26
Inverter Probing for Learning Power Distribution Grids Vassilis Kekatos ([email protected]) October 4 th , 2018 Sid Bhela 1 Acknowledgements NSF-CAREER-1751085 Harsha Veeramachaneni Guido Cavraro

Inverter Probing for Learning Power Distribution Grids Pres Kekatos.pdf · Inverter Probing for Learning Power Distribution Grids Vassilis Kekatos ([email protected] ) October 4th, 2018

Embed Size (px)

Citation preview

Inverter Probing for Learning Power Distribution Grids

Vassilis Kekatos ([email protected])

October 4th, 2018

Sid Bhela

1

Acknowledgements NSF-CAREER-1751085

Harsha Veeramachaneni

Guido Cavraro

Learning in distribution grids

n  Reduced observability due to sheer extent and limited real-time metering

2

n  Leverage smart inverters and smart meters

n  However, load and topology estimates needed for grid optimization, billing, and energy theft detection

From passive to active grid data collection

3

n  For sufficiently rich (pseudo)measurement sets, solve D-PSSE [Dzafic-Jabr-Pal et al’13], [Klauber-Zhu’15], [Gomez-Exposito et al’15]

n  Linear estimator if grid is equipped with micro-PMUs [A. von Meier’15], [Yuan-Ardakanian-Low-Tomlin’16]

n  Load inference from smart meter data [Bhela-Kekatos-Veeramachaneni’17]

Passive data collection

n  Estimating oscillation modes in transmission grids [Trudnowski-Pierre’09]

n  System identification in DC microgrids [Angjelichinoski-Scaglione ’17]

n  Finding grid equivalent Thevenin impedance for one inverter

[Jaksic-Boroyevich-Burgos et al ’17]

Grid perturbation for active data collection

Inverter probing for learning loads

4

(p2, q2)<latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit><latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit><latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit>

(p2, q2)<latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit><latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit><latexit sha1_base64="fq65NCkcnLExIbtgsZfoO6y90QU=">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</latexit>

(p01, q01, |V 0

1 |)<latexit sha1_base64="hQHhVTy3yAinOKYcbzJlMsjdhf0=">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</latexit><latexit sha1_base64="hQHhVTy3yAinOKYcbzJlMsjdhf0=">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</latexit><latexit sha1_base64="hQHhVTy3yAinOKYcbzJlMsjdhf0=">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</latexit>

(p1, q1, |V1|)<latexit sha1_base64="A553HlxFYqp7PenCDGjx/9Zqr6M=">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</latexit><latexit sha1_base64="A553HlxFYqp7PenCDGjx/9Zqr6M=">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</latexit><latexit sha1_base64="A553HlxFYqp7PenCDGjx/9Zqr6M=">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</latexit>

(|V0|, ✓0)<latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit><latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit><latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit>

(|V0|, ✓0)<latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit><latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit><latexit sha1_base64="onxrfgj5DPzF2JdAysQlrYJV0os=">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</latexit>

t<latexit sha1_base64="nO+SkBHFg+Yc6EAecXqPKUnNx+Y=">AAAB53icbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG8FLx5bMLbQhrLZTtq1m03YnQgl9Bd48aDi1b/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6MEmmOfg8kYnuhMyAFAp8FCihk2pgcSihHY5vZ377CbQRibrHSQpBzIZKRIIztFIL+9WaW3fnoKvEK0iNFGj2q1+9QcKzGBRyyYzpem6KQc40Ci5hWullBlLGx2wIXUsVi8EE+fzQKT2zyoBGibalkM7V3xM5i42ZxKHtjBmOzLI3E//zuhlG10EuVJohKL5YFGWSYkJnX9OB0MBRTixhXAt7K+UjphlHm03FhuAtv7xK/Iv6Td1tXdYabpFGmZyQU3JOPHJFGuSONIlPOAHyTF7Jm/PovDjvzseiteQUM8fkD5zPH0jXjLk=</latexit><latexit sha1_base64="nO+SkBHFg+Yc6EAecXqPKUnNx+Y=">AAAB53icbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG8FLx5bMLbQhrLZTtq1m03YnQgl9Bd48aDi1b/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6MEmmOfg8kYnuhMyAFAp8FCihk2pgcSihHY5vZ377CbQRibrHSQpBzIZKRIIztFIL+9WaW3fnoKvEK0iNFGj2q1+9QcKzGBRyyYzpem6KQc40Ci5hWullBlLGx2wIXUsVi8EE+fzQKT2zyoBGibalkM7V3xM5i42ZxKHtjBmOzLI3E//zuhlG10EuVJohKL5YFGWSYkJnX9OB0MBRTixhXAt7K+UjphlHm03FhuAtv7xK/Iv6Td1tXdYabpFGmZyQU3JOPHJFGuSONIlPOAHyTF7Jm/PovDjvzseiteQUM8fkD5zPH0jXjLk=</latexit><latexit sha1_base64="nO+SkBHFg+Yc6EAecXqPKUnNx+Y=">AAAB53icbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG8FLx5bMLbQhrLZTtq1m03YnQgl9Bd48aDi1b/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6MEmmOfg8kYnuhMyAFAp8FCihk2pgcSihHY5vZ377CbQRibrHSQpBzIZKRIIztFIL+9WaW3fnoKvEK0iNFGj2q1+9QcKzGBRyyYzpem6KQc40Ci5hWullBlLGx2wIXUsVi8EE+fzQKT2zyoBGibalkM7V3xM5i42ZxKHtjBmOzLI3E//zuhlG10EuVJohKL5YFGWSYkJnX9OB0MBRTixhXAt7K+UjphlHm03FhuAtv7xK/Iv6Td1tXdYabpFGmZyQU3JOPHJFGuSONIlPOAHyTF7Jm/PovDjvzseiteQUM8fkD5zPH0jXjLk=</latexit>

t0<latexit sha1_base64="1kQfB8dU6wWJpiClyNylgcmchpI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEUG8FLx6rGFtoQ9lsN+3SzSbsToQS+g+8eFDx6k/y5r9x2+agrQ8GHu/NMDMvTKUw6LrfTmlldW19o7xZ2dre2d2r7h88miTTjPsskYluh9RwKRT3UaDk7VRzGoeSt8LRzdRvPXFtRKIecJzyIKYDJSLBKFrpHk971Zpbd2cgy8QrSA0KNHvVr24/YVnMFTJJjel4bopBTjUKJvmk0s0MTykb0QHvWKpozE2Qzy6dkBOr9EmUaFsKyUz9PZHT2JhxHNrOmOLQLHpT8T+vk2F0FeRCpRlyxeaLokwSTMj0bdIXmjOUY0so08LeStiQasrQhlOxIXiLLy8T/7x+XXfvLmoNt0ijDEdwDGfgwSU04Baa4AODCJ7hFd6ckfPivDsf89aSU8wcwh84nz+pI4zq</latexit><latexit sha1_base64="1kQfB8dU6wWJpiClyNylgcmchpI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEUG8FLx6rGFtoQ9lsN+3SzSbsToQS+g+8eFDx6k/y5r9x2+agrQ8GHu/NMDMvTKUw6LrfTmlldW19o7xZ2dre2d2r7h88miTTjPsskYluh9RwKRT3UaDk7VRzGoeSt8LRzdRvPXFtRKIecJzyIKYDJSLBKFrpHk971Zpbd2cgy8QrSA0KNHvVr24/YVnMFTJJjel4bopBTjUKJvmk0s0MTykb0QHvWKpozE2Qzy6dkBOr9EmUaFsKyUz9PZHT2JhxHNrOmOLQLHpT8T+vk2F0FeRCpRlyxeaLokwSTMj0bdIXmjOUY0so08LeStiQasrQhlOxIXiLLy8T/7x+XXfvLmoNt0ijDEdwDGfgwSU04Baa4AODCJ7hFd6ckfPivDsf89aSU8wcwh84nz+pI4zq</latexit><latexit sha1_base64="1kQfB8dU6wWJpiClyNylgcmchpI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEUG8FLx6rGFtoQ9lsN+3SzSbsToQS+g+8eFDx6k/y5r9x2+agrQ8GHu/NMDMvTKUw6LrfTmlldW19o7xZ2dre2d2r7h88miTTjPsskYluh9RwKRT3UaDk7VRzGoeSt8LRzdRvPXFtRKIecJzyIKYDJSLBKFrpHk971Zpbd2cgy8QrSA0KNHvVr24/YVnMFTJJjel4bopBTjUKJvmk0s0MTykb0QHvWKpozE2Qzy6dkBOr9EmUaFsKyUz9PZHT2JhxHNrOmOLQLHpT8T+vk2F0FeRCpRlyxeaLokwSTMj0bdIXmjOUY0so08LeStiQasrQhlOxIXiLLy8T/7x+XXfvLmoNt0ijDEdwDGfgwSU04Baa4AODCJ7hFd6ckfPivDsf89aSU8wcwh84nz+pI4zq</latexit>

S. Bhela, V. Kekatos, and S. Veeramachaneni, "Enhancing Observability in Distribution Grids using Smart Meter Data," IEEE Trans. on Smart Grid, 2018 (early access).

Probing protocol

5

n  Perturb power injections at probing buses [how?]

n  Collect inverter data (injections & voltage magnitudes/phasors)

n  Repeat over T probing actions spaced 1-2’ apart

Coupled power flow (CPF) problem

6

Problem statement: Given inverter data on metered buses and assuming time-invariant injections at buses , find states and non-metered loads on

probing (metered) buses non-metered buses

|M| �2|O|

T

M

O

M

O

O

pn(vt) = ptn 8n 2 Mqn(vt) = qtn 8n 2 Mun(vt) = ut

n 8n 2 M3MT

pn(vt) = pn(vt+1) 8n 2 O

qn(vt) = qn(vt+1) 8n 2 O2O(T � 1)

{vt}Tt=1

Q1) Can non-metered loads be recovered by T-slot probing?

Q2) How to optimally design probing actions?

Q3) How to solve the CPF problem?

n  Counting equations and unknowns yields

probing buses (14) non-metered buses (18)

IEEE 33-bus feeder (T=4)

Q1) Topological observability

7

Theorem: If can be partitioned into T/2 subsets such that each can be matched to , the feeder is topologically observable.

S. Bhela, V. Kekatos, and S. Veeramachameni, “Smart Inverter Probing for Learning Loads: Part I – Identifiability Analysis,” IEEE Trans. on Power Systems, 2018 (under review)

O M<latexit sha1_base64="tqhbY/zoWYZf3oI26GfV3oEfD3Q=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GpLS0S4LbtwIFewDpkPJpJk2NJMZkoxQhn6GGxeKuPVr3Pk3ZtoKKnogcDjnXnLuCVPBtUHowymtrW9sbpW3Kzu7e/sH1cOjrk4yRVmHJiJR/ZBoJrhkHcONYP1UMRKHgvXC6VXh9+6Z0jyRd2aWsiAmY8kjTomxkj+IiZlQIvKb+bBaQy668Bq4CZHrIdzEniV1DyNUh9hFC9TACu1h9X0wSmgWM2moIFr7GKUmyIkynAo2rwwyzVJCp2TMfEsliZkO8kXkOTyzyghGibJPGrhQv2/kJNZ6Fod2soiof3uF+JfnZyZqBjmXaWaYpMuPokxAk8DifjjiilEjZpYQqrjNCumEKEKNbaliS/i6FP5PunUXIxffNmottKqjDE7AKTgHGFyCFrgGbdABFCTgATyBZ8c4j86L87ocLTmrnWPwA87bJ8u8kYc=</latexit><latexit sha1_base64="tqhbY/zoWYZf3oI26GfV3oEfD3Q=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GpLS0S4LbtwIFewDpkPJpJk2NJMZkoxQhn6GGxeKuPVr3Pk3ZtoKKnogcDjnXnLuCVPBtUHowymtrW9sbpW3Kzu7e/sH1cOjrk4yRVmHJiJR/ZBoJrhkHcONYP1UMRKHgvXC6VXh9+6Z0jyRd2aWsiAmY8kjTomxkj+IiZlQIvKb+bBaQy668Bq4CZHrIdzEniV1DyNUh9hFC9TACu1h9X0wSmgWM2moIFr7GKUmyIkynAo2rwwyzVJCp2TMfEsliZkO8kXkOTyzyghGibJPGrhQv2/kJNZ6Fod2soiof3uF+JfnZyZqBjmXaWaYpMuPokxAk8DifjjiilEjZpYQqrjNCumEKEKNbaliS/i6FP5PunUXIxffNmottKqjDE7AKTgHGFyCFrgGbdABFCTgATyBZ8c4j86L87ocLTmrnWPwA87bJ8u8kYc=</latexit><latexit sha1_base64="tqhbY/zoWYZf3oI26GfV3oEfD3Q=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GpLS0S4LbtwIFewDpkPJpJk2NJMZkoxQhn6GGxeKuPVr3Pk3ZtoKKnogcDjnXnLuCVPBtUHowymtrW9sbpW3Kzu7e/sH1cOjrk4yRVmHJiJR/ZBoJrhkHcONYP1UMRKHgvXC6VXh9+6Z0jyRd2aWsiAmY8kjTomxkj+IiZlQIvKb+bBaQy668Bq4CZHrIdzEniV1DyNUh9hFC9TACu1h9X0wSmgWM2moIFr7GKUmyIkynAo2rwwyzVJCp2TMfEsliZkO8kXkOTyzyghGibJPGrhQv2/kJNZ6Fod2soiof3uF+JfnZyZqBjmXaWaYpMuPokxAk8DifjjiilEjZpYQqrjNCumEKEKNbaliS/i6FP5PunUXIxffNmottKqjDE7AKTgHGFyCFrgGbdABFCTgATyBZ8c4j86L87ocLTmrnWPwA87bJ8u8kYc=</latexit><latexit sha1_base64="tqhbY/zoWYZf3oI26GfV3oEfD3Q=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GpLS0S4LbtwIFewDpkPJpJk2NJMZkoxQhn6GGxeKuPVr3Pk3ZtoKKnogcDjnXnLuCVPBtUHowymtrW9sbpW3Kzu7e/sH1cOjrk4yRVmHJiJR/ZBoJrhkHcONYP1UMRKHgvXC6VXh9+6Z0jyRd2aWsiAmY8kjTomxkj+IiZlQIvKb+bBaQy668Bq4CZHrIdzEniV1DyNUh9hFC9TACu1h9X0wSmgWM2moIFr7GKUmyIkynAo2rwwyzVJCp2TMfEsliZkO8kXkOTyzyghGibJPGrhQv2/kJNZ6Fod2soiof3uF+JfnZyZqBjmXaWaYpMuPokxAk8DifjjiilEjZpYQqrjNCumEKEKNbaliS/i6FP5PunUXIxffNmottKqjDE7AKTgHGFyCFrgGbdABFCTgATyBZ8c4j86L87ocLTmrnWPwA87bJ8u8kYc=</latexit>

Q1) Improvement with longer probing

8

n  Conditions checked with max-flow problem

n  Observability improves with phasor data

n  We generalize topological observability to lossy networks

n  How about numerical observability?

probing buses (11) non-metered buses (21)

IEEE 33-bus feeder (T=6)

Q2) Probing injection design

9

n  Step 1: Build a library of inverter-implementable injections

n  Step 2: Reduce library to network-compliant injections

voltage constraints

uncertain loads

S. Bhela, V. Kekatos, and S. Veeramachameni, “Smart Inverter Probing for Learning Loads: Part II – Probing Sequence Design,” IEEE Trans. on Power Systems, 2018 (under review)

