>> OUR NEXT TALK TODAY [INAUDIBLE] WE SHOULD HAVE PUT HIM IN THE MIDDLE. DANIEL BIENSTOCK.

>> GOOD EVENING. [INAUDIBLE] I WOULD LIKE TO THANK THE ORGANIZERS FOR HAVING ME. THIS IS GOING FORTH WITH A COUPLE OF GUYS FROM LOS ALAMOS. THE TALK IS IN THREE PARTS. THE FIRST PART I WILL GO THROUGH BRIEFLY, AND IT OVERLAPS WITH WHAT I TALKED ABOUT LAST YEAR. THERE WAS THIS CONFERENCE THAT BROUGHT A LOT OF ATTENTION TO LARGE FLUCTUATIONS IN WIND, AND THE IMPACT ON THE TRANSMISSION SYSTEM. THE OBSERVATIONS THAT MANY PEOPLE ARE PROBABLY FAMILIAR WITH, POWER DEVIATIONS AND RENEWABLES. POWER SUPPLY FROM OTHER GENERATORS, AND THE RESULTING FLOW PATTERNS IN THE GRID. EXPANDING CAPACITY IS DIFFICULT. PROBLEMS ARE OBSERVED WHEN PENETRATION IS HIGH, AND IN PARTICULAR THE TIME SPAN THAT WAS OBSERVED IN TERMS OF THE RELEVANT DISTURBANCE WAS THIS 15 MINUTE TIME SPAN THAT OVERLAPS WITH THE OPIE OF -- OPF TIME SPAN. OFTEN DUE TO TURBULENCE, A STORM SYSTEM MOVING IN OR OUT OF THE AREA. THE CREATION IN WINS BEING IN THE SAME ORDER OF MAGNITUDE. WHEN PENETRATION IS LARGE, AS OBSERVED, MANY COUNTRIES ARE GETTING INTO THIS REGIME. THIS IS A NORMAL DC-OPF. YOUR MINIMIZING GENERATION COST. WE HAVE A USUAL D.C. POWER FLOW EQUATION, WITH LINE LIMITS. THESE ARE GENERATION BOUND. HOW WOULD NORMAL OPF HANDLE FLUCTUATIONS IN WIND? THAT WOULD BE SIMILAR TO THE WAY THAT NORMAL SHORT-TERM DEMAND FLUCTUATION IS HANDLED. AND THAT IS TO VERY GENERATION INPUT, UP OR DOWN. -- VARY GENERATION INPUT, UP OR DOWN. WIND WOULD BE HANDLED EXACTLY THE SAME WAY. WE HAVE DATA FROM THIS PART, INCLUDING WIND FARM DATA. USING STANDARD OPF, THERE ARE A NUMBER OF LINES HERE. ALL RIGHT. WHAT WE WOULD LIKE TO BE ABLE TO DO IS CAPTURE THE IDEA THAT IS OK TO EXCEED THE LINE LIMIT OCCASIONALLY. THE WAY WE WILL DO THIS IS TO ESTABLISH WHAT ONE WOULD CALL A CONSTRAINT, WHICH IS TO ENFORCE THE PROBABILITY THAT THE LINE LIMIT IS EXCEEDED IS SMALL FOR ANY GIVEN LINE. THE GOAL IS TO HAVE SIMPLE CONTROL, AWARE OF THESE LINE LIMITS, NOT TOO CONSERVATIVE, AND COMPUTATIONALLY PRACTICAL. THIS IS THE KIND OF CONTROL WE HAVE IN MIND, WHICH IS SIMILAR TO NORMAL FREQUENCY CONTROL, WHICH IS TO HAVE GENERATION -- GENERATOR SET UPWARDS OF THE MEAN VALUE. IT IS PER GENERATOR, AND COMPUTED. WE COMPUTE THESE AHEAD OF TIME IN A 15 MINUTE WINDOW, AND THIS IS A CONTROL IMPLEMENTED IN RILL TIME. -- REAL TIME. A FEW TECHNICAL POINTS, WHEN DO YOU ADD WIND TO THE MIX? THE USUAL POWER CODE EQUATIONS ARE ALTERED THIS WAY. NOW THIS IS THE CONTROL. THIS CAN BE SOLVED BY USING THIS MATRIX, THE PSEUDO-INVERSE FOR THE MATRIX. FLOW ENDS UP BEING A LINEAR COMBINATION. YOU GET THIS NICE, UGLY FORMULA THAT INVOLVES EVERYTHING. THE BOLD INDICATES THIS IS A RANDOM VARIABLE, EVEN DEVIATIONS IN WIND ARE RANDOM VARIABLES AS WELL. INDEPENDENCE OF WIND FARMS IS AN OK ASSUMPTION. YOU GET AN EXPRESSION FOR EXPECTATION. A CHANCE CONSTRAINT IS SOMETHING LIKE THE PROBABILITY THAT A LINE FLOW EXCEEDS THE MAXIMUM AND IS SMALL. YOU