the day, we have a cubic polynomial that gives the weatherrelated electricity consumption as a function of the apparent temperature. Figure 2 illustrates how this model works for the months of January and and July. For each hour of the day, the Figure gives the electricity consumption for every day of the month so that there are 31 points for each hour. As this Figure suggests, we have now found the weatherrelated (apparent temperature) component, and we can remove it if desired. B. Lighting Electricity Consumption Component In this Section, we extract the electricity consumption due to residential lighting. The idea is to do a similar analysis as for the extraction of the weather related component, and obtain a function of time that gives the electricity consumption due to lighting. The lighting component is all the electricity that goes to illumination, including inside illumination, backyards, front yards, garages and any other type of light. For each hour of the day, we plot electricity consumption versus the outside illumination due to natural causes. For example, Figure 3 shows this data for 7am. (Each dot represents a day in the year 2008). Again, we interpolate this data with a cubic polynomial, and we shift it down do zero. The reasoning here is as before
3 3 and is heuristically based on the assumption that there is an outside illumination where almost no lighting is necessary. How do we determine outside illumination due to natural causes? The local weather station reports cloud cover once an hour, and the number of cloud layers and their coverage 3 fractions (e.g., 8 to 1 2 for scattered ) give attenuation estimates for sunlight an moonlight. Nielsen s study of twilight effects [3] models the effect of sun (and moon) elevation angle. Estimates for maximum illumination levels for full sunlight, for a full moon, for starlight and air glow are widely available. Moonlight reduction versus the fraction of moon illuminated is available from [4], and sun and moon altitudes and lunar illumination fractions can be computed by Moshier s astronomical ephemeris calculator program [5]. Some of these effects are probably insignificant, but this is hard to know a priori. Once we have the outside illumination, we can get polynomials for each hour of the day. These polynomials model the electricity consumption due to lighting as a function of the outside illumination. This illumination consumption can also be plotted versus time; e.g., results for January and July appear in Figure 4, and in the same Figure we see the electricity consumption due to lighting for the whole year. The red curve shows the typical average, but it is clear that we can also extract the variance for each hour or even higher moments if necessary. This gives us a good model for the energy spent at each hour of the day due to lighting. The use of log(lux) in Figure 3 reflects the widespread belief that human perception of brightness grows less than linearly with actual illumination. Rudd and Popa [6] explain that a subjective measure of perceived brightness is roughly Lux 1/3 but hypothetical neural response is more like log(lux) and exponentiation to give Lux 1/3 may be due to the manner in which observers read out the signal. There is a term directly proportional to Lux, namely the greenhouse effect that reduces heating requirements and increases the need for cooling. Our analysis so far requires this to be captured implicitly via the fact that typical illumination levels are correlated with the hour of the day. After we remove the illuminationrelated electricity consumption from the nonweatherrelated one, we obtain the nonweather and nonillumination related electricity consumption as a function of time. Figure 6 shows that the minimum is between 1am and 4am. This value corresponds to the electricity consumption for timeinvariant processes (all day running processes) such as Freezers and Refrigerators Water Heaters Standby appliances and devices (devices which are not being used but are plugged in) C. Joint extraction of the weather and illumination related components Many applications require nonweather and nonillumination related electricity consumption with no need to distinguish Fig. 3. The nonweather related electricity consumption versus outside illumination for 7am. Fig. 4. The illumination electricity consumption as a function of time localized for January (blue) and July (red). Fig. 5. The illumination electricity consumption as a function of time for the whole year. Fig. 6. The nonweather related and nonillumination related electricity consumption as a function of time.