0 pgn pgn, 8n 2 M<latexit sha1_base64="dPv/PA9xBXmCdF/ywZEbCu0+qTk=">AAAQiHichZfbbts2GIDV7tRlh7rb5W6MBQOGQQjsbUC3iwFN0jUHxI1zctJGXkBStMSFOpiiHDuCnmZPs9vtam8zUpFpS7/UGjBF8/tE8v9JExKOOUtkr/ffo8cffPjRx588+XTjs8+/+PJp59lXoyRKBaEXJOKRuMIooZyF9EIyyelVLCgKMKeX+HZX88sZFQmLwnO5iOk4QF7IJowgqZpuOr/1ug6n0258E/7hPVQdjEQW57rB7jrTFLnLchIJxHk37DpMfQMyuOls9rZ6xacLK/2ysmmVn+HNs6c/OW5E0oCGknCUJNf9XizHGRKSEU7zDSdNaIzILfLotaqGKKCJ7c5YnBTVcVaEXPEyFCTJIsC2R0N9zbvfBUj6Sd3RjU3sOpWTX8YZC+NU0pAoRbFJyrsy6uqUdV0mKJF8oSqICKZm2iU+EohIldiNyjC6cxlFPLHLAe3l5NRV+oFN42TCPLuIwk5STFCsF8KmQezTqe0JFPuMzDVSXjVOGqq8CSSpnQpuI+7FCU1VNiOX6l+Rmpoa4Q4lCzWiHklGsTt5GEtibuv7maRBXpnztURqs4yzeVNmxcJNfB7akt3e2xhxFJJykXQLZ1goRUct2Nx2KVEbRIeTbMUqdkFjjggLPT3gS6rWW9CBaj+OdRSRyByBwts8K8p2w6OFoi5t3STMC/OsKFuMWOTZULTRHSpRnu20YbXsXp65XhvHD8LDpcWRaruoOM5b5zBTW0yt4L2SVLU1HRTxPDttTwUL9FQOgrYOAjRnQRrkas3mrQ4LS4e1JlQnWyvvSvrg6Pc800VbHyn+U0UtI9XNlipb1y6a51lRNho/ZM6EhW6eFaVRhkj9bd2X6mTUm171QyhTuXN4eRXFVeUxpHckCgIUuvro0ymSPp5kKM9rjBvGAcOGYcCIYQQwahgFzDPMA4wZxgCbGTbLQYChgSG4URgmAEsMSwBLDUsBuzPsDrC5YXPAFoYtYBD3Bt6DG2PDYsCmhk0B2zZsG7Adw3YA2zVsF7BXhr0CbM+wPcD2DduHwZ8xL1htU6f4CXoYrjaIMwRbxF924iMJeqq5culKxl36Phsvbf0Q8R7XXbpu1DSLmryym/XqNA6McQDYkWFHgA0MG8CcGjaEq3Ji4Am48dSwU8DODDsD7Nywc8CuDLsC7I1hbwB7a9hbwC4NuwQBSgy2QNNewXUJN0izujRrkNK6lDZIcV2KG6RpXZoCieiJE5+S23dMHGtpfV83xTaqDzaCe1nii7p10dDVsC4NG6STunTSIG3Xpe2G4EbV4EYNyqyqwGXDOrJ1BcaF9cquKylMENaxrzswcqwjX1dg3L4eaf2EgzvI10u6rsAlDUh5+hPE4fEfkL0VBQd5QI5WFBw26gVqRQcgCQE5XuFjePPrFX0N6XBFh5CerSg8dopZm9SWUwfpV8/MleeMKpZ+wXHEXf36E6lnp6IJLrV6T2E8Cqtu2QgfpXxWFVUDXNPCKlYVqGr4ei7OV3HCY/bQxHgI/1aH9b/VYX0rOyQQ1/2x8uhcFi9YmXoozbPNfrU35eGah3lKS1G9ZvfrL9WwMvpxq9/b6p/8vPmiX75wP7G+sb61vrf61nPrhbVvDa0Li1h/WX9b/1j/djY6vc7zzq8P6uNH5T1fW5VPZ+d/bLFFUw==</latexit><latexit sha1_base64="dPv/PA9xBXmCdF/ywZEbCu0+qTk=">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</latexit><latexit sha1_base64="dPv/PA9xBXmCdF/ywZEbCu0+qTk=">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</latexit><latexit sha1_base64="dPv/PA9xBXmCdF/ywZEbCu0+qTk=">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</latexit>

0 (pgn)2 + (qgn)

2 (sgn)2, 8n 2 M

<latexit sha1_base64="rU1XBCB043fGxF12OERQu91zuTM=">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</latexit><latexit sha1_base64="rU1XBCB043fGxF12OERQu91zuTM=">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</latexit><latexit sha1_base64="rU1XBCB043fGxF12OERQu91zuTM=">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</latexit><latexit sha1_base64="rU1XBCB043fGxF12OERQu91zuTM=">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</latexit>

q  random draws

q  unknown loads lie in box (pl,ql) 2 Bl<latexit sha1_base64="Uw0EndRX39LmlunUmhCKictknNM=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0WoUEoigi6LblxWsA9oaphMJ+3QySTOTIQS8hNu/BU3LhRxK7jzb5ykEbT1wMC559zL3Hu8iFGpLOvLKC0tr6yuldcrG5tb2zvm7l5HhrHApI1DFoqehyRhlJO2ooqRXiQICjxGut7kMvO790RIGvIbNY3IIEAjTn2KkdKSa9ZrToDU2POTKL1l9Z/iThfHDuUwFzBiyUXqMtesWg0rB1wkdkGqoEDLNT+dYYjjgHCFGZKyb1uRGiRIKIoZSStOLEmE8ASNSF9TjgIiB0l+VQqPtDKEfij04wrm6u+JBAVSTgNPd2ZLynkvE//z+rHyzwcJ5VGsCMezj/yYQRXCLCI4pIJgxaaaICyo3hXiMRIIKx1kRYdgz5+8SDonDdtq2Nen1aZdxFEGB+AQ1IANzkATXIEWaAMMHsATeAGvxqPxbLwZ77PWklHM7IM/MD6+ASRen14=</latexit><latexit sha1_base64="Uw0EndRX39LmlunUmhCKictknNM=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0WoUEoigi6LblxWsA9oaphMJ+3QySTOTIQS8hNu/BU3LhRxK7jzb5ykEbT1wMC559zL3Hu8iFGpLOvLKC0tr6yuldcrG5tb2zvm7l5HhrHApI1DFoqehyRhlJO2ooqRXiQICjxGut7kMvO790RIGvIbNY3IIEAjTn2KkdKSa9ZrToDU2POTKL1l9Z/iThfHDuUwFzBiyUXqMtesWg0rB1wkdkGqoEDLNT+dYYjjgHCFGZKyb1uRGiRIKIoZSStOLEmE8ASNSF9TjgIiB0l+VQqPtDKEfij04wrm6u+JBAVSTgNPd2ZLynkvE//z+rHyzwcJ5VGsCMezj/yYQRXCLCI4pIJgxaaaICyo3hXiMRIIKx1kRYdgz5+8SDonDdtq2Nen1aZdxFEGB+AQ1IANzkATXIEWaAMMHsATeAGvxqPxbLwZ77PWklHM7IM/MD6+ASRen14=</latexit><latexit sha1_base64="Uw0EndRX39LmlunUmhCKictknNM=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0WoUEoigi6LblxWsA9oaphMJ+3QySTOTIQS8hNu/BU3LhRxK7jzb5ykEbT1wMC559zL3Hu8iFGpLOvLKC0tr6yuldcrG5tb2zvm7l5HhrHApI1DFoqehyRhlJO2ooqRXiQICjxGut7kMvO790RIGvIbNY3IIEAjTn2KkdKSa9ZrToDU2POTKL1l9Z/iThfHDuUwFzBiyUXqMtesWg0rB1wkdkGqoEDLNT+dYYjjgHCFGZKyb1uRGiRIKIoZSStOLEmE8ASNSF9TjgIiB0l+VQqPtDKEfij04wrm6u+JBAVSTgNPd2ZLynkvE//z+rHyzwcJ5VGsCMezj/yYQRXCLCI4pIJgxaaaICyo3hXiMRIIKx1kRYdgz5+8SDonDdtq2Nen1aZdxFEGB+AQ1IANzkATXIEWaAMMHsATeAGvxqPxbLwZ77PWklHM7IM/MD6+ASRen14=</latexit><latexit sha1_base64="Uw0EndRX39LmlunUmhCKictknNM=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0WoUEoigi6LblxWsA9oaphMJ+3QySTOTIQS8hNu/BU3LhRxK7jzb5ykEbT1wMC559zL3Hu8iFGpLOvLKC0tr6yuldcrG5tb2zvm7l5HhrHApI1DFoqehyRhlJO2ooqRXiQICjxGut7kMvO790RIGvIbNY3IIEAjTn2KkdKSa9ZrToDU2POTKL1l9Z/iThfHDuUwFzBiyUXqMtesWg0rB1wkdkGqoEDLNT+dYYjjgHCFGZKyb1uRGiRIKIoZSStOLEmE8ASNSF9TjgIiB0l+VQqPtDKEfij04wrm6u+JBAVSTgNPd2ZLynkvE//z+rHyzwcJ5VGsCMezj/yYQRXCLCI4pIJgxaaaICyo3hXiMRIIKx1kRYdgz5+8SDonDdtq2Nen1aZdxFEGB+AQ1IANzkATXIEWaAMMHsATeAGvxqPxbLwZ77PWklHM7IM/MD6+ASRen14=</latexit>

q  voltages should lie in box v � 1 = R(pg � pl) +X(qg � ql) 2 Bv<latexit sha1_base64="RfZH7rFlbeO0NJ5uMx0lBzLa9Xo=">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</latexit><latexit sha1_base64="RfZH7rFlbeO0NJ5uMx0lBzLa9Xo=">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</latexit><latexit sha1_base64="RfZH7rFlbeO0NJ5uMx0lBzLa9Xo=">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</latexit><latexit sha1_base64="RfZH7rFlbeO0NJ5uMx0lBzLa9Xo=">AAACW3icbZFdS8MwFIbTTt2cU6vilTfBIUxkoxVBb4ShN15OcR+wzZFm6RaWpl2SDkbpn/RKL/wrYroPnJsHAs95z3vIyYkbMiqVbX8aZmZreyeb283vFfYPDq2j44YMIoFJHQcsEC0XScIoJ3VFFSOtUBDku4w03dFjWm9OiJA04K9qGpKujwacehQjpaWeJTo+UkPXiydJeYlOcr/El6S0xDB5G5RXEnZ5tcxav67xqmucujqUw5mAEYsfkt6kZxXtij0LuAnOAopgEbWe9d7pBzjyCVeYISnbjh2qboyEopiRJN+JJAkRHqEBaWvkyCeyG892k8ALrfShFwh9uIIzdbUjRr6UU9/VznRIuV5Lxf9q7Uh5d92Y8jBShOP5RV7EoApgumjYp4JgxaYaEBZUzwrxEAmElf6OvF6Cs/7kTWhcVxy74jzfFKvOYh05cAbOQQk44BZUwROogTrA4AN8G1kjZ3yZGTNvFuZW01j0nIA/YZ7+AMytt54=</latexit>

q  given check if (pg,qg),<latexit sha1_base64="ew/rUw0cV2OSvN7eRoPQxywv50k=">AAACBXicbVDLSgMxFL1TX7W+Rl3qIliECqXMiKDLghuXFewD2loyaaYNzWTGJCOUoRs3/oobF4q49R/c+Tdm2hG09UDg3HPuJfceL+JMacf5snJLyyura/n1wsbm1vaOvbvXUGEsCa2TkIey5WFFORO0rpnmtBVJigOP06Y3ukz95j2VioXiRo8j2g3wQDCfEayN1LMPS50A66HnJ9HkdlD+Ke5McVLu2UWn4kyBFombkSJkqPXsz04/JHFAhSYcK9V2nUh3Eyw1I5xOCp1Y0QiTER7QtqECB1R1k+kVE3RslD7yQ2me0Giq/p5IcKDUOPBMZ7qlmvdS8T+vHWv/opswEcWaCjL7yI850iFKI0F9JinRfGwIJpKZXREZYomJNsEVTAju/MmLpHFacZ2Ke31WrLpZHHk4gCMogQvnUIUrqEEdCDzAE7zAq/VoPVtv1vusNWdlM/vwB9bHNzhsmFI=</latexit><latexit sha1_base64="ew/rUw0cV2OSvN7eRoPQxywv50k=">AAACBXicbVDLSgMxFL1TX7W+Rl3qIliECqXMiKDLghuXFewD2loyaaYNzWTGJCOUoRs3/oobF4q49R/c+Tdm2hG09UDg3HPuJfceL+JMacf5snJLyyura/n1wsbm1vaOvbvXUGEsCa2TkIey5WFFORO0rpnmtBVJigOP06Y3ukz95j2VioXiRo8j2g3wQDCfEayN1LMPS50A66HnJ9HkdlD+Ke5McVLu2UWn4kyBFombkSJkqPXsz04/JHFAhSYcK9V2nUh3Eyw1I5xOCp1Y0QiTER7QtqECB1R1k+kVE3RslD7yQ2me0Giq/p5IcKDUOPBMZ7qlmvdS8T+vHWv/opswEcWaCjL7yI850iFKI0F9JinRfGwIJpKZXREZYomJNsEVTAju/MmLpHFacZ2Ke31WrLpZHHk4gCMogQvnUIUrqEEdCDzAE7zAq/VoPVtv1vusNWdlM/vwB9bHNzhsmFI=</latexit><latexit sha1_base64="ew/rUw0cV2OSvN7eRoPQxywv50k=">AAACBXicbVDLSgMxFL1TX7W+Rl3qIliECqXMiKDLghuXFewD2loyaaYNzWTGJCOUoRs3/oobF4q49R/c+Tdm2hG09UDg3HPuJfceL+JMacf5snJLyyura/n1wsbm1vaOvbvXUGEsCa2TkIey5WFFORO0rpnmtBVJigOP06Y3ukz95j2VioXiRo8j2g3wQDCfEayN1LMPS50A66HnJ9HkdlD+Ke5McVLu2UWn4kyBFombkSJkqPXsz04/JHFAhSYcK9V2nUh3Eyw1I5xOCp1Y0QiTER7QtqECB1R1k+kVE3RslD7yQ2me0Giq/p5IcKDUOPBMZ7qlmvdS8T+vHWv/opswEcWaCjL7yI850iFKI0F9JinRfGwIJpKZXREZYomJNsEVTAju/MmLpHFacZ2Ke31WrLpZHHk4gCMogQvnUIUrqEEdCDzAE7zAq/VoPVtv1vusNWdlM/vwB9bHNzhsmFI=</latexit><latexit sha1_base64="ew/rUw0cV2OSvN7eRoPQxywv50k=">AAACBXicbVDLSgMxFL1TX7W+Rl3qIliECqXMiKDLghuXFewD2loyaaYNzWTGJCOUoRs3/oobF4q49R/c+Tdm2hG09UDg3HPuJfceL+JMacf5snJLyyura/n1wsbm1vaOvbvXUGEsCa2TkIey5WFFORO0rpnmtBVJigOP06Y3ukz95j2VioXiRo8j2g3wQDCfEayN1LMPS50A66HnJ9HkdlD+Ke5McVLu2UWn4kyBFombkSJkqPXsz04/JHFAhSYcK9V2nUh3Eyw1I5xOCp1Y0QiTER7QtqECB1R1k+kVE3RslD7yQ2me0Giq/p5IcKDUOPBMZ7qlmvdS8T+vHWv/opswEcWaCjL7yI850iFKI0F9JinRfGwIJpKZXREZYomJNsEVTAju/MmLpHFacZ2Ke31WrLpZHHk4gCMogQvnUIUrqEEdCDzAE7zAq/VoPVtv1vusNWdlM/vwB9bHNzhsmFI=</latexit>

