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Flow Analysis of a Five-Bus Power System

This application analyzes this five-bus power system, and calculates the voltages and real and reactive powers at each 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inject power (distribution substations) and loads remove power. The power flow equations are derived from nodal power balances at each bus i using Kirchoff's Current 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wherevoltage magnitude, Vivoltage phase angle, \316\264ireal power, Pireactive power, QiThese variables are known for each bus (where the suffix G and D stand for generation and demand respectively). Bus typeUnknown variableKnown variable SlackAlways has a generator|V|, \316\264, PD, QDPG, QGPQPG, QG, PD, QD|V|, \316\264PV|V|, PG, PD, QD\316\264, QG Larger power systems (for example, those modeling real-life distribution systems) can consist of tens of thousands of buses; the number of nonlinear equations needed to model these systems is similarly large. For these, power systems engineers use dedicated tools with specialized solvers.Smaller systems can be modeled and studied in Maple, perhaps to experiment with or optimize different topologies, investigate numerical techniques, or reinforce the underlying theory.This Maple application demonstrates two approaches to numerically solving the load flow equations. The symbolic real and reactive load flow equations are generated, and solved with fsolve (a powerful numerical equation solver). This is quick to set up, but may be slower for larger systems.The system Jacobian is constructed, and solved with Newton-Raphson iteration (this method requires more initial setup, but may be faster for larger systems).restart:

Each row of the following matrix contains the properties of a transmission lineColumnData1Sending end bus2Receiving end bus3Series impedance in pu4Shunt admittance in pu Series := Matrix([[1, 2, 0.042 + I * 0.168, I * 0.020 ] ,[1, 5, 0.031 + I * 0.126, I * 0.0014] ,[2, 3, 0.031 + I * 0.126, I * 0.014 ] ,[3, 4, 0.084 + I * 0.336, I * 0.04 ] ,[3, 5, 0.053 + I * 0.21, I * 0.024 ] ,[4, 5, 0.063 + I * 0.252, I * 0.06 ]]):

If no shunt equipment is available, enter any bus number in the first column and 0 in the second column of the shunt array. Shunt elements may represent reactive power compensation at certain system buses. In this case, the reactance of the corresponding compensation device (capacitor or inductor) is calculated and its admittance is entered in the above array. If lines are compensated by series capacitors, the reactance of the capacitor is subtracted from the line series reactance and the net reactance is included in the array Series. Shunt := Matrix([1, 0]):

Bus numberBus := Vector([1, 2, 3, 4, 5]): Generated real power on the bus (per unit) P__G := Vector([0, 0, 1.8, 0, 0]): Real load power on the bus (per unit)P__L := Vector([0.65, 1.15, 0.7, 0.7, 0.85]): Reactive load power on the bus (per unit)Q__L := Vector([0.3, 0.6, 0.4, 0.3, 0.4]): Magnitude of the bus voltage (per unit). This is fixed for the slack (first value) and PV buses (third value), and is the initial guess for PQ buses.V := Vector([1.04, 0.961, 1.02, 0.92, 0.968]): Voltage phase angle in radians. This is normally 0 for the slack bus (the first value). The remaining values are the initial guess of the bus voltage phase angles for the other busesdelta := Vector([0, 0, 0, 0, 0]): Bus type (0 for the slack bus, 1 for PV bus, 2 for PQ-bus)bus_type := Vector(["slack", "PQ", "PV", "PQ", "PQ"]):

Number of buses. series elements and shunt elementsN_bus := numelems(Bus): N_ser := numelems(Series[.., 1]): N_sh := numelems(Shunt[.., 1]): Terminals of series and shunt equipment Is := Re(Series[.., 1]): Js := Re(Series[.., 2]): Ish := Re(Shunt [.., 1]): Real power injection (net) Pb := P__G - P__L

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Reactive power injection (net) Qb := -Q__L