4 4 between the two. Joint treatment of weather and illumination is conceptually more accurate since these components could possibly be positively correlated. Both the weather and illumination components are very much dependent on the hour of the day. For this reason it is better to do different fittings for each hour separately as we explained in Section II and IIB. More specifically, we fix the hour of the day, and for each data point, we plot in three dimensional space the vectors with coordinates apparent temperature, illumination and consumption. Using the Matlab procedure gridfit.m we find the surface that best fits these data points. See Figure 7 to see the data set and the fitted surface for 18h. We see that the fitted surface behaves convexly in the temperature for a fixed illumination, and it is increasing in the illumination component (as it gets darker) for a fixed temperature. Except for possible greenhouse effect at high illumination levels, this is the expected behavior of the consumption as a function of these two variables. We shift this surface down so that its minimum is zero as we did before. The shifted surface will be our function model for the weatherrelated and illumination electricity consumption as a function of the apparent temperature and illumination. Subtracting this function from the total consumption gives us the nonweather related and nonillumination related electricity consumption for 18h. We repeat this procedure for each hour of the day to get a complete model. Note that, as before, for each hour of the day we have a slightly different model. Fig. 7. Data points and fitted surface for 18h. The coordinates are illumination, apparent temperature and electricity consumption. D. Different Surface Fitting Algorithms It can be awkward to use Matlab for a large application, particularly if other languages like C or C++ are heavily used. Furthermore, gridfit.m is not documented in a manner that facilitates modification or reimplementation. Hence we implemented a rectangular array of bicubic surface patches, using a Bspline representation that guarantees C 2 continuity and uses (m + 3)(n + 3) degrees of freedom for m n patches. Since m = 3 and n = 6 prove to be sufficient for the illumination and temperature (respectively), a vector v R 54 determines the surface, where each point on the surface is a weighted average of 16 entries of v. The least squares problem is to choose v so as to minimize Av b 2, where each temperature, illumination, consumption data point (x i, y i, z i ) generates one row of matrix A. It has 16 nonzeros for the weighted average of v entries that gives the surface height at (x i, y i ), with z i as the corresponding entry in vector b. There are also rows of A for regularization terms that penalize for high second derivatives 2 z and 2 z 2 y. For each patch corner 2 i, j, there are row vectors ξ i,j and η i,j in R 54 (each with 9 nonzeros) such that the values of 2 z and 2 z 2 y at i, j are ξ 2 i,j v and η i,j v. Regularization terms for second derivatives at the edges of the surface involve A rows σ x ξ i,0 and σ x ξ i,n σ y η 0,j and σ y η m,j for 0 i m for 0 j n where σ x and σ y are scale factors and the 2m + 2n + 4 corresponding b entries are zeros. The regularization terms for 2 z on patch i, j generate four 2 rows of A: σ x 16 (ξ i,j + ξ i+1,j + ξ i,j+1 + ξ i+1,j+1 ), σ x 48 (ξ i,j ± (ξ i+1,j ξ i,j+1 ) ξ i+1,j+1 ), σ x 144 (ξ i,j ξ i+1,j ξ i,j+1 + ξ i+1,j+1 ), (2) where σ x is a scale factor and the corresponding b entries are all zero. Replacing ξ i,j... ξ i+1,j+1 by η i,j... η i+1,j+1 and σ x by another scale factor σ y gives terms for 2 z z 2. The motivation for (2) is that we treat 2 z and 2 z 2 y as approximately bilinear, and then integrate this bilinear approximation 2 over the surface patch. We tried this method on 24 surfaces with σ x = σ y = 364/18 and σx = σ y = /(18 28), where 364 is the number of (x i, y i, z i ) data points. This gave slightly better fits than gridfit.m (RMS error ranging from 82.1% to 99.3% as much), even though gridfit.m used about 800 degrees of freedom per surface. This large dimensionality slows down the least squares solver it is a parameter that can easily be reduced, but at the cost of surface roughness. The C 2 bicubic splines remain smooth regardless of how many patches are used. III. STREET LIGHTING Electricity consumption that goes into street lighting is a different category that considers energy spent by a city to illuminate its streets. Repeating an analysis similar to that in Section IIB produces a model for each hour of the day that gives energy for street illumination as a function of the outside illumination. Figure 8 gives this for 6am. Putting all the pieces together as we did in Section IIB, we can see in Figure 9 also the typical energy spent on street illumination for each hour of the day. The red curve shows the typical average energy consumption.
5 5 Fig. 8. In this Figure we see the street lighting electricity consumption versus outside illumination for 6am. Fig. 10. Data taken from the U.S. Energy Information Administration  Official Energy Statistics from the U.S. Government. Fig. 9. In this Figure we see the street lighting consumption as a function of time for the whole year. We expect the variations due to weather to be less strong for the industrial and commercial sectors. To test this, we repeated the analysis done before for the commercial and industrial electricity consumption. Again, for each hour of the day we obtain a surface that gives the consumption as a function of the natural illumination and apparent temperature. Figure 11 shows this dependence for the industrial sector. A similar behavior is presented in the commercial sector. IV. COMMERCIAL AND INDUSTRIAL ELECTRICITY CONSUMPTION As we saw, the residential electricity consumption varies