Bl ✓ Bv<latexit sha1_base64="Rnnrsij5RwvCz5STpW2lej3NzWc=">AAACDXicbVBNS8NAEN34WetX1KOXxSp4KokIeix68VjBfkAbwmY7aZduNnF3Uyihf8CLf8WLB0W8evfmv3HT5lBbHww83pthZl6QcKa04/xYK6tr6xubpa3y9s7u3r59cNhUcSopNGjMY9kOiALOBDQ00xzaiQQSBRxawfA291sjkIrF4kGPE/Ai0hcsZJRoI/n2aTciekAJz24mPu+qNFCg4RHPyyPfrjhVZwq8TNyCVFCBum9/d3sxTSMQmnKiVMd1Eu1lRGpGOUzK3VRBQuiQ9KFjqCARKC+bfjPBZ0bp4TCWpoTGU3V+IiORUuMoMJ35kWrRy8X/vE6qw2svYyJJNQg6WxSmHOsY59HgHpNANR8bQqhk5lZMB0QSqk2AZROCu/jyMmleVF2n6t5fVmpuEUcJHaMTdI5cdIVq6A7VUQNR9IRe0Bt6t56tV+vD+py1rljFzBH6A+vrF4jinGo=</latexit><latexit sha1_base64="Rnnrsij5RwvCz5STpW2lej3NzWc=">AAACDXicbVBNS8NAEN34WetX1KOXxSp4KokIeix68VjBfkAbwmY7aZduNnF3Uyihf8CLf8WLB0W8evfmv3HT5lBbHww83pthZl6QcKa04/xYK6tr6xubpa3y9s7u3r59cNhUcSopNGjMY9kOiALOBDQ00xzaiQQSBRxawfA291sjkIrF4kGPE/Ai0hcsZJRoI/n2aTciekAJz24mPu+qNFCg4RHPyyPfrjhVZwq8TNyCVFCBum9/d3sxTSMQmnKiVMd1Eu1lRGpGOUzK3VRBQuiQ9KFjqCARKC+bfjPBZ0bp4TCWpoTGU3V+IiORUuMoMJ35kWrRy8X/vE6qw2svYyJJNQg6WxSmHOsY59HgHpNANR8bQqhk5lZMB0QSqk2AZROCu/jyMmleVF2n6t5fVmpuEUcJHaMTdI5cdIVq6A7VUQNR9IRe0Bt6t56tV+vD+py1rljFzBH6A+vrF4jinGo=</latexit><latexit sha1_base64="Rnnrsij5RwvCz5STpW2lej3NzWc=">AAACDXicbVBNS8NAEN34WetX1KOXxSp4KokIeix68VjBfkAbwmY7aZduNnF3Uyihf8CLf8WLB0W8evfmv3HT5lBbHww83pthZl6QcKa04/xYK6tr6xubpa3y9s7u3r59cNhUcSopNGjMY9kOiALOBDQ00xzaiQQSBRxawfA291sjkIrF4kGPE/Ai0hcsZJRoI/n2aTciekAJz24mPu+qNFCg4RHPyyPfrjhVZwq8TNyCVFCBum9/d3sxTSMQmnKiVMd1Eu1lRGpGOUzK3VRBQuiQ9KFjqCARKC+bfjPBZ0bp4TCWpoTGU3V+IiORUuMoMJ35kWrRy8X/vE6qw2svYyJJNQg6WxSmHOsY59HgHpNANR8bQqhk5lZMB0QSqk2AZROCu/jyMmleVF2n6t5fVmpuEUcJHaMTdI5cdIVq6A7VUQNR9IRe0Bt6t56tV+vD+py1rljFzBH6A+vrF4jinGo=</latexit><latexit sha1_base64="Rnnrsij5RwvCz5STpW2lej3NzWc=">AAACDXicbVBNS8NAEN34WetX1KOXxSp4KokIeix68VjBfkAbwmY7aZduNnF3Uyihf8CLf8WLB0W8evfmv3HT5lBbHww83pthZl6QcKa04/xYK6tr6xubpa3y9s7u3r59cNhUcSopNGjMY9kOiALOBDQ00xzaiQQSBRxawfA291sjkIrF4kGPE/Ai0hcsZJRoI/n2aTciekAJz24mPu+qNFCg4RHPyyPfrjhVZwq8TNyCVFCBum9/d3sxTSMQmnKiVMd1Eu1lRGpGOUzK3VRBQuiQ9KFjqCARKC+bfjPBZ0bp4TCWpoTGU3V+IiORUuMoMJ35kWrRy8X/vE6qw2svYyJJNQg6WxSmHOsY59HgHpNANR8bQqhk5lZMB0QSqk2AZROCu/jyMmleVF2n6t5fVmpuEUcJHaMTdI5cdIVq6A7VUQNR9IRe0Bt6t56tV+vD+py1rljFzBH6A+vrF4jinGo=</latexit>

q  Farkas’ lemma for containment of polytopes

Q2) Probing injection design

10

n  Step 3: Find T most diverse injection vectors

0 5 10 15 20 25 30 35Bus number

0.9

1

1.1

volta

ge m

agni

tude

(p.u

.)

Probing scheme for T=2

-0.1

0

0.1

volta

ge a

ngle

(rad

)

Condition numbers improved by an order of magnitude via injection design

q  linearized grid model u ' Ks<latexit sha1_base64="q4/6JjQ39lSM5i9PRe1Y2kcS3kY=">AAACDnicbZDLSsNAFIYnXmu9RV26GSwFVyURQZcFN4KbCvYCTSmT6Uk7dCaJMxOhhDyBG1/FjQtF3Lp259s4aVPQ1h8GPv5zDnPO78ecKe0439bK6tr6xmZpq7y9s7u3bx8ctlSUSApNGvFIdnyigLMQmpppDp1YAhE+h7Y/vsrr7QeQikXhnZ7E0BNkGLKAUaKN1berniB65AdpknmKCbjHc+Mmm5PK+nbFqTlT4WVwC6igQo2+/eUNIpoICDXlRKmu68S6lxKpGeWQlb1EQUzomAyhazAkAlQvnZ6T4apxBjiIpHmhxlP390RKhFIT4ZvOfEO1WMvN/2rdRAeXvZSFcaIhpLOPgoRjHeE8GzxgEqjmEwOESmZ2xXREJKHaJFg2IbiLJy9D66zmOjX39rxSd4s4SugYnaBT5KILVEfXqIGaiKJH9Ixe0Zv1ZL1Y79bHrHXFKmaO0B9Znz/Wup0s</latexit><latexit sha1_base64="q4/6JjQ39lSM5i9PRe1Y2kcS3kY=">AAACDnicbZDLSsNAFIYnXmu9RV26GSwFVyURQZcFN4KbCvYCTSmT6Uk7dCaJMxOhhDyBG1/FjQtF3Lp259s4aVPQ1h8GPv5zDnPO78ecKe0439bK6tr6xmZpq7y9s7u3bx8ctlSUSApNGvFIdnyigLMQmpppDp1YAhE+h7Y/vsrr7QeQikXhnZ7E0BNkGLKAUaKN1berniB65AdpknmKCbjHc+Mmm5PK+nbFqTlT4WVwC6igQo2+/eUNIpoICDXlRKmu68S6lxKpGeWQlb1EQUzomAyhazAkAlQvnZ6T4apxBjiIpHmhxlP390RKhFIT4ZvOfEO1WMvN/2rdRAeXvZSFcaIhpLOPgoRjHeE8GzxgEqjmEwOESmZ2xXREJKHaJFg2IbiLJy9D66zmOjX39rxSd4s4SugYnaBT5KILVEfXqIGaiKJH9Ixe0Zv1ZL1Y79bHrHXFKmaO0B9Znz/Wup0s</latexit><latexit sha1_base64="q4/6JjQ39lSM5i9PRe1Y2kcS3kY=">AAACDnicbZDLSsNAFIYnXmu9RV26GSwFVyURQZcFN4KbCvYCTSmT6Uk7dCaJMxOhhDyBG1/FjQtF3Lp259s4aVPQ1h8GPv5zDnPO78ecKe0439bK6tr6xmZpq7y9s7u3bx8ctlSUSApNGvFIdnyigLMQmpppDp1YAhE+h7Y/vsrr7QeQikXhnZ7E0BNkGLKAUaKN1berniB65AdpknmKCbjHc+Mmm5PK+nbFqTlT4WVwC6igQo2+/eUNIpoICDXlRKmu68S6lxKpGeWQlb1EQUzomAyhazAkAlQvnZ6T4apxBjiIpHmhxlP390RKhFIT4ZvOfEO1WMvN/2rdRAeXvZSFcaIhpLOPgoRjHeE8GzxgEqjmEwOESmZ2xXREJKHaJFg2IbiLJy9D66zmOjX39rxSd4s4SugYnaBT5KILVEfXqIGaiKJH9Ixe0Zv1ZL1Y79bHrHXFKmaO0B9Znz/Wup0s</latexit><latexit sha1_base64="q4/6JjQ39lSM5i9PRe1Y2kcS3kY=">AAACDnicbZDLSsNAFIYnXmu9RV26GSwFVyURQZcFN4KbCvYCTSmT6Uk7dCaJMxOhhDyBG1/FjQtF3Lp259s4aVPQ1h8GPv5zDnPO78ecKe0439bK6tr6xmZpq7y9s7u3bx8ctlSUSApNGvFIdnyigLMQmpppDp1YAhE+h7Y/vsrr7QeQikXhnZ7E0BNkGLKAUaKN1berniB65AdpknmKCbjHc+Mmm5PK+nbFqTlT4WVwC6igQo2+/eUNIpoICDXlRKmu68S6lxKpGeWQlb1EQUzomAyhazAkAlQvnZ6T4apxBjiIpHmhxlP390RKhFIT4ZvOfEO1WMvN/2rdRAeXvZSFcaIhpLOPgoRjHeE8GzxgEqjmEwOESmZ2xXREJKHaJFg2IbiLJy9D66zmOjX39rxSd4s4SugYnaBT5KILVEfXqIGaiKJH9Ixe0Zv1ZL1Y79bHrHXFKmaO0B9Znz/Wup0s</latexit>

q  distance between induced voltages ku� u0k22 ' (sg � s0g)>K>K(sg � s0g)

<latexit sha1_base64="D64iNQagYEkAas9QD3Ya35G1ayY=">AAACZHicfVFbS8MwGE3rfd6q4pMgwSHOh412CPoo+CL4MsE5Ya0lzdItLGlqkgqj9k/65qMv/g7TXZg3/CBwzvm+Q76cRCmjSrvum2UvLC4tr6yuVdY3Nre2nZ3deyUyiUkbCybkQ4QUYTQhbU01Iw+pJIhHjHSi4VXZ7zwTqahI7vQoJQFH/YTGFCNtpNDJ/RefIz2I4jwr6nN44r+EzcemrygnT7WZroqwX/9KTk4ffS3SmXRTjCmc8/+soVN1G+644G/gTUEVTKsVOq9+T+CMk0RjhpTqem6qgxxJTTEjRcXPFEkRHqI+6RqYIE5UkI9DKuCxUXowFtKcRMOx+tWRI67UiEdmstxS/eyV4l+9bqbjiyCnSZppkuDJRXHGoBawTBz2qCRYs5EBCEtqdoV4gCTC2vxLxYTg/Xzyb3DfbHhuw7s9q1560zhWwQE4AjXggXNwCa5BC7QBBu/WiuVYO9aHvWHv2fuTUduaevbAt7IPPwFdC7on</latexit><latexit sha1_base64="D64iNQagYEkAas9QD3Ya35G1ayY=">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</latexit><latexit sha1_base64="D64iNQagYEkAas9QD3Ya35G1ayY=">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</latexit><latexit sha1_base64="D64iNQagYEkAas9QD3Ya35G1ayY=">AAACZHicfVFbS8MwGE3rfd6q4pMgwSHOh412CPoo+CL4MsE5Ya0lzdItLGlqkgqj9k/65qMv/g7TXZg3/CBwzvm+Q76cRCmjSrvum2UvLC4tr6yuVdY3Nre2nZ3deyUyiUkbCybkQ4QUYTQhbU01Iw+pJIhHjHSi4VXZ7zwTqahI7vQoJQFH/YTGFCNtpNDJ/RefIz2I4jwr6nN44r+EzcemrygnT7WZroqwX/9KTk4ffS3SmXRTjCmc8/+soVN1G+644G/gTUEVTKsVOq9+T+CMk0RjhpTqem6qgxxJTTEjRcXPFEkRHqI+6RqYIE5UkI9DKuCxUXowFtKcRMOx+tWRI67UiEdmstxS/eyV4l+9bqbjiyCnSZppkuDJRXHGoBawTBz2qCRYs5EBCEtqdoV4gCTC2vxLxYTg/Xzyb3DfbHhuw7s9q1560zhWwQE4AjXggXNwCa5BC7QBBu/WiuVYO9aHvWHv2fuTUduaevbAt7IPPwFdC7on</latexit>

q  select T injection vectors from library with largest pair-wise distances

Q3) CPF as penalized SDR

11

n  Solve CPF as relaxed and penalized SDP

Madani, Ashpraphijuo, Lavaei, & Baldick, ‘PSSE with limited number of measurements,’ IEEE CDC’16.

n  Lift states to psd and rank-one matrices {vt} {Vt = vtvH

t}Tt=1

n  Rank-one solutions guaranteed for and lightly loaded system M = �B

minX

t2TTr(MVt)

s.t. Tr(MnVt) = stn, n 2 Mt, t 2 T

Tr(MnVt) = Tr(MnVt+1), n 2 O, t 2 T0

Vt ⌫ 0, t 2 T

rk(Vt) = 1, t 2 T<latexit sha1_base64="aONBoaSFD8gN4R2SLWfMxPdHxAg=">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</latexit><latexit sha1_base64="aONBoaSFD8gN4R2SLWfMxPdHxAg=">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</latexit><latexit sha1_base64="aONBoaSFD8gN4R2SLWfMxPdHxAg=">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</latexit><latexit sha1_base64="aONBoaSFD8gN4R2SLWfMxPdHxAg=">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</latexit>

Q3) CPSSE as penalized SDR

12

n  Inverter data are noisy and non-metered injections may vary during probing

n  Data fitting options

- weighted least-squares (WLS)