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Initialize the nodal admittance matrix. Y__bus := Matrix(N_bus, N_bus): Add series elements to the nodal admittance matric for i from 1 to N_ser do Y__bus[Is[i], Is[i]] := Y__bus[Is[i], Is[i]] + Series[i, 4] + 1/Series[i, 3]; Y__bus[Is[i], Js[i]] := Y__bus[Is[i], Js[i]] - 1/Series[i, 3]; Y__bus[Js[i], Is[i]] := Y__bus[Js[i], Is[i]] - 1/Series[i, 3]; Y__bus[Js[i], Js[i]] := Y__bus[Js[i], Js[i]] + Series[i, 4] + 1/Series[i, 3]; end do: Y__bus

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Add shunt elements for i from 1 to N_sh do Y__bus[Is[i], Is[i]] := Y__bus[Is[i], Is[i]] + Shunt[i, 2]; Y__bus[Js[i], Js[i]] := Y__bus[Js[i], Js[i]] + Shunt[i, 2]; end do: Y__bus

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Real power flow function. In the function arguments, k is the bus number, v is the voltage magnitude and d is the voltage phase. P := (k, v, d) -> add(v[k] * v[i] * abs(Y__bus[k, i]) * cos(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]), 0) + d[i] - d[k]),i = 1..N_bus):

Warning, (in P) `i` is implicitly declared local

Reactive power flow function Q := (k, v, d) -> -add(v[k] * v[i] * abs(Y__bus[k, i]) * sin(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]), 0) + d[i] - d[k]),i = 1..N_bus):

Warning, (in Q) `i` is implicitly declared local

Generate the real power flow equations. real_power_eq := seq(Pb[i] = P(i, v, d), i in {2, 3, 4, 5})

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Generate the reactive power flow equations reactive_power_eq := seq(Qb[i] = Q(i, v, d), i in {2, 4, 5})

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v[1], d[1] and v[3] are known values (slack bus and PV-buses) and are substituted into the equation system sys := eval({real_power_eq, reactive_power_eq}, {v[1] = 1.04, d[1] = 0, v[3] = 1.02})

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Numerically solve the real and reactive power flow equations. sys := fsolve(sys, {v[2] = 0.961, v[4] = 0.92, v[5] = 0.968, d[2] = 0, d[3] = 0, d[4] = 0,d[5] = 0}, fulldigits)

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Check the correctness of the solution eval([real_power_eq, reactive_power_eq],{v[1] = 1.04, d[1] = 0, v[3] = 1.02} union sys)

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Jac := Matrix(8, 8): for k from 2 to N_bus do for n from 2 to N_bus do Jac[k - 1, n + N_bus - 2] := V[k] * abs(Y__bus[k, n]) * cos(ifelse(Y__bus[k, n] <> 0, argument(Y__bus[k, n]), 0)) + delta[n] - delta[k]; Jac[k - 1, k + N_bus - 2] := add(V[i] * abs(Y__bus[k, i]) * cos(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]), 0)) + delta[i] - delta[k], i = 1..N_bus) + V[k] * abs(Y__bus[k, k]) * cos(argument(Y__bus[k, k])): Jac[k + N_bus - 2, n + N_bus - 2] := -V[k] * abs(Y__bus[k,n])* sin(ifelse(Y__bus[k, n] <>0, argument(Y__bus[k, n]), 0)) + delta[n] - delta[k]: Jac[k + N_bus - 2, k + N_bus -2 ] := add(-V[i] * abs(Y__bus[k,i]) * sin(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]), 0)) + delta[i] - delta[k], i = 1..N_bus) - V[k] * abs(Y__bus[k, k]) * sin(argument(Y__bus[k, k])): Jac[k - 1, n - 1] := -V[k] * V[n] * abs(Y__bus[k, n]) * sin(ifelse(Y__bus[k, n] <> 0,argument(Y__bus[k, n]),0)) + delta[i] - delta[k]; Jac[k - 1, k - 1] := add(V[k] * V[i] * abs(Y__bus[k, i]) * sin(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]), 0)) + delta[i] - delta[k], i = 1..N_bus) - V[k]^2 * abs(Y__bus[k, k]) * sin(argument(Y__bus[k, k])); Jac[k + N_bus - 2, n - 1] := -V[k] * V[n] * abs(Y__bus[k, n]) * cos(ifelse(Y__bus[k, n] <> 0, argument(Y__bus[k, n]), 0) + delta[n] - delta[k]); Jac[k + N_bus - 2, k - 1] := add(V[k] * V[i] * abs(Y__bus[k, i]) *cos(ifelse(Y__bus[k, i] <> 0, argument(Y__bus[k, i]),0) + delta[i] - delta[k]),i = 1..N_bus) - V[k]^2 * abs(Y__bus[k, k]) * cos(argument(Y__bus[k, k])); end do: end do: for k from 2 to N_bus do for n from 2 to N_bus do Jac[k - 1, n + N_bus - 2] := ifelse(bus_type[n] = "PV", 0, Jac[k - 1, n + N_bus - 2]); Jac[k + N_bus - 2, n - 1] := ifelse(bus_type[k] = "PV", 0, Jac[k + N_bus - 2, n - 1]): Jac[k + N_bus - 2, n + N_bus - 2] := ifelse(bus_type[k] = "PV", 0, Jac[k + N_bus - 2, n + N_bus - 2]): Jac[k + N_bus - 2, n + N_bus - 2] := ifelse(bus_type[n] = "PV", 0, Jac[k + N_bus - 2, n + N_bus - 2]): Jac[k + N_bus - 2, k + N_bus - 2] := ifelse(bus_type[k] = "PV", 1, Jac[k + N_bus - 2, k + N_bus - 2]) end do: end do: Hence the complete Jacobian matrix is Jac