CAN GET CONSERVATIVE APPROXIMATIONS IF YOU ASSUME THE WHEN DEVIATIONS -- APPROXIMATIONS. IF YOU ASSUME THE WHEN DEVIATIONS ARE NORMAL, YOU CAN EXPECT AN EXACT. YOU CAN USE A NORMAL DISTRIBUTION TO APPROXIMATE OUTER DISTRIBUTION. PART OF WHAT THE REPORT SAYS IS THAT ESTIMATING THE DISTRIBUTIONS OF THE WIND DEVIATIONS IN A NARROW TIME FRAME IS VERY DIFFICULT. IT IS VERY NOISY DATA. IT'S NOT ANY KIND OF CLEAN DISTRIBUTION. SO WE GET THIS FORMULATION, WHICH IS -- I KEPT A FEW DETAILS. THIS CHANCE CONSTRAINT PROBLEM INCLUDING THE EPSILON, AND THIS IS THE INVERSE OF THE CALCIUM DISTRIBUTION. THIS IS A CONVEX OPTIMIZE ASIAN PROBLEM, AN EXACT REPRESENTATION OF A CHANCE CONSTRAINT -- OPTIMIZE ASIAN -- OPTIMIZATION PROBLEM, AN EXACT REPRESENTATION OF A CHANCE CONSTRAINT PROBLEM. EVEN THOUGH IT IS A CONVEX PROBLEM, THERE ARE EXTREME DIFFICULTIES. THEY CANNOT SOLVE THEM. THE STORY THAT YOU WILL BELIEVE IN AND I WILL REPEAT LATER, CONVEX OPTIMIZATION IS EASY. THIS IS ONE SUCH THING. THIS IS REPEATED OVER AND OVER AGAIN. WE IMPLEMENTED A LOW-TECH CUTTING PLANE ALGORITHM. ICONIC RESTRAINTS ARE APPROXIMATED BY THE TANGENT. THIS SYSTEM WORKS VERY QUICKLY IN ALL THE EXAMPLES WE LOOKED AT. BACK TO THIS MOTIVATING EXAMPLE, WITH STANDARD OPF, WE GET A SOLUTION WITH THIS COST AND THE NUMBER OF LINES. WITH A LITTLE MORE COST, WE GET THAT EVERY LINE IS SAFE MOST OF THE TIME IN THE COMPUTER IN 10 SECONDS. THIS IS A SUMMARY. THIS IS THE PAST. CAN WE HANDLE POWER FLOWS MORE ACCURATELY? THIS IS THE MODEL WE ARE LOOKING AT. ACTIVE POWER AND LOSSLESS. WE GET THESE NONLINEAR AND NON- CONVEX EQUATIONS. WE WOULD LIKE TO SOLVE THIS AND INCORPORATE THE CHANCE CONSTRAINT AS WELL. OUT OF THE BLUE, AND SINCE THE RESULTS OF BOYD, SUPPOSE YOU SOLVE THIS PROBLEM -- I'M NOT TELL YOU WHAT THIS TRIED AND FUNCTION IS -- TRIDENT FUNCTION IS -- I FORGET THE NAME FOR THE GREEK LETTER. THIS LOOKS LIKE A POWER FLOW EQUATION. THIS IS OUTGOING FLOWS MINUS INCOMING FLOWS. THEY ARE PUTTING A LIMIT. IF YOU THINK OF IT AS BEING THE SIGN, IT HAS TO BE AT MOST ONE. SO WHAT IS AC┬ŻAI -- IT? THIS IS WHAT IT IS. BOYD PROVED THAT IF WE SOLVE THIS PROBLEM AND YOU LOOK AT THE DUO VARIABLES, YOU SOLVE THE POWER FLOW EQUATION THAT YOU HAD BEFORE. WHAT IS THE BIG DEAL? THIS IS A CONVEX OPTIMIZE ASIAN -- OPTIMIZATION PROBLEM. IT IS CONVEX BECAUSE IF YOU TAKE THE SECOND DERIVATIVE, YOU GET THE DERIVATIVE OF THE ARC SINE -- SIGN. THIS PROBLEM, IN THEORY -- WE WOULD EXPECT THE NUMERICALLY WE WOULD HAVE TO EXERT SOME MUSCLE. THIS IS A CONVEX PROBLEM. IT IS REALLY NEAT. THIS IS NOT OPF. HOWEVER, THIS IS OUR RESULT. THIS IS BACK TO THE USUAL OPF OBJECTIVE FUNCTION. THIS IS THE TRIDENT FUNCTION WITH A MULTIPLIER, AND THIS IS A BARRIER PROBLEM. WHY DO WE NEED THE BARRIER FUNCTION? THE PROBLEM I HAD BEFORE WAS POORLY DEFINED. IF THE PROBLEM HAS AN OPTIMAL SOLUTION, WHICH IS WHEN THE SIGN IS GOING TO ONE. WE ADD THIS OVER HERE WITH A D VARIABLE -- DELTA VARIABLE. WE PENALIZE THAT IN THE OBJECTIVE