- weighted least-absolute value (WLAV)

min �X

t2T

Tr(MVt) +X

t2T

X

n2M

fn(✏tn) +

X

t2T 0

X

k2O

fn(✏tn)

s.to Tr(MnVt) + ✏tn = stn, n 2 M, t 2 T

Tr(MnVt) = Tr(MnVt+1) + ✏tn, n 2 O, t 2 T0

Vt ⌫ 0, t 2 T<latexit sha1_base64="CaP97BI9jr7r0KFLlkuQJqI86dY=">AAAEYHicjVNda9swFHWTbM2yjybb2/YiFtamJAS7DLaXQmEveyntIEkLUWpkRUlELNmR5LEglL+57XU/ZEx23NE47RaBzOHce4/OkVEQh1Qq1/25VypXHj3erz6pPX32/MVBvfFyIKNEYNLHURiJ6wBJElJO+oqqkFzHgiAWhOQqmH9K61dfiZA04j21jMmIoSmnE4qRspTfKC0goxyswOEKwCliDAEoE+ZrBSmHDKkZRqHuGaOhIt+UYLonTCsrBBN9bm7RwPjq2ID2A9NrlgNLg7/8uTFg4vNWppxF0YKM7UkkljS07viNMscPaR7lovNN0YudRCGsZRM2j+yqyIDV4er+gD7fiNj+p+4pnCGlpblRPu8AABcJGuffrewdUAhkPe3q4nSHLq3anvmP386GxaLHi22PR7nJO2bSv4MxWYBb0i3IFiRqfr3pdt1sgW3g5aDp5OvSr3+H4wgnjHCFQyTl0HNjNdJIKIpDYmowkSRGeI6mZGghR4zIkc4yG/DOMmMwiYTdXIGMvTuhEZNyyQLbmXqUxVpKdqxzfF/DMFGTjyNNeZwowvH6tEkSAhWB9LWBMRUEq3BpAcKCWsMAz5BAWNk3md6EV8y9DQYnXc/tel/eN8+8/E6qzhvnrdNyPOeDc+Z8di6dvoNLP0q/y/vlavlXpVo5qDTWraW9fOaVs7Eqr/8AruZ+fA==</latexit><latexit sha1_base64="CaP97BI9jr7r0KFLlkuQJqI86dY=">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</latexit><latexit sha1_base64="CaP97BI9jr7r0KFLlkuQJqI86dY=">AAAEYHicjVNda9swFHWTbM2yjybb2/YiFtamJAS7DLaXQmEveyntIEkLUWpkRUlELNmR5LEglL+57XU/ZEx23NE47RaBzOHce4/OkVEQh1Qq1/25VypXHj3erz6pPX32/MVBvfFyIKNEYNLHURiJ6wBJElJO+oqqkFzHgiAWhOQqmH9K61dfiZA04j21jMmIoSmnE4qRspTfKC0goxyswOEKwCliDAEoE+ZrBSmHDKkZRqHuGaOhIt+UYLonTCsrBBN9bm7RwPjq2ID2A9NrlgNLg7/8uTFg4vNWppxF0YKM7UkkljS07viNMscPaR7lovNN0YudRCGsZRM2j+yqyIDV4er+gD7fiNj+p+4pnCGlpblRPu8AABcJGuffrewdUAhkPe3q4nSHLq3anvmP386GxaLHi22PR7nJO2bSv4MxWYBb0i3IFiRqfr3pdt1sgW3g5aDp5OvSr3+H4wgnjHCFQyTl0HNjNdJIKIpDYmowkSRGeI6mZGghR4zIkc4yG/DOMmMwiYTdXIGMvTuhEZNyyQLbmXqUxVpKdqxzfF/DMFGTjyNNeZwowvH6tEkSAhWB9LWBMRUEq3BpAcKCWsMAz5BAWNk3md6EV8y9DQYnXc/tel/eN8+8/E6qzhvnrdNyPOeDc+Z8di6dvoNLP0q/y/vlavlXpVo5qDTWraW9fOaVs7Eqr/8AruZ+fA==</latexit><latexit sha1_base64="CaP97BI9jr7r0KFLlkuQJqI86dY=">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</latexit>

fn(✏) =✏2

�2n

<latexit sha1_base64="5jNb93veFoRU+v93K/Vx/00pwCI=">AAACFnicbZDLSsNAFIZP6q3WW9Slm2AR6sKadKMboeDGZQV7gaaGyXTSDp1MwsxEKKFP4cZXcSOiiAs34s4H8D2ctBW09YeBb/5zDjPn92NGpbLtTyO3sLi0vJJfLaytb2xumds7DRklApM6jlgkWj6ShFFO6ooqRlqxICj0GWn6g/Os3rwhQtKIX6lhTDoh6nEaUIyUtjzzKPB4ySWxpCzih2duIBBOf+7XlVHqStoLkZdxoeCZRbtsj2XNgzOFYvX46+0RAGqe+eF2I5yEhCvMkJRtx45VJ0VCUczIqOAmksQID1CPtDVyFBLZScdrjawD7XStIBL6cGWN3d8TKQqlHIa+7gyR6svZWmb+V2snKjjtpJTHiSIcTx4KEmapyMoysrpUEKzYUAPCguq/WriPdDJKJ5mF4MyuPA+NStmxy86lTsOBifKwB/tQAgdOoAoXUIM6YLiFe3iCZ+POeDBejNdJa86YzuzCHxnv38jgofo=</latexit><latexit sha1_base64="zyhQYqGZN57hvqNBpdjZFM/U7ok=">AAACFnicbZDLSgMxFIYz9VbrbdSlm8EiVMQ6041uhIIblxXsBTrjkEkzbWiSGZKMUIY+hS58FUFcKOLCjbjrA/geZtoKav0h8OU/55CcP4gpkcq2R0Zubn5hcSm/XFhZXVvfMDe3GjJKBMJ1FNFItAIoMSUc1xVRFLdigSELKG4G/bOs3rzGQpKIX6pBjD0Gu5yEBEGlLd88DH1ecnEsCY34/qkbCojS7/tVZZi6knQZ9DMuFHyzaJftsaxZcKZQrB59vj0c3I5qvvnhdiKUMMwVolDKtmPHykuhUARRPCy4icQxRH3YxW2NHDIsvXS81tDa007HCiOhD1fW2P05kUIm5YAFupNB1ZN/a5n5X62dqPDESwmPE4U5mjwUJtRSkZVlZHWIwEjRgQaIBNF/tVAP6mSUTjILwfm78iw0KmXHLjsXOg0HTJQHO2AXlIADjkEVnIMaqAMEbsA9eALPxp3xaLwYr5PWnDGd2Qa/ZLx/AeQAo44=</latexit><latexit sha1_base64="zyhQYqGZN57hvqNBpdjZFM/U7ok=">AAACFnicbZDLSgMxFIYz9VbrbdSlm8EiVMQ6041uhIIblxXsBTrjkEkzbWiSGZKMUIY+hS58FUFcKOLCjbjrA/geZtoKav0h8OU/55CcP4gpkcq2R0Zubn5hcSm/XFhZXVvfMDe3GjJKBMJ1FNFItAIoMSUc1xVRFLdigSELKG4G/bOs3rzGQpKIX6pBjD0Gu5yEBEGlLd88DH1ecnEsCY34/qkbCojS7/tVZZi6knQZ9DMuFHyzaJftsaxZcKZQrB59vj0c3I5qvvnhdiKUMMwVolDKtmPHykuhUARRPCy4icQxRH3YxW2NHDIsvXS81tDa007HCiOhD1fW2P05kUIm5YAFupNB1ZN/a5n5X62dqPDESwmPE4U5mjwUJtRSkZVlZHWIwEjRgQaIBNF/tVAP6mSUTjILwfm78iw0KmXHLjsXOg0HTJQHO2AXlIADjkEVnIMaqAMEbsA9eALPxp3xaLwYr5PWnDGd2Qa/ZLx/AeQAo44=</latexit><latexit sha1_base64="mYN97UF+HVS1jCIT23BDIC1n3Bw=">AAACFnicbZBNS8MwGMfT+TbrW9Wjl+AQ5sHR7qIXYeDF4wT3AussaZZuYWlSklQYZZ/Ci1/FiwdFvIo3v43pVkE3/xD45f88D8nzDxNGlXbdL6u0srq2vlHetLe2d3b3nP2DthKpxKSFBROyGyJFGOWkpalmpJtIguKQkU44vsrrnXsiFRX8Vk8S0o/RkNOIYqSNFThnUcCrPkkUZYKfXvqRRDj7ud/Vp5mv6DBGQc62HTgVt+bOBJfBK6ACCjUD59MfCJzGhGvMkFI9z010P0NSU8zI1PZTRRKEx2hIegY5ionqZ7O1pvDEOAMYCWkO13Dm/p7IUKzUJA5NZ4z0SC3WcvO/Wi/V0UU/ozxJNeF4/lCUMqgFzDOCAyoJ1mxiAGFJzV8hHiGTjDZJ5iF4iysvQ7te89yad+NWGl4RRxkcgWNQBR44Bw1wDZqgBTB4AE/gBbxaj9az9Wa9z1tLVjFzCP7I+vgG6NqfGw==</latexit>

fn(✏) =|✏|�n

<latexit sha1_base64="VHqwBd/IIuVx0xhnWTUnxrGrNOE=">AAACEnicbVDLSsNAFL2pr1pfUZdugkVoQUriRjdCwY3LCvYBTYiT6aQdOpmEmYlQ0n6DG0HwP9y4UMStK3f+jdOHoK0HLhzOuZd77wkSRqWy7S8jt7S8srqWXy9sbG5t75i7ew0ZpwKTOo5ZLFoBkoRRTuqKKkZaiSAoChhpBv2Lsd+8JULSmF+rQUK8CHU5DSlGSku+WQ59XnJJIimLefncDQXC2fBHGI4yV9JuhHw+8s2iXbEnsBaJMyPF6vHD4w0A1Hzz0+3EOI0IV5ghKduOnSgvQ0JRzMio4KaSJAj3UZe0NeUoItLLJi+NrCOtdKwwFrq4sibq74kMRVIOokB3Rkj15Lw3Fv/z2qkKz7yM8iRVhOPpojBlloqtcT5WhwqCFRtogrCg+lYL95BORekUCzoEZ/7lRdI4qTh2xbnSaTgwRR4O4BBK4MApVOESalAHDHfwBC/watwbz8ab8T5tzRmzmX34A+PjG2t6oLQ=</latexit><latexit sha1_base64="9LYXCjfrsyHYpPnloMGgUkae/dQ=">AAACEnicbVDLSsNAFJ34rPVVdekmWIQWpCRudCMU3LisYB/QhDCZTtqh8wgzE6Gk+QY3guBXuHHjQhG3rtz5G36B04egrQcuHM65l3vvCWNKlHacT2thcWl5ZTW3ll/f2NzaLuzsNpRIJMJ1JKiQrRAqTAnHdU00xa1YYshCipth/3zkN6+xVETwKz2Isc9gl5OIIKiNFBTKUcBLHo4VoYKXz7xIQpQOf4RhlnqKdBkMeBYUik7FGcOeJ+6UFKtHd/f92sNXLSh8eB2BEoa5RhQq1XadWPsplJogirO8lygcQ9SHXdw2lEOGlZ+OX8rsQ6N07EhIU1zbY/X3RAqZUgMWmk4GdU/NeiPxP6+d6OjUTwmPE405miyKEmprYY/ysTtEYqTpwBCIJDG32qgHTSrapJg3IbizL8+TxnHFdSrupUnDBRPkwD44ACXgghNQBRegBuoAgRvwCJ7Bi3VrPVmv1tukdcGazuyBP7DevwETGaKx</latexit><latexit sha1_base64="9LYXCjfrsyHYpPnloMGgUkae/dQ=">AAACEnicbVDLSsNAFJ34rPVVdekmWIQWpCRudCMU3LisYB/QhDCZTtqh8wgzE6Gk+QY3guBXuHHjQhG3rtz5G36B04egrQcuHM65l3vvCWNKlHacT2thcWl5ZTW3ll/f2NzaLuzsNpRIJMJ1JKiQrRAqTAnHdU00xa1YYshCipth/3zkN6+xVETwKz2Isc9gl5OIIKiNFBTKUcBLHo4VoYKXz7xIQpQOf4RhlnqKdBkMeBYUik7FGcOeJ+6UFKtHd/f92sNXLSh8eB2BEoa5RhQq1XadWPsplJogirO8lygcQ9SHXdw2lEOGlZ+OX8rsQ6N07EhIU1zbY/X3RAqZUgMWmk4GdU/NeiPxP6+d6OjUTwmPE405miyKEmprYY/ysTtEYqTpwBCIJDG32qgHTSrapJg3IbizL8+TxnHFdSrupUnDBRPkwD44ACXgghNQBRegBuoAgRvwCJ7Bi3VrPVmv1tukdcGazuyBP7DevwETGaKx</latexit><latexit sha1_base64="mmWmMv4DP9LqT6R8PP6IBwZLqV0=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSK0m5K40Y1QcOOygn1AE8JkOmmHziPMTISS5hvc+CtuXCji1pU7/8ZpG0FbD1w4nHMv994TJZQo7bpfVmltfWNzq7xd2dnd2z+wD486SqQS4TYSVMheBBWmhOO2JpriXiIxZBHF3Wh8PfO791gqIvidniQ4YHDISUwQ1EYK7Xoc8pqPE0Wo4PUrP5YQZdMfYZpnviJDBkOeh3bVbbhzOKvEK0gVFGiF9qc/EChlmGtEoVJ9z010kEGpCaI4r/ipwglEYzjEfUM5ZFgF2fyl3DkzysCJhTTFtTNXf09kkCk1YZHpZFCP1LI3E//z+qmOL4OM8CTVmKPFojiljhbOLB9nQCRGmk4MgUgSc6uDRtCkok2KFROCt/zyKumcNzy34d261aZXxFEGJ+AU1IAHLkAT3IAWaAMEHsATeAGv1qP1bL1Z74vWklXMHIM/sD6+AbwJnrc=</latexit>

Zhu & Giannakis, ‘Robust power system state estimation for the nonlinear AC flow model,’ NAPS’12

Numerical tests

13

n  Tracking O=6 buses on the IEEE 33-bus feeder

n  Actual load/solar data from the Pecan Str project

T=4 T=6

Grid topology learning

n  Grid topology also needed for feeder monitoring and control

14

n  Exact voltage covariance [Bolognani’13], [Deka-Backhaus-Chertkov’15], [Li-Scaglione-Poor’15]

n  Graphical models and tree learning [Weng-Liao-Rajagopal’17], [Sevlian-Rajagopal’17]

n  Deep neural networks [Zhao-Chen-Poor’17]

n  LS fit with micro-PMU data [Yuan-Ardakanian-Low-Tomlin’17]

Passive data collection

topology detection topology identification

n  So far, grid topology has been assumed known

Approximate grid model

15

n  Single-phase radial grid with N+1 nodes and N lines

n  Linear distribution flow (LDF) model [Baran-Wu’89], [Deka et al’17]

vn

pn, qnpm, qm

vmnode m

node n

inverse of reduced graph Laplacian

a` =

2

66664

0+10�10

3

77775

r` + jx`

n  Differential quantities

v ' Rp+Xq+ 1N<latexit sha1_base64="YWwLwcea/jGQVSoNy9ZLqtRAXpk=">AAACHXicbZDLSsNAFIYn9VbrLerSzWARBKEkoqi7ghtXUsXYQhPCZDpph04unZkUSsiTuPFV3LhQceFGfBsnbYra+sPAx3/OYc75vZhRIQ3jSystLC4tr5RXK2vrG5tb+vbOvYgSjomFIxbxlocEYTQklqSSkVbMCQo8Rppe/zKvN4eECxqFd3IUEydA3ZD6FCOpLFc/tQMke56fDjNb0IAM4NS4jbOjKbcGP2xm7rWrV42aMRacB7OAKijUcPUPuxPhJCChxAwJ0TaNWDop4pJiRrKKnQgSI9xHXdJWGKKACCcdn5fBA+V0oB9x9UIJx+7viRQFQowCT3XmO4rZWm7+V2sn0j93UhrGiSQhnnzkJwzKCOZZwQ7lBEs2UoAwp2pXiHuIIyxVohUVgjl78jxYx7WLmnlzUq2bRRplsAf2wSEwwRmogyvQABbA4AE8gRfwqj1qz9qb9j5pLWnFzC74I+3zG2OMo2g=</latexit><latexit sha1_base64="YWwLwcea/jGQVSoNy9ZLqtRAXpk=">AAACHXicbZDLSsNAFIYn9VbrLerSzWARBKEkoqi7ghtXUsXYQhPCZDpph04unZkUSsiTuPFV3LhQceFGfBsnbYra+sPAx3/OYc75vZhRIQ3jSystLC4tr5RXK2vrG5tb+vbOvYgSjomFIxbxlocEYTQklqSSkVbMCQo8Rppe/zKvN4eECxqFd3IUEydA3ZD6FCOpLFc/tQMke56fDjNb0IAM4NS4jbOjKbcGP2xm7rWrV42aMRacB7OAKijUcPUPuxPhJCChxAwJ0TaNWDop4pJiRrKKnQgSI9xHXdJWGKKACCcdn5fBA+V0oB9x9UIJx+7viRQFQowCT3XmO4rZWm7+V2sn0j93UhrGiSQhnnzkJwzKCOZZwQ7lBEs2UoAwp2pXiHuIIyxVohUVgjl78jxYx7WLmnlzUq2bRRplsAf2wSEwwRmogyvQABbA4AE8gRfwqj1qz9qb9j5pLWnFzC74I+3zG2OMo2g=</latexit><latexit sha1_base64="YWwLwcea/jGQVSoNy9ZLqtRAXpk=">AAACHXicbZDLSsNAFIYn9VbrLerSzWARBKEkoqi7ghtXUsXYQhPCZDpph04unZkUSsiTuPFV3LhQceFGfBsnbYra+sPAx3/OYc75vZhRIQ3jSystLC4tr5RXK2vrG5tb+vbOvYgSjomFIxbxlocEYTQklqSSkVbMCQo8Rppe/zKvN4eECxqFd3IUEydA3ZD6FCOpLFc/tQMke56fDjNb0IAM4NS4jbOjKbcGP2xm7rWrV42aMRacB7OAKijUcPUPuxPhJCChxAwJ0TaNWDop4pJiRrKKnQgSI9xHXdJWGKKACCcdn5fBA+V0oB9x9UIJx+7viRQFQowCT3XmO4rZWm7+V2sn0j93UhrGiSQhnnzkJwzKCOZZwQ7lBEs2UoAwp2pXiHuIIyxVohUVgjl78jxYx7WLmnlzUq2bRRplsAf2wSEwwRmogyvQABbA4AE8gRfwqj1qz9qb9j5pLWnFzC74I+3zG2OMo2g=</latexit><latexit sha1_base64="YWwLwcea/jGQVSoNy9ZLqtRAXpk=">AAACHXicbZDLSsNAFIYn9VbrLerSzWARBKEkoqi7ghtXUsXYQhPCZDpph04unZkUSsiTuPFV3LhQceFGfBsnbYra+sPAx3/OYc75vZhRIQ3jSystLC4tr5RXK2vrG5tb+vbOvYgSjomFIxbxlocEYTQklqSSkVbMCQo8Rppe/zKvN4eECxqFd3IUEydA3ZD6FCOpLFc/tQMke56fDjNb0IAM4NS4jbOjKbcGP2xm7rWrV42aMRacB7OAKijUcPUPuxPhJCChxAwJ0TaNWDop4pJiRrKKnQgSI9xHXdJWGKKACCcdn5fBA+V0oB9x9UIJx+7viRQFQowCT3XmO4rZWm7+V2sn0j93UhrGiSQhnnzkJwzKCOZZwQ7lBEs2UoAwp2pXiHuIIyxVohUVgjl78jxYx7WLmnlzUq2bRRplsAf2wSEwwRmogyvQABbA4AE8gRfwqj1qz9qb9j5pLWnFzC74I+3zG2OMo2g=</latexit>