significantly with the weather. For example, it has been shown by the U.S. Energy Information Administration that an average home in the Pacific region (consisting of California, Oregon, and Washington) consumes 35% less energy than a home in the South Central region. Most of the regional differences can be explained by climate. The heavily populated coastal areas of the Pacific states experience generally mild winters and summers, reducing the need for both home heating and air conditioning. The warm, humid climates of the South Central and South Atlantic regions lead to higher electricity usage, while the cold winters experienced in the Northeast and North Central regions result in much higher consumption of natural gas and heating oil. Another reason for regional differences is the variety of building codes and environmental regulations found at the local and state level. California has some of the strictest environmental laws and building codes in the country, which may contribute to the fact that its perhousehold energy consumption is lower than all other states except Hawaii. Major U.S. cities also show significant variation in per capita energy consumption. In addition to differences in regional climates and variations in building code standards, factors affecting energy use in cities include population density and building design. Townhouses are more energy efficient than singlefamily homes because less heat, for example, is wasted per person. Fig. 11. Data points and shifted down fitted surface for 7h for the industrial sector. The coordinates are illumination, apparent temperature and industrial electricity consumption. It is also instructive to examine this city s total industrial and commercial KWh per month as shown in Figure 12. The market sectors that are less temperature dependent also show less seasonal variation. Note that industrial electricity consumption refers to the aggregation of the consumption of all industrial buildings in the city, and commercial consumption is defined analogously. V. REAL CASE PERFORMANCE TEST In this Section we test the performance of our weather and illumination algorithm. Given the hourly consumption
6 6 Fig. 12. Monthly consumption for the residential, commercial and industrial (units 10 7 W) and the apparent temperature in yellow (units C). cons (2007), apparent temperature and natural illumination from 2007 we obtain a model for the weather and illumination related consumption cwir (2007) as explained in Section IIC. Now using the hourly apparent temperature and natural illumination from 2008 we predict the weather and illumination related consumption for the year We denote this function by pcwir (2008). The difference between the total consumption and the weather and illumination related consumption is the residual for 2007 res (2007) = cons (2007) cwir (2007). We estimate the hourly total consumption for 2008 denoted by pcons (2008) as pcons (2008) = res (2007) + pcwir (2008) taking care to use corresponding days of the week from res (2007). We can repeat the same experiment to predict the hourly consumption of 2009 from the 2008 data. The 2008 real hourly consumption figures range from 68,343Kw to 679,134Kw with a median of 261,505Kw, and the 2009 figures are very similar with ranges going from 68,477Kw to 677,887Kw with a median of 256,860Kw. Taking the RMS mean of the prediction errors for each hour of the year gives values between of 44,003Kw for the 2008 prediction and 44,080Kw for the one of We can see in Figure 13 the real and predicted consumption for some period of time in 2008 (others periods are similar). In order to put the prediction errors into context, we would like to highlight that one needs to consider both the time granularity of the predictions and the prediction horizon, and the size of the population responsible for the electricity consumption. In the analysis we have shared with you, we predict the hourly consumption for an entire year. If one looks at more aggregate consumption levels like daily or weekly consumption, as we demonstrate below, the error terms improve significantly. The standard deviation of percentage prediction error 1 for hourly, daily and weekly consumption is 14.9%, 6.5%, and 4.1%, respectively. Another dimension is the population size. As the size of the population increases, we expect to have significantly lower prediction errors. The error statistics we have shown you are for a city of 100,000 people. If you consider predicting electricity consumption at an hourly basis for an entire country which has a population of 50 million, the errors one can expect with our method would be much lower. Suppose the population is divided into 500 regions with 100,000 people in each region, and consider using the numerical methods from this paper to predict hourly consumption for each region. If each region has errors like that for the US city, and if these errors are independent, then the standard deviation of percentage prediction error for hourly consumption would be 0.7%. Moreover, almost 89% of all hourly predictions would be at most 1% off the actual consumption. Finally, we would also like to mention that the RECS survey [21] shows that air conditioning and heating contributes to 31.5% of the total electricity consumption (for the geographical region where our middle sized city is located). Our model predicts that for the 2008 data the weather related electricity consumption is % of the total consumption which is incredible close to the real case. VI. CONCLUSIONS We have presented a new method for computing weatherrelated electricity consumption and illuminationrelated electricity consumption. More precisely, given the hourly total consumption, we obtain the hourly weatherrelated and illuminationrelated electricity consumption. Analysis of the residual consumption is reserved