WITH OUR BARRIER. HERE IS ANOTHER WORD I HAVE NOT USED TOO OFTEN, THE SERUM. -- THEOREM. IF YOU SOLVE THIS PROBLEM, WHICH IS A CONVEX OPTIMIZATION PROBLEM -- IF YOU SOLVE IT YOU APPROXIMATELY SOLVE THE POWER FLOW EQUATION. PRICE SI, THAT'S PRECISELY, IT IS ABOUT THE MAGNITUDE OF THE ERROR. IT SAYS, YOU ARE ACTUALLY SOLVING THE POWER FLOW PROBLEM. THIS CAN BE MADE PRECISE. YOU ALSO GET THE POWER FLOW EQUATION, APPROXIMATELY. FROM A TACTICAL -- PRACTICAL PERSPECTIVE, IF WE WANTED TO IMPLEMENT THIS, TO SOLVE A CONVEX OPTIMIZATION PROBLEM, YOU NEED AN OUTER ENVELOPE APPROXIMATION TO BE VERY NONLINEAR FUNCTION. AND THEN YOU NEED TO ITERATE, LETTING D GO TO ZERO. ITERATION IN THIS SENSE IS RELATED TO A QUESTION THAT WAS ASKED BEFORE, YOU WANT TO SOLVE THIS NONLINEAR PROBLEM AND YOU NEED TO ITERATE EITHER WAY, APPROXIMATING A FEASIBLE REGION FORWARD BY MODIFYING THE OBJECTIVE. THERE IS A RICH LITERATURE, AND MORE THAN LITERATURE. SOMETHING IMPORTANT, THE PROBLEM MAY BE INFEASIBLE, AS HAS BEEN BROUGHT UP BEFORE. BEING ABLE TO DETECT THAT EARLY IS IMPORTANT. THE CAPABILITY IN THE CONTEXT OF THE PROBLEM MEANS THAT THE ROW IN THE SOLUTION GOES TO ONE. IN ORDER TO BE FEASIBLE, SOME LINE HAS TO HAVE A VERY BIG PHASE ANGLE DIFFERENT. THE OBJECTIVE IS GOING TO PLUS INFINITY. YOU WANT TO ABLE TO DETECT THAT QUICKLY, OTHERWISE IT KEEPS GOING TO INFINITY AND NOTHING CONVERTS. HOW MUCH TIME DO I HAVE? FIVE. THIS IS MY COLLEAGUE AND TWO OTHER GUYS. THEY STUDIED WHAT THEY CALL AN APPROXIMATION TO WHAT I HAD BEFORE. THIS IS WHAT WOULD BE THE DC POWER FLOW PROBLEM, ALTHOUGH IN ADDITION TO BOUNDING THE FLOW ON A LINE WITH A THERMAL LIMIT, THEY ALSO BOUND IT THIS WAY WITH SYNCHRONIZATION. THE HIM. REALLY OBSERVED -- THEY EMPIRICALLY OBSERVED THAT WHEN THEY SOLVE THIS PROBLEM, THEY GET VERY CLOSE TO THE ACTIVE POWER SOLUTION, NOT NECESSARILY TO THE PHASE ANGLE. THE VARIABLES ARE NOT APPROXIMATIONS TO THE PHASE ANGLES. THIS IS A GOOD APPROXIMATION FOR ACTIVE POWER FLOW. THIS IS SOMETHING THEY EMPIRICALLY OBSERVED. THIS IS EXACT FOR A RADIAL NETWORK. NOW WE WOULD LIKE TO TAKE THIS MODEL, AND ADD CHANCE CONSTRAINT. THE SYNCHRONIZATION CONSTRAINT, THE PHASE ANGLE IS SMALLER THAN ONE NUMBER -- EITHER ONE IS GOING TO BE SO CASTAIC -- STO CHASTIC. WE ARE GOING TO IMPOSE, ESPECIALLY FOR THIS GUY, WE'RE GOING TO IMPOSE A CHANCE CONSTRAINT AND SAY THIS IS EXTREMELY SMALL, THE PROBABILITY THAT THE PHASE ANGLE DIFFERENCE IS TOO BIG FOR THAT LINE IN A VERY SMALL NUMBER. THIS IS THE CONTROL I DESCRIBED BEFORE. WE GET TO CHANCE CONSTRAINT, THERMAL AND SINGLE WHERE. -- SYNC-AWARE. DC-OPF. IT HAS CHANCE CONSTRAINTS FOR EVERY LINE OF TWO KINDS. SYNCHRONIZATION AND THERMAL LINE CONSTRAINT. THERE IS CONSTRAINT FOR EVERY GENERATOR. THE EPSILON'S HAVE THIS RADIATION HIERARCHY. THE THERMAL LIMIT PROBABILITY, THOUGH SMALL, IS STILL MUCH HIGHER THAN THE PROBABILITY THAT A GENERATOR EXCEEDS THIS LIMIT. THAT IS ALSO SO MUCH HIGHER THAN THE SYNCHRONIZATION CONSTRAINT.