vt = Rpt +Xqt + nt

pt := pt � pt�1<latexit sha1_base64="r4F+sxmiLvk8h6U/zFL0Em+vR+g=">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</latexit><latexit sha1_base64="r4F+sxmiLvk8h6U/zFL0Em+vR+g=">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</latexit><latexit sha1_base64="r4F+sxmiLvk8h6U/zFL0Em+vR+g=">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</latexit><latexit sha1_base64="r4F+sxmiLvk8h6U/zFL0Em+vR+g=">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</latexit>

R :=

X

`2L

1

2r`a`a

>`

!�1

<latexit sha1_base64="PpeOcI/nA80l6xouw/RLS9lWKoU=">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</latexit><latexit sha1_base64="PpeOcI/nA80l6xouw/RLS9lWKoU=">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</latexit><latexit sha1_base64="PpeOcI/nA80l6xouw/RLS9lWKoU=">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</latexit><latexit sha1_base64="PpeOcI/nA80l6xouw/RLS9lWKoU=">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</latexit>

q 

q 

Probing for topology identification

16

vt = Rpt +Xqt + nt<latexit sha1_base64="vmEG1zfch+CMsYiAagGUqdZcfYU=">AAACTXicbVFNSwMxEM3Wj9b6VfXoJVgEQSi7IqgHoeDFYxWrhbaUbDprQ7PZNZktlGV/oRfx5s/w4kFFTD8UrQ4E3rz3hkxe/FgKg6775OTm5hcW84Wl4vLK6tp6aWPz2kSJ5lDnkYx0w2cGpFBQR4ESGrEGFvoSbvz+2Ui/GYA2IlJXOIyhHbJbJQLBGVqqU4IWCtmFtBUy7PlBOsiyDtJTSr+Iy2zGEY8d+9+GxqzhbmL4apXtOqWyW3HHRf8CbwrKZFq1Tumx1Y14EoJCLpkxTc+NsZ0yjYJLyIqtxEDMeJ/dQtNCxUIw7XQcR0Z3LdOlQaTtUUjH7M+JlIXGDEPfOkc7mlltRP6nNRMMjtupUHGCoPjkoiCRFCM6ypZ2hQaOcmgB41rYXSnvMc042h8o2hC82Sf/BfWDyknFuzgsV71pGgWyTXbIHvHIEamSc1IjdcLJPXkmr+TNeXBenHfnY2LNOdOZLfKrcvlPu/63RQ==</latexit><latexit sha1_base64="vmEG1zfch+CMsYiAagGUqdZcfYU=">AAACTXicbVFNSwMxEM3Wj9b6VfXoJVgEQSi7IqgHoeDFYxWrhbaUbDprQ7PZNZktlGV/oRfx5s/w4kFFTD8UrQ4E3rz3hkxe/FgKg6775OTm5hcW84Wl4vLK6tp6aWPz2kSJ5lDnkYx0w2cGpFBQR4ESGrEGFvoSbvz+2Ui/GYA2IlJXOIyhHbJbJQLBGVqqU4IWCtmFtBUy7PlBOsiyDtJTSr+Iy2zGEY8d+9+GxqzhbmL4apXtOqWyW3HHRf8CbwrKZFq1Tumx1Y14EoJCLpkxTc+NsZ0yjYJLyIqtxEDMeJ/dQtNCxUIw7XQcR0Z3LdOlQaTtUUjH7M+JlIXGDEPfOkc7mlltRP6nNRMMjtupUHGCoPjkoiCRFCM6ypZ2hQaOcmgB41rYXSnvMc042h8o2hC82Sf/BfWDyknFuzgsV71pGgWyTXbIHvHIEamSc1IjdcLJPXkmr+TNeXBenHfnY2LNOdOZLfKrcvlPu/63RQ==</latexit><latexit sha1_base64="vmEG1zfch+CMsYiAagGUqdZcfYU=">AAACTXicbVFNSwMxEM3Wj9b6VfXoJVgEQSi7IqgHoeDFYxWrhbaUbDprQ7PZNZktlGV/oRfx5s/w4kFFTD8UrQ4E3rz3hkxe/FgKg6775OTm5hcW84Wl4vLK6tp6aWPz2kSJ5lDnkYx0w2cGpFBQR4ESGrEGFvoSbvz+2Ui/GYA2IlJXOIyhHbJbJQLBGVqqU4IWCtmFtBUy7PlBOsiyDtJTSr+Iy2zGEY8d+9+GxqzhbmL4apXtOqWyW3HHRf8CbwrKZFq1Tumx1Y14EoJCLpkxTc+NsZ0yjYJLyIqtxEDMeJ/dQtNCxUIw7XQcR0Z3LdOlQaTtUUjH7M+JlIXGDEPfOkc7mlltRP6nNRMMjtupUHGCoPjkoiCRFCM6ypZ2hQaOcmgB41rYXSnvMc042h8o2hC82Sf/BfWDyknFuzgsV71pGgWyTXbIHvHIEamSc1IjdcLJPXkmr+TNeXBenHfnY2LNOdOZLfKrcvlPu/63RQ==</latexit><latexit sha1_base64="vmEG1zfch+CMsYiAagGUqdZcfYU=">AAACTXicbVFNSwMxEM3Wj9b6VfXoJVgEQSi7IqgHoeDFYxWrhbaUbDprQ7PZNZktlGV/oRfx5s/w4kFFTD8UrQ4E3rz3hkxe/FgKg6775OTm5hcW84Wl4vLK6tp6aWPz2kSJ5lDnkYx0w2cGpFBQR4ESGrEGFvoSbvz+2Ui/GYA2IlJXOIyhHbJbJQLBGVqqU4IWCtmFtBUy7PlBOsiyDtJTSr+Iy2zGEY8d+9+GxqzhbmL4apXtOqWyW3HHRf8CbwrKZFq1Tumx1Y14EoJCLpkxTc+NsZ0yjYJLyIqtxEDMeJ/dQtNCxUIw7XQcR0Z3LdOlQaTtUUjH7M+JlIXGDEPfOkc7mlltRP6nNRMMjtupUHGCoPjkoiCRFCM6ypZ2hQaOcmgB41rYXSnvMc042h8o2hC82Sf/BfWDyknFuzgsV71pGgWyTXbIHvHIEamSc1IjdcLJPXkmr+TNeXBenHfnY2LNOdOZLfKrcvlPu/63RQ==</latexit>

n  Probe grid over T>M periods and ignore noise

n  Active (vs. reactive) probing

qt = 0<latexit sha1_base64="ghuZXV4NPoXR1sOdY1/j2SC09Gs=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhbBVUlE0I1QcOOygn1AU8JkMmmHTh7O3AglZOnGX3HjQhG3foI7/8ZJG0FbDwycOede7r3HSwRXYFlfRmVpeWV1rbpe29jc2t4xd/c6Kk4lZW0ai1j2PKKY4BFrAwfBeolkJPQE63rjq8Lv3jOpeBzdwiRhg5AMIx5wSkBLrnnoABc+y5yQwMgLsrs8d+Hy52flrlm3GtYUeJHYJamjEi3X/HT8mKYhi4AKolTfthIYZEQCp4LlNSdVLCF0TIasr2lEQqYG2fSQHB9rxcdBLPWLAE/V3x0ZCZWahJ6uLDZU814h/uf1UwguBhmPkhRYRGeDglRgiHGRCva5ZBTERBNCJde7YjoiklDQ2dV0CPb8yYukc9qwrYZ9c1Zv2mUcVXSAjtAJstE5aqJr1EJtRNEDekIv6NV4NJ6NN+N9Vloxyp599AfGxzfjEJpz</latexit><latexit sha1_base64="ghuZXV4NPoXR1sOdY1/j2SC09Gs=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhbBVUlE0I1QcOOygn1AU8JkMmmHTh7O3AglZOnGX3HjQhG3foI7/8ZJG0FbDwycOede7r3HSwRXYFlfRmVpeWV1rbpe29jc2t4xd/c6Kk4lZW0ai1j2PKKY4BFrAwfBeolkJPQE63rjq8Lv3jOpeBzdwiRhg5AMIx5wSkBLrnnoABc+y5yQwMgLsrs8d+Hy52flrlm3GtYUeJHYJamjEi3X/HT8mKYhi4AKolTfthIYZEQCp4LlNSdVLCF0TIasr2lEQqYG2fSQHB9rxcdBLPWLAE/V3x0ZCZWahJ6uLDZU814h/uf1UwguBhmPkhRYRGeDglRgiHGRCva5ZBTERBNCJde7YjoiklDQ2dV0CPb8yYukc9qwrYZ9c1Zv2mUcVXSAjtAJstE5aqJr1EJtRNEDekIv6NV4NJ6NN+N9Vloxyp599AfGxzfjEJpz</latexit><latexit sha1_base64="ghuZXV4NPoXR1sOdY1/j2SC09Gs=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhbBVUlE0I1QcOOygn1AU8JkMmmHTh7O3AglZOnGX3HjQhG3foI7/8ZJG0FbDwycOede7r3HSwRXYFlfRmVpeWV1rbpe29jc2t4xd/c6Kk4lZW0ai1j2PKKY4BFrAwfBeolkJPQE63rjq8Lv3jOpeBzdwiRhg5AMIx5wSkBLrnnoABc+y5yQwMgLsrs8d+Hy52flrlm3GtYUeJHYJamjEi3X/HT8mKYhi4AKolTfthIYZEQCp4LlNSdVLCF0TIasr2lEQqYG2fSQHB9rxcdBLPWLAE/V3x0ZCZWahJ6uLDZU814h/uf1UwguBhmPkhRYRGeDglRgiHGRCva5ZBTERBNCJde7YjoiklDQ2dV0CPb8yYukc9qwrYZ9c1Zv2mUcVXSAjtAJstE5aqJr1EJtRNEDekIv6NV4NJ6NN+N9Vloxyp599AfGxzfjEJpz</latexit><latexit sha1_base64="ghuZXV4NPoXR1sOdY1/j2SC09Gs=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhbBVUlE0I1QcOOygn1AU8JkMmmHTh7O3AglZOnGX3HjQhG3foI7/8ZJG0FbDwycOede7r3HSwRXYFlfRmVpeWV1rbpe29jc2t4xd/c6Kk4lZW0ai1j2PKKY4BFrAwfBeolkJPQE63rjq8Lv3jOpeBzdwiRhg5AMIx5wSkBLrnnoABc+y5yQwMgLsrs8d+Hy52flrlm3GtYUeJHYJamjEi3X/HT8mKYhi4AKolTfthIYZEQCp4LlNSdVLCF0TIasr2lEQqYG2fSQHB9rxcdBLPWLAE/V3x0ZCZWahJ6uLDZU814h/uf1UwguBhmPkhRYRGeDglRgiHGRCva5ZBTERBNCJde7YjoiklDQ2dV0CPb8yYukc9qwrYZ9c1Zv2mUcVXSAjtAJstE5aqJr1EJtRNEDekIv6NV4NJ6NN+N9Vloxyp599AfGxzfjEJpz</latexit>

pt = IM�t : zero entries at non-probing buses O<latexit sha1_base64="2SK7p6mQhwtaV9ggX8EaFOlaMi0=">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</latexit><latexit sha1_base64="2SK7p6mQhwtaV9ggX8EaFOlaMi0=">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</latexit><latexit sha1_base64="2SK7p6mQhwtaV9ggX8EaFOlaMi0=">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</latexit><latexit sha1_base64="2SK7p6mQhwtaV9ggX8EaFOlaMi0=">AAACb3icbVFdaxQxFM2MVev6tdYHH1okuAj64DJTChVBKPSlPogV3Lawsyx3Mne2oZlkSO6Iaxgf/YG+9T/0xX9gZrqKtl4IOTn33Jybm7xW0lGSnEfxjbWbt26v3xncvXf/wcPho40jZxorcCKMMvYkB4dKapyQJIUntUWocoXH+dl+lz/+jNZJoz/RssZZBQstSymAAjUffs9IqgJ9VgGd5qWv23ZOb3+f3rXzHgpQ/n2b5UYVblmFzWcFKoKgfZMRfiH/Fa3hqMlKdByIa6Nf1dbkUi943jh07bde2HfsLRat/3Pzh+A5HCXjpA9+HaQrMGKrOJwPf2SFEU0VLIUC56ZpUtPMgyUpFLaDLHjWIM5ggdMANVToZr53b/nzwBS8NDYsTbxn/67wULnunUHZ9eiu5jryf7lpQ+XrmZe6bgi1uDQqG8XJ8G74vJAWBallACCsDL1ycQoWBIUvGoQhpFeffB0cbY/TZJx+3Bntba/Gsc422TP2gqVsl+2xA3bIJkywi2gj2oy2op/xk/hpzC+lcbSqecz+ifjlL0NZwYo=</latexit>