for future work. It has daily, weekly and seasonal variations, and it could also be analyzed in terms of electrical appliances. The twostage data processing model and algorithms presented in this work can be used to construct a tool for predicting shortterm residential electricity consumption. APPENDIX A JUSTIFICATION FOR SHIFTING FITTED CURVES Fig. 13. Real Consumption of 2008 (blue) versus the predicted one from 2007 (red). We have a more elaborate treatment for the predicted residual using spectral analysis techniques but is out of the scope of this paper and we will leave for a future work. Consider consumption as a random function of the hour h = 1,..., 24 and the temperature T so consumption is F (h, T ). Represent F (h, T ) as a sum F (h, T ) = C wr (h, T ) + C nwr (h, T ), 1 Percentage prediction error is defined as prediction  actual consumption actual consumption 100
7 7 where C wr (h, T ) is the portion of consumption due to HVAC, and C nwr (h, T ) is the rest of the consumption. Now choose an hour h 0. For any T, consider the expectation E(C nwr (h 0, T )) = c nwr (h 0, T ) of the random variable C nwr (h 0, T ). It is reasonable to assume that it does not depend on T. In other words, c nwr (h 0, T ) = c nwr (h 0 ). Then for each hour h 0 and temperature T, the expectation E(F (h 0, T )) of the random variable F (h 0, T ) is equal to E(F (h 0, T )) = c wr (h 0, T ) + c nwr (h 0 ). Our main assumption for the shifting down procedure is that for every hour h 0, there exists a comfort temperature T min (h 0 ) such that c wr (h 0, T min (h 0 )) = 0, and thus no electricity is consumed for heating and/or air conditioning. Approximating E(F (h 0, T )) by a polynomial P (h 0, T ) in T with coefficients depending on h 0 gives [15] Tae Young Jung, Ordered Logit model for residential electricity demand in Korea, Energy Economics, pp , [16] Tsuji K., Sano F., Ueno T., and Saeki O., Bottomup Simulation Model for Estimating EndUse Energy Demand profiles in Residential Houses, Proc. of ACEEE Summer Study on Energy Efficiency in Buildings, California, ISBN , [17] Jessen Page, Darren Robinson and JeanLouis Scartezzini, Stochastic Simulations of Occupant Presence and Behaviour in Buildings, Proceedings: Building Simulation, [18] Yoshiyuki Shimoda, Takahiro Asahi, Ayako Taniguchi, Minoru Mizuno, Evaluation of cityscale impact of residential energy conservation measures using the detailed enduse simulation model, Energy, 32:9, pp , [19] American Time Use Survey, [20] Minkowski addition, addition [21] Residential Energy Consumption Survey, (2005 data), BIOGRAPHIES c wr (h 0, T ) = P (h 0, T ) min(p (h 0, T )) = P (h 0, T ) P (h 0, T c (h 0 )). ACKNOWLEDGMENT We would like to thank Yuliy Baryshnikov and Mustafa Dogru for many valuable discussions and comments. This work was partially funded by the Gachon Energy Research Institute in South Korea. REFERENCES [1] Robert G. Steadman, A Universal Scale of Apparent Temperature, J. of Applied Meteorology, vol. 23 no. 12, pp , [2] Kennneth G. Libbrecht, Properties of Ice, atomic/snowcrystals/ice/ice.htm. [3] Eric Tetens Nielsen, Illumination at Twilight, Oikos, vol. 14, no. 1, pp. 9 21, [4] C. D. Courter, How Bright is Moonlight?, kitathome/lunarlight/moonlight/gallery/technique/moonbright.htm, [5] Stephen L. Moshier, Selfcontained Ephemeris Calculator, [6] Michael E Rudd, Dorin Popa, Steven s Brightness Law, Contrast Gain Control and Edge Integration in Achromatic Color Perception: A Unified Model, J. Opt Soc. Am, Vol 23, No 9, pp , [7] A. Capasso, W. Grattieri, A. Prudenzi, A bottomup approach to residential load modeling, IEEE Transactions on Power Systems, 9:2, pp , May [8] Jukka V. Paatero and Peter D. Lund, A model for generating household electricity load profiles, International Journal of Energy Research, vol. 30, no. 5, pp , [9] Jeffrey A. Dubin and Daniel L. McFadden, An Econometric Analysis of Residential Electric Appliance Holdings and Consumption, Econometrica, vol. 52, no. 2, pp , [10] Anssi Seppälä, Load research and load estimation in electricity distribution, VTT Energy, [11] Peter C. Reiss, Matthew W. White, Household Electricity Demand Revisited, Working Paper 8687, [12] Michael Parti, Cyntia Parti, The total and appliancespecific conditional demand for electricity in the household sector, Bell Journal of Economics, 11:1, pp , [13] Bodil Merethe Larsen, Runa Nesbakken, Household electricity consumption for different end uses, [14] Bodil Merethe Larsen, Runa Nesbakken, Household electricity enduse consumption: results from econometric and engineering models, Energy Economics, 26:2, pp , John D. Hobby received his B.S. in Mathematics and Computer Science from the University of Washington in 1980, and his Ph.D. in Computer Science from Stanford University in Since then, he has been at Bell Labs (now part of Alcatel Lucent) in Murray Hill, New Jersey, and he is now a Distinguished Member of Technical Staff. His interests include graphics algorithms, computational geometry, document image analysis, scientific computing, and modeling and optimization, especially optmizing wireless systems. Gabriel H. Tucci was born in Montevideo, Uruguay. He received a B.Sc. in Mathematics and an Electrical Engineering Degree (M.Sc. equivalent) from the Universidad de la República in Uruguay in 2002 and 2003 respectively. He also received a Ph.D. in Mathematics from Texas A&M University in He later joined Bell Laboratories, AlcatelLucent in Murray Hill, where he is a member of Technical Staff in the Industrial Mathematics and Operations Research department. His interests are in analysis, probability, free probability, random matrices, hyperbolic geometry and their applications to wireless communications, coding theory, information theory, compress sensing and network optimization.
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