>> [INAUDIBLE]

>> THIS IS EXACTLY HOW WE STATE. I GUESS THAT IS SEPARATE.

>> [INAUDIBLE]

>> WE WRITE THEM SEPARATE GRID . WE HAVE MULTIPLE CONSTRAINTS. FOR THE POLISH SYSTEM, THIS IS THOUSANDS OF CONSTRAINTS. THEY CANNOT DO ANYTHING. MY GUESS IS THEY CANNOT DO ANYTHING NOT BECAUSE THESE PROBLEMS ARE INHERENTLY DIFFICULT. IT IS JUST NOT SET UP TO HANDLE IN SOME WAY. I DON'T KNOW WHAT IT IS. THERE IS EVIDENCE THAT THE PROBLEM IS NOT DIFFICULT. AN INTERESTING POINT, WE HAVE THESE COMMON CONSTRAINTS THAT MODEL THE CHANCE CONSTRAINTS, FOR EXAMPLE. THESE ARE VERY NONLINEAR CONSTRAINTS, WHICH WE ARE PROGRESSIVELY APPROXIMATING. WHEN A LINE IS A LINE THAT IS AT RISK, THAT'S WHEN THE CHANCE CONSTRAINT BECOMES ACTIVE IN THE ALGORITHM AND THAT'S WHEN WE NEED TO APPROXIMATING MORE ACCURATELY. THE ALGORITHM IS DISCOVERING THE LINES THAT REALLY MATTER. THE ALGORITHM IS WORKING ON THOSE HANDFUL OF LINES ALL THE TIME. AND THAT'S IT. THANK YOU VERY MUCH. [APPLAUSE]