V = RIM�<latexit sha1_base64="DZ30uX3tUVvVMugOgN9nTN6DYk4=">AAACNXicbZDLSsNAFIYn3q23qEs3wSK4KokIuhEKulBQULG10IQwmZ7o4OTCzIlQwryUG9/DlS5cKOLWV3B6EbT1h4GP/5zDnPNHueAKXffFmpicmp6ZnZuvLCwuLa/Yq2tNlRWSQYNlIpOtiCoQPIUGchTQyiXQJBJwHd0d9urX9yAVz9Ir7OYQJPQm5TFnFI0V2qc+ctGB0k8o3kZx2dT64Icv9Q+d6HDQwagoz7QeGfKPQCDVOrSrbs3tyxkHbwhVMtR5aD/5nYwVCaTIBFWq7bk5BiWVyJkAXfELBTlld/QG2gZTmoAKyv7V2tkyTseJM2leik7f/T1R0kSpbhKZzt6earTWM/+rtQuM94OSp3mBkLLBR3EhHMycXoROh0tgKLoGKJPc7OqwWyopQxN0xYTgjZ48Ds2dmufWvIvdan1nGMcc2SCbZJt4ZI/UyTE5Jw3CyAN5Jm/k3Xq0Xq0P63PQOmENZ9bJH1lf31qvrr0=</latexit><latexit sha1_base64="DZ30uX3tUVvVMugOgN9nTN6DYk4=">AAACNXicbZDLSsNAFIYn3q23qEs3wSK4KokIuhEKulBQULG10IQwmZ7o4OTCzIlQwryUG9/DlS5cKOLWV3B6EbT1h4GP/5zDnPNHueAKXffFmpicmp6ZnZuvLCwuLa/Yq2tNlRWSQYNlIpOtiCoQPIUGchTQyiXQJBJwHd0d9urX9yAVz9Ir7OYQJPQm5TFnFI0V2qc+ctGB0k8o3kZx2dT64Icv9Q+d6HDQwagoz7QeGfKPQCDVOrSrbs3tyxkHbwhVMtR5aD/5nYwVCaTIBFWq7bk5BiWVyJkAXfELBTlld/QG2gZTmoAKyv7V2tkyTseJM2leik7f/T1R0kSpbhKZzt6earTWM/+rtQuM94OSp3mBkLLBR3EhHMycXoROh0tgKLoGKJPc7OqwWyopQxN0xYTgjZ48Ds2dmufWvIvdan1nGMcc2SCbZJt4ZI/UyTE5Jw3CyAN5Jm/k3Xq0Xq0P63PQOmENZ9bJH1lf31qvrr0=</latexit><latexit sha1_base64="DZ30uX3tUVvVMugOgN9nTN6DYk4=">AAACNXicbZDLSsNAFIYn3q23qEs3wSK4KokIuhEKulBQULG10IQwmZ7o4OTCzIlQwryUG9/DlS5cKOLWV3B6EbT1h4GP/5zDnPNHueAKXffFmpicmp6ZnZuvLCwuLa/Yq2tNlRWSQYNlIpOtiCoQPIUGchTQyiXQJBJwHd0d9urX9yAVz9Ir7OYQJPQm5TFnFI0V2qc+ctGB0k8o3kZx2dT64Icv9Q+d6HDQwagoz7QeGfKPQCDVOrSrbs3tyxkHbwhVMtR5aD/5nYwVCaTIBFWq7bk5BiWVyJkAXfELBTlld/QG2gZTmoAKyv7V2tkyTseJM2leik7f/T1R0kSpbhKZzt6earTWM/+rtQuM94OSp3mBkLLBR3EhHMycXoROh0tgKLoGKJPc7OqwWyopQxN0xYTgjZ48Ds2dmufWvIvdan1nGMcc2SCbZJt4ZI/UyTE5Jw3CyAN5Jm/k3Xq0Xq0P63PQOmENZ9bJH1lf31qvrr0=</latexit><latexit sha1_base64="DZ30uX3tUVvVMugOgN9nTN6DYk4=">AAACNXicbZDLSsNAFIYn3q23qEs3wSK4KokIuhEKulBQULG10IQwmZ7o4OTCzIlQwryUG9/DlS5cKOLWV3B6EbT1h4GP/5zDnPNHueAKXffFmpicmp6ZnZuvLCwuLa/Yq2tNlRWSQYNlIpOtiCoQPIUGchTQyiXQJBJwHd0d9urX9yAVz9Ir7OYQJPQm5TFnFI0V2qc+ctGB0k8o3kZx2dT64Icv9Q+d6HDQwagoz7QeGfKPQCDVOrSrbs3tyxkHbwhVMtR5aD/5nYwVCaTIBFWq7bk5BiWVyJkAXfELBTlld/QG2gZTmoAKyv7V2tkyTseJM2leik7f/T1R0kSpbhKZzt6earTWM/+rtQuM94OSp3mBkLLBR3EhHMycXoROh0tgKLoGKJPc7OqwWyopQxN0xYTgjZ48Ds2dmufWvIvdan1nGMcc2SCbZJt4ZI/UyTE5Jw3CyAN5Jm/k3Xq0Xq0P63PQOmENZ9bJH1lf31qvrr0=</latexit>

n  Complete data: voltages collected at all buses, recover

n  Partial data: voltages collected only at probing buses, recover

RM =

RM,M

RO,M

<latexit sha1_base64="b2vJMDwGm5KkKoUoxaFmwS2rJAY=">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</latexit><latexit sha1_base64="b2vJMDwGm5KkKoUoxaFmwS2rJAY=">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</latexit><latexit sha1_base64="b2vJMDwGm5KkKoUoxaFmwS2rJAY=">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</latexit><latexit sha1_base64="b2vJMDwGm5KkKoUoxaFmwS2rJAY=">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</latexit>

RM,M<latexit sha1_base64="cGk9ScV80SxvZGY7YUNCULyD11c=">AAACIHicbZDLSgMxFIYz9VbrbdSlm2ARXEiZEaEuC27cCFXsBdpSMumZNjRzITkjlmEexY2v4saFIrrTpzG9LGrrgcDH/5+TnPxeLIVGx/m2ciura+sb+c3C1vbO7p69f1DXUaI41HgkI9X0mAYpQqihQAnNWAELPAkNb3g19hsPoLSIwnscxdAJWD8UvuAMjdS1y22ER5zck3oygSxtBwwHnp/eZd0pcybTm+xsjrOMdu2iU3ImRZfBnUGRzKratb/avYgnAYTIJdO65ToxdlKmUHAJWaGdaIgZH7I+tAyGLADdSSeLZfTEKD3qR8qcEOlEnZ9IWaD1KPBM53hLveiNxf+8VoL+ZScVYZwghHz6kJ9IihEdp0V7QgFHOTLAuBJmV8oHTDGOJtOCCcFd/PIy1M9LrlNyby+KFXcWR54ckWNySlxSJhVyTaqkRjh5Ii/kjbxbz9ar9WF9Tltz1mzmkPwp6+cXElilSQ==</latexit><latexit sha1_base64="cGk9ScV80SxvZGY7YUNCULyD11c=">AAACIHicbZDLSgMxFIYz9VbrbdSlm2ARXEiZEaEuC27cCFXsBdpSMumZNjRzITkjlmEexY2v4saFIrrTpzG9LGrrgcDH/5+TnPxeLIVGx/m2ciura+sb+c3C1vbO7p69f1DXUaI41HgkI9X0mAYpQqihQAnNWAELPAkNb3g19hsPoLSIwnscxdAJWD8UvuAMjdS1y22ER5zck3oygSxtBwwHnp/eZd0pcybTm+xsjrOMdu2iU3ImRZfBnUGRzKratb/avYgnAYTIJdO65ToxdlKmUHAJWaGdaIgZH7I+tAyGLADdSSeLZfTEKD3qR8qcEOlEnZ9IWaD1KPBM53hLveiNxf+8VoL+ZScVYZwghHz6kJ9IihEdp0V7QgFHOTLAuBJmV8oHTDGOJtOCCcFd/PIy1M9LrlNyby+KFXcWR54ckWNySlxSJhVyTaqkRjh5Ii/kjbxbz9ar9WF9Tltz1mzmkPwp6+cXElilSQ==</latexit><latexit sha1_base64="cGk9ScV80SxvZGY7YUNCULyD11c=">AAACIHicbZDLSgMxFIYz9VbrbdSlm2ARXEiZEaEuC27cCFXsBdpSMumZNjRzITkjlmEexY2v4saFIrrTpzG9LGrrgcDH/5+TnPxeLIVGx/m2ciura+sb+c3C1vbO7p69f1DXUaI41HgkI9X0mAYpQqihQAnNWAELPAkNb3g19hsPoLSIwnscxdAJWD8UvuAMjdS1y22ER5zck3oygSxtBwwHnp/eZd0pcybTm+xsjrOMdu2iU3ImRZfBnUGRzKratb/avYgnAYTIJdO65ToxdlKmUHAJWaGdaIgZH7I+tAyGLADdSSeLZfTEKD3qR8qcEOlEnZ9IWaD1KPBM53hLveiNxf+8VoL+ZScVYZwghHz6kJ9IihEdp0V7QgFHOTLAuBJmV8oHTDGOJtOCCcFd/PIy1M9LrlNyby+KFXcWR54ckWNySlxSJhVyTaqkRjh5Ii/kjbxbz9ar9WF9Tltz1mzmkPwp6+cXElilSQ==</latexit><latexit sha1_base64="cGk9ScV80SxvZGY7YUNCULyD11c=">AAACIHicbZDLSgMxFIYz9VbrbdSlm2ARXEiZEaEuC27cCFXsBdpSMumZNjRzITkjlmEexY2v4saFIrrTpzG9LGrrgcDH/5+TnPxeLIVGx/m2ciura+sb+c3C1vbO7p69f1DXUaI41HgkI9X0mAYpQqihQAnNWAELPAkNb3g19hsPoLSIwnscxdAJWD8UvuAMjdS1y22ER5zck3oygSxtBwwHnp/eZd0pcybTm+xsjrOMdu2iU3ImRZfBnUGRzKratb/avYgnAYTIJdO65ToxdlKmUHAJWaGdaIgZH7I+tAyGLADdSSeLZfTEKD3qR8qcEOlEnZ9IWaD1KPBM53hLveiNxf+8VoL+ZScVYZwghHz6kJ9IihEdp0V7QgFHOTLAuBJmV8oHTDGOJtOCCcFd/PIy1M9LrlNyby+KFXcWR54ckWNySlxSJhVyTaqkRjh5Ii/kjbxbz9ar9WF9Tltz1mzmkPwp6+cXElilSQ==</latexit>

matrix partition

R =

RM,M RM,O

RO,M RO,O

<latexit sha1_base64="9ok78/VM8IhwodZMgXmwg3yaPeM=">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</latexit><latexit sha1_base64="9ok78/VM8IhwodZMgXmwg3yaPeM=">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</latexit><latexit sha1_base64="9ok78/VM8IhwodZMgXmwg3yaPeM=">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</latexit><latexit sha1_base64="9ok78/VM8IhwodZMgXmwg3yaPeM=">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</latexit>

Topology identifiability

17

Theorem 1: With complete data, the topology upstream of all probing buses is identifiable

Corollary: With complete data, if all leaf buses are probed, grid topology is identifiable

Theorem 2: With partial data, if all leaf nodes are probed, a reduced grid can be identified

actual recovered

Root and Branch (R&B) algorithm

18

n  Find the level sets from

G. Cavraro and V. Kekatos, ‘Graph Algorithms for Topology Identification using Power Grid Probing,’ IEEE Control Systems Letters, October 2018.

Recursive graph algorithm

S1) Find k-depth root as intersection of

depth (k-1)

depth k {N k

m}<latexit sha1_base64="MpUYTE5h5fpY1A87q8iEEGWGQh0=">AAAB+3icbVBPS8MwHE3nvzn/VXf0EhyCp9GKoN4GXjzJBOsGay1plm1hSVqSVCilfhUvHlS8+kW8+W1Mtx5080Hg8d7vx+/lRQmjSjvOt1VbWV1b36hvNra2d3b37P2DexWnEhMPxyyW/QgpwqggnqaakX4iCeIRI71oelX6vUciFY3Fnc4SEnA0FnREMdJGCu2mn/sc6QlGLL8pQv4w9YvQbjltZwa4TNyKtECFbmh/+cMYp5wIjRlSauA6iQ5yJDXFjBQNP1UkQXiKxmRgqECcqCCfhS/gsVGGcBRL84SGM/X3Ro64UhmPzGQZVC16pfifN0j16CLIqUhSTQSeHxqlDOoYlk3AIZUEa5YZgrCkJivEEyQR1qavhinBXfzyMvFO25dt9/as1XGrNurgEByBE+CCc9AB16ALPIBBBp7BK3iznqwX6936mI/WrGqnCf7A+vwBqluU7Q==</latexit><latexit sha1_base64="MpUYTE5h5fpY1A87q8iEEGWGQh0=">AAAB+3icbVBPS8MwHE3nvzn/VXf0EhyCp9GKoN4GXjzJBOsGay1plm1hSVqSVCilfhUvHlS8+kW8+W1Mtx5080Hg8d7vx+/lRQmjSjvOt1VbWV1b36hvNra2d3b37P2DexWnEhMPxyyW/QgpwqggnqaakX4iCeIRI71oelX6vUciFY3Fnc4SEnA0FnREMdJGCu2mn/sc6QlGLL8pQv4w9YvQbjltZwa4TNyKtECFbmh/+cMYp5wIjRlSauA6iQ5yJDXFjBQNP1UkQXiKxmRgqECcqCCfhS/gsVGGcBRL84SGM/X3Ro64UhmPzGQZVC16pfifN0j16CLIqUhSTQSeHxqlDOoYlk3AIZUEa5YZgrCkJivEEyQR1qavhinBXfzyMvFO25dt9/as1XGrNurgEByBE+CCc9AB16ALPIBBBp7BK3iznqwX6936mI/WrGqnCf7A+vwBqluU7Q==</latexit><latexit sha1_base64="MpUYTE5h5fpY1A87q8iEEGWGQh0=">AAAB+3icbVBPS8MwHE3nvzn/VXf0EhyCp9GKoN4GXjzJBOsGay1plm1hSVqSVCilfhUvHlS8+kW8+W1Mtx5080Hg8d7vx+/lRQmjSjvOt1VbWV1b36hvNra2d3b37P2DexWnEhMPxyyW/QgpwqggnqaakX4iCeIRI71oelX6vUciFY3Fnc4SEnA0FnREMdJGCu2mn/sc6QlGLL8pQv4w9YvQbjltZwa4TNyKtECFbmh/+cMYp5wIjRlSauA6iQ5yJDXFjBQNP1UkQXiKxmRgqECcqCCfhS/gsVGGcBRL84SGM/X3Ro64UhmPzGQZVC16pfifN0j16CLIqUhSTQSeHxqlDOoYlk3AIZUEa5YZgrCkJivEEyQR1qavhinBXfzyMvFO25dt9/as1XGrNurgEByBE+CCc9AB16ALPIBBBp7BK3iznqwX6936mI/WrGqnCf7A+vwBqluU7Q==</latexit><latexit sha1_base64="MpUYTE5h5fpY1A87q8iEEGWGQh0=">AAAB+3icbVBPS8MwHE3nvzn/VXf0EhyCp9GKoN4GXjzJBOsGay1plm1hSVqSVCilfhUvHlS8+kW8+W1Mtx5080Hg8d7vx+/lRQmjSjvOt1VbWV1b36hvNra2d3b37P2DexWnEhMPxyyW/QgpwqggnqaakX4iCeIRI71oelX6vUciFY3Fnc4SEnA0FnREMdJGCu2mn/sc6QlGLL8pQv4w9YvQbjltZwa4TNyKtECFbmh/+cMYp5wIjRlSauA6iQ5yJDXFjBQNP1UkQXiKxmRgqECcqCCfhS/gsVGGcBRL84SGM/X3Ro64UhmPzGQZVC16pfifN0j16CLIqUhSTQSeHxqlDOoYlk3AIZUEa5YZgrCkJivEEyQR1qavhinBXfzyMvFO25dt9/as1XGrNurgEByBE+CCc9AB16ALPIBBBp7BK3iznqwX6936mI/WrGqnCf7A+vwBqluU7Q==</latexit>