>> TO QUESTIONS. -- TWO QUESTIONS. YOU HAVE A LOT OF EXPERIENCE ABOUT THIS GRID. THIS IS A DIFFICULT NETWORK COMPARED TO A LOT OF THE NETWORKS WE SEE HERE BECAUSE THERE ARE A LOT OF STABILITY CONSTRAINTS. THEY ARE NOT SIMPLE THERMAL CONSTRAINTS. WHO GIVES YOU THIS? IT IS NOT A NICE NETWORK TO WORK TO APPLY YOUR METHOD, BECAUSE A LOT OF PECULIAR CONSTRAINTS, MANY MORE THAN WE HAVE HERE. SECOND QUESTION, IT HAS TO DO, WHEN YOU COME UP WITH A SET OF THE POWER FLOWS, THE DISTRIBUTION OF THE WIND, DO YOU MAKE ANY ASSUMPTIONS ABOUT THE PROBABILITY OF THE WIND?

>> AS YOU KNOW, IT'S HARD FOR US TO GET A LOT OF REAL DATA. THERE ARE THREE OR FOUR SNAPSHOTS OF THE GRID THAT COME WITH. -- POWER. I DON'T THINK WE HAVE ANY OF THE DATA. THE MODEL IS NOT REALLY CAPTURING THAT.

>> [INAUDIBLE]

>> NO. I DON'T KNOW WHAT IT IS.

>> [INAUDIBLE]

>> THE SECOND QUESTION IS A VERY GOOD QUESTION. THE WAY THE CONTROL WORKS, YOU CAN THINK ABOUT HAVING A BASE POWER FLOW, WHICH IS ADAPTED IN REAL TIME. IN REAL TIME YOU GET A COMBINATION OF TWO THINGS. WE USE A NORMAL DISTRIBUTION FOR EACH FARM. HOWEVER -- NO, NOT OVER. [LAUGHTER] AS PART OF WHAT WE DO IN THE PAPER, YOU CAN USE A NORMAL DISTRIBUTION PLUS WHAT YOU WOULD CALL ROBUSTNESS. NORMAL DISTRIBUTION, THAT ALLOWS FOR THE ACTUAL WIND AND REAL- TIME TO DEVIATE FROM IT IN A MEASURED WAY. THE MEAN IN THE EXCEEDED AND THE CENTER DEVIATION. -- STANDARD DEVIATION. THIS IS AN EXTREMELY GOOD QUESTION COMPS -- QUESTION, SOMETHING THAT MUST BE DONE.

>> [INAUDIBLE]

>> IT IS WHAT WE CALL A ROBUST WIND ESTIMATION. THIS IS A VERY GOOD QUESTION.

>> I WAS WONDERING HOW YOU GOT THE 10 TO THE MINUS FOUR VALUE.

>> THIS IS BASICALLY SAYING, EXTREMELY LOW PROBABILITY. THINK ABOUT A 15 MINUTE TIME SPAN. 10 TO THE MINUS FOUR FRACTION OF THAT IS HOW MUCH? A FRACTION OF A SECOND. THAT'S THE WHOLE POINT. THE SMALLER WE COULD MAKE IT, THE BETTER.

>> DANIEL, I THINK I MADE THIS RECOMMENDATION. THERE ARE GADGETS NOW WHICH WOULD BE REALLY NICE. THEY ARE MONITORING IN A VERY DIFFERENT WAY WHEN IS A LINE OVERLOADED, AND COMPARE THAT WITH A [INAUDIBLE] I THINK SOMEBODY SHOULD BE THINKING ABOUT WHAT IS HAPPENING IN THE REAL WORLD.

>> I DO NOT KNOW IF THEY ARE MAKING THEM WITH DATA AVAILABLE.

>> [INAUDIBLE]

>> FANTASTIC. I WOULD LOVE THAT.

>> THANK YOU, DAN. [APPLAUSE] . .