S2) Connect k- to (k-1)-depth root with resistance

S3) Group buses with identical and recurse N km

<latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit>

n  Group entries of m-th column of with equal values

n  Buses grouped together comprise a level set N km

<latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit><latexit sha1_base64="kEEC7bzA7ZcValjfBXFBUV6/R5I=">AAAB93icbVBNS8NAFHypX7V+NOrRS7AInkoignorePEkFYwttDFstpt26e4m7G6EGvpLvHhQ8epf8ea/cdPmoK0DC8PMe7zZiVJGlXbdb6uysrq2vlHdrG1t7+zW7b39e5VkEhMfJyyR3QgpwqggvqaakW4qCeIRI51ofFX4nUciFU3EnZ6kJOBoKGhMMdJGCu16nyM9wojlN9OQP4xDu+E23RmcZeKVpAEl2qH91R8kOONEaMyQUj3PTXWQI6kpZmRa62eKpAiP0ZD0DBWIExXks+BT59goAydOpHlCOzP190aOuFITHpnJIqZa9ArxP6+X6fgiyKlIM00Enh+KM+boxClacAZUEqzZxBCEJTVZHTxCEmFtuqqZErzFLy8T/7R52fRuzxotr2yjCodwBCfgwTm04Bra4AOGDJ7hFd6sJ+vFerc+5qMVq9w5gD+wPn8AbXmTFQ==</latexit>

RM<latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit>

RM<latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit><latexit sha1_base64="CogYWJivdTTzvL5TaUgFd99TSEk=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSLopiQi6LLgxo1QxT6gDWEynbRDJw9mbsQy5Bvc+CtuXCji1pU7/8ZJmoW2Hhg4nPs6c/yEMwm2/W1UlpZXVteq67WNza3tHXN3ryPjVBDaJjGPRc/HknIW0TYw4LSXCIpDn9OuP7nM6917KiSLozuYJtQN8ShiASMYtOSZJwOgD1DsUT7HZJKpQYhh7AfqNvMKSjBX11nmmXW7YRewFolTkjoq0fLMr8EwJmlIIyAcS9l37ARchQUwwmlWG6SSJvoiHtG+phEOqXRVYSWzjrQytIJY6BeBVai/JxQOpZyGvu7MPcr5Wi7+V+unEFy4ikVJCjQis0NByi2IrTwfa8gEJcCnmmAimPZqkTEWmIBOsaZDcOa/vEg6pw3Hbjg3Z/WmU8ZRRQfoEB0jB52jJrpCLdRGBD2iZ/SK3own48V4Nz5mrRWjnNlHf2B8/gC8YZ9T</latexit>

n  What if voltage data are collected only at

R&B with partial and noisy data

19

n  Level sets of probing nodes are partially observed

where

n  Modified R&B recovers a reduced grid

ü  radial grid

ü  effective resistances

ü  recovers non-metered buses with at least 2 probed children

M?<latexit sha1_base64="qXZZsxNsg4kjChMzgqIZzrURdlI=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6M6CGzdCBfuAJpTJ9KYdOpmEmYlQQn/DjQtF3Poz7vwbp20W2npg4HDOvdwzJ0wF18Z1v53S2vrG5lZ5u7Kzu7d/UD08auskUwxbLBGJ6oZUo+ASW4Ybgd1UIY1DgZ1wfDvzO0+oNE/ko5mkGMR0KHnEGTVW8v2YmhGjIr+f3vSrNbfuzkFWiVeQGhRo9qtf/iBhWYzSMEG17nluaoKcKsOZwGnFzzSmlI3pEHuWShqjDvJ55ik5s8qARImyTxoyV39v5DTWehKHdnKWUS97M/E/r5eZ6DrIuUwzg5ItDkWZICYhswLIgCtkRkwsoUxxm5WwEVWUGVtTxZbgLX95lbQv6p5b9x4uaw2vqKMMJ3AK5+DBFTTgDprQAgYpPMMrvDmZ8+K8Ox+L0ZJT7BzDHzifPwZrkZs=</latexit><latexit sha1_base64="qXZZsxNsg4kjChMzgqIZzrURdlI=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6M6CGzdCBfuAJpTJ9KYdOpmEmYlQQn/DjQtF3Poz7vwbp20W2npg4HDOvdwzJ0wF18Z1v53S2vrG5lZ5u7Kzu7d/UD08auskUwxbLBGJ6oZUo+ASW4Ybgd1UIY1DgZ1wfDvzO0+oNE/ko5mkGMR0KHnEGTVW8v2YmhGjIr+f3vSrNbfuzkFWiVeQGhRo9qtf/iBhWYzSMEG17nluaoKcKsOZwGnFzzSmlI3pEHuWShqjDvJ55ik5s8qARImyTxoyV39v5DTWehKHdnKWUS97M/E/r5eZ6DrIuUwzg5ItDkWZICYhswLIgCtkRkwsoUxxm5WwEVWUGVtTxZbgLX95lbQv6p5b9x4uaw2vqKMMJ3AK5+DBFTTgDprQAgYpPMMrvDmZ8+K8Ox+L0ZJT7BzDHzifPwZrkZs=</latexit><latexit sha1_base64="qXZZsxNsg4kjChMzgqIZzrURdlI=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6M6CGzdCBfuAJpTJ9KYdOpmEmYlQQn/DjQtF3Poz7vwbp20W2npg4HDOvdwzJ0wF18Z1v53S2vrG5lZ5u7Kzu7d/UD08auskUwxbLBGJ6oZUo+ASW4Ybgd1UIY1DgZ1wfDvzO0+oNE/ko5mkGMR0KHnEGTVW8v2YmhGjIr+f3vSrNbfuzkFWiVeQGhRo9qtf/iBhWYzSMEG17nluaoKcKsOZwGnFzzSmlI3pEHuWShqjDvJ55ik5s8qARImyTxoyV39v5DTWehKHdnKWUS97M/E/r5eZ6DrIuUwzg5ItDkWZICYhswLIgCtkRkwsoUxxm5WwEVWUGVtTxZbgLX95lbQv6p5b9x4uaw2vqKMMJ3AK5+DBFTTgDprQAgYpPMMrvDmZ8+K8Ox+L0ZJT7BzDHzifPwZrkZs=</latexit><latexit sha1_base64="qXZZsxNsg4kjChMzgqIZzrURdlI=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6M6CGzdCBfuAJpTJ9KYdOpmEmYlQQn/DjQtF3Poz7vwbp20W2npg4HDOvdwzJ0wF18Z1v53S2vrG5lZ5u7Kzu7d/UD08auskUwxbLBGJ6oZUo+ASW4Ybgd1UIY1DgZ1wfDvzO0+oNE/ko5mkGMR0KHnEGTVW8v2YmhGjIr+f3vSrNbfuzkFWiVeQGhRo9qtf/iBhWYzSMEG17nluaoKcKsOZwGnFzzSmlI3pEHuWShqjDvJ55ik5s8qARImyTxoyV39v5DTWehKHdnKWUS97M/E/r5eZ6DrIuUwzg5ItDkWZICYhswLIgCtkRkwsoUxxm5WwEVWUGVtTxZbgLX95lbQv6p5b9x4uaw2vqKMMJ3AK5+DBFTTgDprQAgYpPMMrvDmZ8+K8Ox+L0ZJT7BzDHzifPwZrkZs=</latexit>

VM = RMM�<latexit sha1_base64="zCD9P8S54wPkj6zkKcx/ZvtOF1A=">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</latexit><latexit sha1_base64="zCD9P8S54wPkj6zkKcx/ZvtOF1A=">AAACVnicbZFNS8NAEIY3qR+1fkU9egkWwVNJRNCLIOjBi1DFVqEJZbOZ6OLmg92JWJb8Sb3oT/EibtMitTqw8PDOOzuzs1EhuELP+7DsxsLi0nJzpbW6tr6x6Wxt91VeSgY9lotc3kdUgeAZ9JCjgPtCAk0jAXfR0/k4f/cMUvE8u8VRAWFKHzKecEbRSEMnDZCLGHSQUnyMEt2vqmHNjAp9VZ0GCC9Yt9GRKKHSP84b49Qz1hmsqrlbgwsQSE2B0/Y6Xh3uX/Cn0CbT6A6d1yDOWZlChkxQpQa+V2CoqUTOBFStoFRQUPZEH2BgMKMpqFDX81buvlFiN8mlORm6tTpboWmq1CiNjHM8p5rPjcX/coMSk5NQ86woETI2aZSUwsXcHe/YjbkEhmJkgDLJzawue6SSMjQ/0TJL8Oef/Bf6hx3f6/jXR+0zf7qOJtkle+SA+OSYnJFL0iU9wsgb+bRsq2G9W1/2or08sdrWtGaH/Arb+Qb1HLkh</latexit><latexit sha1_base64="zCD9P8S54wPkj6zkKcx/ZvtOF1A=">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</latexit><latexit sha1_base64="zCD9P8S54wPkj6zkKcx/ZvtOF1A=">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</latexit>

R =

RM,M RM,O

RO,M RO,O

<latexit sha1_base64="SyacWIzBU12AqXld/VMwPaxnUow=">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</latexit><latexit sha1_base64="SyacWIzBU12AqXld/VMwPaxnUow=">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</latexit><latexit sha1_base64="SyacWIzBU12AqXld/VMwPaxnUow=">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</latexit><latexit sha1_base64="SyacWIzBU12AqXld/VMwPaxnUow=">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</latexit>

Numerical tests

20

IEEE 37-bus feeder

EP: probability of finding erroneous topology

MPE: mean percentage error for line resistances

Topology Identification (limited info known)

Limited and noisy probing data

21

n  Graph algorithms rely on recovery of

V = RIM�+N<latexit sha1_base64="MR02b6Ff5b12EJbUeCX3aRjgDhc=">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</latexit><latexit sha1_base64="MR02b6Ff5b12EJbUeCX3aRjgDhc=">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</latexit><latexit sha1_base64="MR02b6Ff5b12EJbUeCX3aRjgDhc=">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</latexit><latexit sha1_base64="MR02b6Ff5b12EJbUeCX3aRjgDhc=">AAACOXicbZDLSgMxFIYz3q23qks3wSIIQpkRQTeCoAtdKFVsFTqlZNIzGpq5kJwRypDXcuNbuBPcuFDErS9gehOt/hD4+M855Jw/SKXQ6LpPztj4xOTU9MxsYW5+YXGpuLxS00mmOFR5IhN1HTANUsRQRYESrlMFLAokXAXtw2796g6UFkl8iZ0UGhG7iUUoOENrNYsVH4VsQe5HDG+DMK8Zsz/kCzOkE9PsIWcyPzV0aPtHIJEZuvXtnJlmseSW3Z7oX/AGUCIDVZrFR7+V8CyCGLlkWtc9N8VGzhQKLsEU/ExDynib3UDdYswi0I28d7mhG9Zp0TBR9sVIe+7PiZxFWneiwHZ2N9Sjta75X62eYbjXyEWcZggx738UZpJiQrsx0pZQwFF2LDCuhN2V8lumGEcbdsGG4I2e/Bdq22XPLXvnO6UDbxDHDFkj62STeGSXHJBjUiFVwsk9eSav5M15cF6cd+ej3zrmDGZWyS85n1+6da8l</latexit>

RM (RM,M)<latexit sha1_base64="OEMjQOhmIcUd/pCSbA4x8Yd6xx0=">AAACKHicdVDLSsNAFJ34rPUVdelmsAgVpCQi6M6CGzdCFfuANoTJdNIOnUzCzEQoIf6NG3/FjYgi3folTtIsaqsHBg7nnMvce7yIUaksa2IsLa+srq2XNsqbW9s7u+befkuGscCkiUMWio6HJGGUk6aiipFOJAgKPEba3ug689uPREga8gc1jogToAGnPsVIack1r3oBUkPPT+5TN8k5Riy5TdOn6j/O6WzqxDUrVs3KAReJXZAKKNBwzfdeP8RxQLjCDEnZta1IOQkSimJG0nIvliRCeIQGpKspRwGRTpIfmsJjrfShHwr9uIK5OjuRoEDKceDpZLaknPcy8S+vGyv/0kkoj2JFOJ5+5McMqhBmrcE+FQQrNtYEYUH1rhAPkUBY6W7LugR7/uRF0jqr2VbNvjuv1O2ijhI4BEegCmxwAergBjRAE2DwDF7BB/g0Xow348uYTKNLRjFzAH7B+P4BFj2oWg==</latexit><latexit sha1_base64="OEMjQOhmIcUd/pCSbA4x8Yd6xx0=">AAACKHicdVDLSsNAFJ34rPUVdelmsAgVpCQi6M6CGzdCFfuANoTJdNIOnUzCzEQoIf6NG3/FjYgi3folTtIsaqsHBg7nnMvce7yIUaksa2IsLa+srq2XNsqbW9s7u+befkuGscCkiUMWio6HJGGUk6aiipFOJAgKPEba3ug689uPREga8gc1jogToAGnPsVIack1r3oBUkPPT+5TN8k5Riy5TdOn6j/O6WzqxDUrVs3KAReJXZAKKNBwzfdeP8RxQLjCDEnZta1IOQkSimJG0nIvliRCeIQGpKspRwGRTpIfmsJjrfShHwr9uIK5OjuRoEDKceDpZLaknPcy8S+vGyv/0kkoj2JFOJ5+5McMqhBmrcE+FQQrNtYEYUH1rhAPkUBY6W7LugR7/uRF0jqr2VbNvjuv1O2ijhI4BEegCmxwAergBjRAE2DwDF7BB/g0Xow348uYTKNLRjFzAH7B+P4BFj2oWg==</latexit><latexit sha1_base64="OEMjQOhmIcUd/pCSbA4x8Yd6xx0=">AAACKHicdVDLSsNAFJ34rPUVdelmsAgVpCQi6M6CGzdCFfuANoTJdNIOnUzCzEQoIf6NG3/FjYgi3folTtIsaqsHBg7nnMvce7yIUaksa2IsLa+srq2XNsqbW9s7u+befkuGscCkiUMWio6HJGGUk6aiipFOJAgKPEba3ug689uPREga8gc1jogToAGnPsVIack1r3oBUkPPT+5TN8k5Riy5TdOn6j/O6WzqxDUrVs3KAReJXZAKKNBwzfdeP8RxQLjCDEnZta1IOQkSimJG0nIvliRCeIQGpKspRwGRTpIfmsJjrfShHwr9uIK5OjuRoEDKceDpZLaknPcy8S+vGyv/0kkoj2JFOJ5+5McMqhBmrcE+FQQrNtYEYUH1rhAPkUBY6W7LugR7/uRF0jqr2VbNvjuv1O2ijhI4BEegCmxwAergBjRAE2DwDF7BB/g0Xow348uYTKNLRjFzAH7B+P4BFj2oWg==</latexit><latexit sha1_base64="OEMjQOhmIcUd/pCSbA4x8Yd6xx0=">AAACKHicdVDLSsNAFJ34rPUVdelmsAgVpCQi6M6CGzdCFfuANoTJdNIOnUzCzEQoIf6NG3/FjYgi3folTtIsaqsHBg7nnMvce7yIUaksa2IsLa+srq2XNsqbW9s7u+befkuGscCkiUMWio6HJGGUk6aiipFOJAgKPEba3ug689uPREga8gc1jogToAGnPsVIack1r3oBUkPPT+5TN8k5Riy5TdOn6j/O6WzqxDUrVs3KAReJXZAKKNBwzfdeP8RxQLjCDEnZta1IOQkSimJG0nIvliRCeIQGpKspRwGRTpIfmsJjrfShHwr9uIK5OjuRoEDKceDpZLaknPcy8S+vGyv/0kkoj2JFOJ5+5McMqhBmrcE+FQQrNtYEYUH1rhAPkUBY6W7LugR7/uRF0jqr2VbNvjuv1O2ijhI4BEegCmxwAergBjRAE2DwDF7BB/g0Xow348uYTKNLRjFzAH7B+P4BFj2oWg==</latexit>

n  Resort to data fitting under limited/noisy data

min⇥

k⇥V � IM�k2F

s.to ⇥ � 0

⇥m,n 0, 8m 6= n

k⇥k0,o↵ = 2N<latexit sha1_base64="TzYQ1iKpBkWRiwNOQ5PW6SWR2gA=">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</latexit><latexit sha1_base64="TzYQ1iKpBkWRiwNOQ5PW6SWR2gA=">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</latexit><latexit sha1_base64="TzYQ1iKpBkWRiwNOQ5PW6SWR2gA=">AAAC3HicbVJNb9QwEHXCV1m+FjhysViBOJRVskKil0qVQAgOoCJ1t0XrbeQ4k65V20ljB7Hyulw4gBBXfhg3fgd/ACcNFewykqWnN+95xjNOS8G1iaKfQXjh4qXLVzau9q5dv3HzVv/2nYku6orBmBWiqA5SqkFwBWPDjYCDsgIqUwH76fGzJr//HirNC7VnFiXMJD1SPOeMGk8l/V9EcpVYIqmZp7kle3Mw1LnTh6dkuUISw0UG59KJc4//4FcuaSGjwr52577nIBrfMnlxOCKk1/KVtHpoirbCyv26ZgxHXrieS6zcVI4IOMHRJjmpaYZJXlRUCCyJ8qzqbGtNLxPrHQY+GFvkuXPbozdJfxANozbwOog7MEBd7Cb9HyQrWC1BGSao1tM4Ks3M0spwJsD1SK2hpOyYHsHUQ0Ul6Jltl+PwA89k2DfrjzK4Zf92WCq1XsjUK5vW9WquIf+Xm9Ym35pZrsragGJnhfJaYFPgZtM44xUwIxYeUFZx3ytmc1pRZvx/6PkhxKtPXgeT0TCOhvHbJ4OduBvHBrqH7qNHKEZP0Q56iXbRGLHgXfAx+Bx8CQ/DT+HX8NuZNAw6z130T4TffwNcb+tM</latexit><latexit sha1_base64="TzYQ1iKpBkWRiwNOQ5PW6SWR2gA=">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</latexit>

convex relaxation

min⇥

k⇥V � IM�k2F + �k⇥k1 � µ log |⇥|

s.to ⇥ ⌫ 0

⇥m,n 0, 8m 6= n<latexit sha1_base64="9NGf5LDK1NvvwwiHLOinSTQGHqs=">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</latexit><latexit sha1_base64="9NGf5LDK1NvvwwiHLOinSTQGHqs=">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</latexit><latexit sha1_base64="9NGf5LDK1NvvwwiHLOinSTQGHqs=">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</latexit><latexit sha1_base64="9NGf5LDK1NvvwwiHLOinSTQGHqs=">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</latexit>

n  ADMM-based solver with closed-form updates for each step

G. Cavraro and V. Kekatos, "Probing for distribution grid topology processing," IEEE Trans. on Control of Network Systems, (under review).

n  Recover tree through heuristics (e.g., minimum spanning tree)

Numerical tests

22

n  Resistive Laplacian matrix for IEEE 13-bus feeder

n  Identification/verification results for IEEE 37-bus feeder

Conclusions

23

Take-home: Inverter probing as active data collection paradigm for grid learning

þ  topological observability (meshed)

þ  probing sequence design

þ  solvers for noiseless/noisy data

Finding non-metered loads

þ  identifiability with full/partial data

þ  graph algorithms

þ  approximate solvers for limited data

Learning grid topologies

Related publications

24

Thank You!

n  S. Bhela, V. Kekatos, and S. Veeramachaneni, "Enhancing observability in distribution grids using smart meter data," IEEE Trans. on Smart Grid, 2018 (early access).

n  S. Bhela, V. Kekatos, and S. Veeramachameni, “Smart inverter probing for learning loads: Part I – Identifiability analysis,” IEEE Trans. on Power Systems, 2018 (under review).

n  S. Bhela, V. Kekatos, and S. Veeramachameni, “Smart inverter probing for learning loads: Part II – Probing sequence design,” IEEE Trans. on Power Systems, 2018 (under review).

n  G. Cavraro, V. Kekatos, and S. Veeramachaneni, "Voltage analytics for power distribution network topology verification," IEEE Trans. on Smart Grid, 2018 (early access).

n  G. Cavraro and V. Kekatos, “Graph algorithms for topology identification using power grid probing,” IEEE Control Systems Letters, Vol. 2, No. 4, pp. 689-694, Oct. 2018.

n  G. Cavraro and V. Kekatos, "Probing for distribution grid topology processing," IEEE Trans. on Control of Network Systems, (under review).

Grid topology detection

25

n  Similar formulation if Laplacian is parameterized in terms of lines

n  Topology detection through data fitting

⇥(b) =X

`2L

b`2r`

a`a>`

<latexit sha1_base64="oARjPXU7eJjKXc9vBo9R6T+OeHM=">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</latexit><latexit sha1_base64="oARjPXU7eJjKXc9vBo9R6T+OeHM=">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</latexit><latexit sha1_base64="oARjPXU7eJjKXc9vBo9R6T+OeHM=">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</latexit><latexit sha1_base64="oARjPXU7eJjKXc9vBo9R6T+OeHM=">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</latexit>

b` =

⇢1 , line ` is energized0 , otherwise

<latexit sha1_base64="e4heVNzJcjXyfiDC3Mqcubm0DfY=">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</latexit><latexit sha1_base64="e4heVNzJcjXyfiDC3Mqcubm0DfY=">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</latexit><latexit sha1_base64="e4heVNzJcjXyfiDC3Mqcubm0DfY=">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</latexit><latexit sha1_base64="e4heVNzJcjXyfiDC3Mqcubm0DfY=">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</latexit>

where

convex relaxation

Q1) Invertibility of CPF equations

n  [Tutte]: A matrix is generically full column-rank iff columns can be matched to rows

26 Tutte, ‘The factorization of linear graphs,’ Journal of the London Mathematical Society, 1947.

n  Inverse function theorem: CPF equations are locally invertible iff invertible Jacobian

quadratic equations

states (voltages)

(coupled) power flow

equations

n  We generalize topological observability to lossy power networks (T=1 and above)

n  How about numerical observability?

J(v) =

2

66664

0 ⇤ ⇤ ⇤⇤ 0 ⇤ 00 ⇤ 0 ⇤0 0 0 ⇤0 0 ⇤ 0

3

77775

<latexit sha1_base64="3KazVGxRUxtAfOpsWawgfWDulyU=">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</latexit><latexit sha1_base64="3KazVGxRUxtAfOpsWawgfWDulyU=">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</latexit><latexit sha1_base64="3KazVGxRUxtAfOpsWawgfWDulyU=">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</latexit><latexit sha1_base64="3KazVGxRUxtAfOpsWawgfWDulyU=">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</latexit>

linear gradient

Jacobian J(v) depends on unknown states

pn(v) = vHMnv<latexit sha1_base64="cr5UgRk5nyBIhTVMo7sOmwZlBi0=">AAACGHicbZBLS8NAFIUn9VXrK+rSTbAIFUpJiqAboeCmG6GCfUAbw2Q6bYdOJsPMpFBC/of4Y8RddSW48984qZHS1ruZj3POwL3H55RIZdvfRm5jc2t7J79b2Ns/ODwyj09aMowEwk0U0lB0fCgxJQw3FVEUd7jAMPApbvvju9RvT7CQJGSPasqxG8AhIwOCoNKSZ1a5x0q9AKqRP4gnyeXtgp/qf3yfeGyhe2bRrtjzsdbByaAIsml45mevH6IowEwhCqXsOjZXbgyFIojipNCLJOYQjeEQdzUyGGBZ7k8Il3N04/mZS7kYBlJOA788xCx9E+si3U+uZlLxP68bqcGNGxPGI4UZ0hHtDSJqqdBKa7L6RGCk6FQDRILoTS00ggIipcvUDTir965Dq1px7IrzcFWsOVkXeXAGzkEJOOAa1EAdNEATIPAMXsEMvBsvxpsxMz5+ozkj+3MKlsb4+gHAvaKL</latexit><latexit sha1_base64="cr5UgRk5nyBIhTVMo7sOmwZlBi0=">AAACGHicbZBLS8NAFIUn9VXrK+rSTbAIFUpJiqAboeCmG6GCfUAbw2Q6bYdOJsPMpFBC/of4Y8RddSW48984qZHS1ruZj3POwL3H55RIZdvfRm5jc2t7J79b2Ns/ODwyj09aMowEwk0U0lB0fCgxJQw3FVEUd7jAMPApbvvju9RvT7CQJGSPasqxG8AhIwOCoNKSZ1a5x0q9AKqRP4gnyeXtgp/qf3yfeGyhe2bRrtjzsdbByaAIsml45mevH6IowEwhCqXsOjZXbgyFIojipNCLJOYQjeEQdzUyGGBZ7k8Il3N04/mZS7kYBlJOA788xCx9E+si3U+uZlLxP68bqcGNGxPGI4UZ0hHtDSJqqdBKa7L6RGCk6FQDRILoTS00ggIipcvUDTir965Dq1px7IrzcFWsOVkXeXAGzkEJOOAa1EAdNEATIPAMXsEMvBsvxpsxMz5+ozkj+3MKlsb4+gHAvaKL</latexit><latexit sha1_base64="cr5UgRk5nyBIhTVMo7sOmwZlBi0=">AAACGHicbZBLS8NAFIUn9VXrK+rSTbAIFUpJiqAboeCmG6GCfUAbw2Q6bYdOJsPMpFBC/of4Y8RddSW48984qZHS1ruZj3POwL3H55RIZdvfRm5jc2t7J79b2Ns/ODwyj09aMowEwk0U0lB0fCgxJQw3FVEUd7jAMPApbvvju9RvT7CQJGSPasqxG8AhIwOCoNKSZ1a5x0q9AKqRP4gnyeXtgp/qf3yfeGyhe2bRrtjzsdbByaAIsml45mevH6IowEwhCqXsOjZXbgyFIojipNCLJOYQjeEQdzUyGGBZ7k8Il3N04/mZS7kYBlJOA788xCx9E+si3U+uZlLxP68bqcGNGxPGI4UZ0hHtDSJqqdBKa7L6RGCk6FQDRILoTS00ggIipcvUDTir965Dq1px7IrzcFWsOVkXeXAGzkEJOOAa1EAdNEATIPAMXsEMvBsvxpsxMz5+ozkj+3MKlsb4+gHAvaKL</latexit><latexit sha1_base64="cr5UgRk5nyBIhTVMo7sOmwZlBi0=">AAACGHicbZBLS8NAFIUn9VXrK+rSTbAIFUpJiqAboeCmG6GCfUAbw2Q6bYdOJsPMpFBC/of4Y8RddSW48984qZHS1ruZj3POwL3H55RIZdvfRm5jc2t7J79b2Ns/ODwyj09aMowEwk0U0lB0fCgxJQw3FVEUd7jAMPApbvvju9RvT7CQJGSPasqxG8AhIwOCoNKSZ1a5x0q9AKqRP4gnyeXtgp/qf3yfeGyhe2bRrtjzsdbByaAIsml45mevH6IowEwhCqXsOjZXbgyFIojipNCLJOYQjeEQdzUyGGBZ7k8Il3N04/mZS7kYBlJOA788xCx9E+si3U+uZlLxP68bqcGNGxPGI4UZ0hHtDSJqqdBKa7L6RGCk6FQDRILoTS00ggIipcvUDTir965Dq1px7IrzcFWsOVkXeXAGzkEJOOAa1EAdNEATIPAMXsEMvBsvxpsxMz5+ozkj+3MKlsb4+gHAvaKL</latexit>

rpn(v) = 2Mnv<latexit sha1_base64="P3eKeQUQnP39cuCWNRFEi3DxNps=">AAACFHicbZDLSsNAFIYnXmu9RV26CRahQilJEXQjFNy4ESrYCzQhTKaTduhkMsxMCiXkLcSHEXfVra59Gyc1Im09m/nm/P+Bc/6AUyKVbX8Za+sbm1vbpZ3y7t7+waF5dNyRcSIQbqOYxqIXQIkpYbitiKK4xwWGUUBxNxjf5np3goUkMXtUU469CA4ZCQmCSrd8s+4yGFBocZ9V3QiqURCmk+zipvH7uc989if4ZsWu2/OyVsEpoAKKavnmpzuIURJhphCFUvYdmysvhUIRRHFWdhOJOURjOMR9jQxGWNYGE8LlHL10fuKCL4WRlNMoqA0xy9/MOs/3k8uevPmf1k9UeO2lhPFEYYa0RWthQi0VW3lE1oAIjBSdaoBIEL2phUZQQKR0kDoBZ/neVeg06o5ddx4uK02nyKIETsEZqAIHXIEmuAMt0AYIPIEXMANvxrPxasyM9x/rmlHMnICFMj6+AXwqoC0=</latexit><latexit sha1_base64="P3eKeQUQnP39cuCWNRFEi3DxNps=">AAACFHicbZDLSsNAFIYnXmu9RV26CRahQilJEXQjFNy4ESrYCzQhTKaTduhkMsxMCiXkLcSHEXfVra59Gyc1Im09m/nm/P+Bc/6AUyKVbX8Za+sbm1vbpZ3y7t7+waF5dNyRcSIQbqOYxqIXQIkpYbitiKK4xwWGUUBxNxjf5np3goUkMXtUU469CA4ZCQmCSrd8s+4yGFBocZ9V3QiqURCmk+zipvH7uc989if4ZsWu2/OyVsEpoAKKavnmpzuIURJhphCFUvYdmysvhUIRRHFWdhOJOURjOMR9jQxGWNYGE8LlHL10fuKCL4WRlNMoqA0xy9/MOs/3k8uevPmf1k9UeO2lhPFEYYa0RWthQi0VW3lE1oAIjBSdaoBIEL2phUZQQKR0kDoBZ/neVeg06o5ddx4uK02nyKIETsEZqAIHXIEmuAMt0AYIPIEXMANvxrPxasyM9x/rmlHMnICFMj6+AXwqoC0=</latexit><latexit sha1_base64="P3eKeQUQnP39cuCWNRFEi3DxNps=">AAACFHicbZDLSsNAFIYnXmu9RV26CRahQilJEXQjFNy4ESrYCzQhTKaTduhkMsxMCiXkLcSHEXfVra59Gyc1Im09m/nm/P+Bc/6AUyKVbX8Za+sbm1vbpZ3y7t7+waF5dNyRcSIQbqOYxqIXQIkpYbitiKK4xwWGUUBxNxjf5np3goUkMXtUU469CA4ZCQmCSrd8s+4yGFBocZ9V3QiqURCmk+zipvH7uc989if4ZsWu2/OyVsEpoAKKavnmpzuIURJhphCFUvYdmysvhUIRRHFWdhOJOURjOMR9jQxGWNYGE8LlHL10fuKCL4WRlNMoqA0xy9/MOs/3k8uevPmf1k9UeO2lhPFEYYa0RWthQi0VW3lE1oAIjBSdaoBIEL2phUZQQKR0kDoBZ/neVeg06o5ddx4uK02nyKIETsEZqAIHXIEmuAMt0AYIPIEXMANvxrPxasyM9x/rmlHMnICFMj6+AXwqoC0=</latexit><latexit sha1_base64="P3eKeQUQnP39cuCWNRFEi3DxNps=">AAACFHicbZDLSsNAFIYnXmu9RV26CRahQilJEXQjFNy4ESrYCzQhTKaTduhkMsxMCiXkLcSHEXfVra59Gyc1Im09m/nm/P+Bc/6AUyKVbX8Za+sbm1vbpZ3y7t7+waF5dNyRcSIQbqOYxqIXQIkpYbitiKK4xwWGUUBxNxjf5np3goUkMXtUU469CA4ZCQmCSrd8s+4yGFBocZ9V3QiqURCmk+zipvH7uc989if4ZsWu2/OyVsEpoAKKavnmpzuIURJhphCFUvYdmysvhUIRRHFWdhOJOURjOMR9jQxGWNYGE8LlHL10fuKCL4WRlNMoqA0xy9/MOs/3k8uevPmf1k9UeO2lhPFEYYa0RWthQi0VW3lE1oAIjBSdaoBIEL2phUZQQKR0kDoBZ/neVeg06o5ddx4uK02nyKIETsEZqAIHXIEmuAMt0AYIPIEXMANvxrPxasyM9x/rmlHMnICFMj6+AXwqoC0=</latexit>