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R0ZULyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmpqbEZMRk9GUkZARkAvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJ0ZDRkYvJSVvcGVuR1EnJmxzcWI7RicvJSZjbG9zZUdRJyZyc3FiO0YnRmBcbQ==

Invert the Jacobian matrix Jinv := Jac^(-1)

LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXJvd0dGJDYnLUknbXRhYmxlR0YkNj8tSSRtdHJHRiQ2MC1JJG10ZEdGJDYrLUkqbWNvbXBsZXRlR0YkNiQtRiw2JC1JI21uR0YkNiZRJTAuMTJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSpyZWZlcmVuY2VHUSQxOThGJy8lMmRpc3BsYXlfcmVmZXJlbmNlR1EkMTk3RidGQC1GPTYmUTIwLjEyMjg0MDc2MTYyNTY5MUYnRkBGQ0ZGLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHUSZmYWxzZUYnLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0ZXLyUrZ3JvdXBhbGlnbkdGVy8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0Zobi1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjA5RidGQEZDRkZGQC1GPTYmUTMwLjA4NzQ3NTA2OTA2MzQ3NTBGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjA2RidGQEZDRkZGQC1GPTYmUTMwLjA2MDkzNTM3NDI2OTk5NjlGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjA0RidGQEZDRkZGQC1GPTYmUTMwLjAzODc5NTc4MjIwNDMyNjZGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JS1JI21vR0YkNi9RKiZ1bWludXMwO0YnRkBGQ0ZGLyUmZmVuY2VHRlQvJSpzZXBhcmF0b3JHRlQvJSlzdHJldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRlQvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGaXItRj02JlElMC4wMkYnRkBGQ0ZGRkAtRiw2J0ZlcS1GPTYmUTMwLjAxNjQ2OTQ0NTExOTYxNTVGJ0ZARkNGRkZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjAwRidGQEZDRkZGQC1GLDYnRmVxLUY9NiZRIzAuRidGQEZDRkZGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkRmhzLUY9NiZRNDAuMDAzNTQ3ODk3MjgxNjk3NTJGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE0MC4wMDE2ODk1MzY3NTE2MTM1N0YnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRkxGT0ZSRlVGWEZaLUYyNjAtRjU2Ky1GODYkRl9vLUY9NiZRMzAuMDg4MjQwMDE1NTEyNTQ1OUYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JC1GLDYkLUY9NiZRJTAuMTVGJ0ZARkNGRkZALUY9NiZRMjAuMTU0MzA2MDMzMzY3Mzg2RidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkLUYsNiQtRj02JlElMC4xMUYnRkBGQ0ZGRkAtRj02JlEyMC4xMDc0OTAwMDgyNTA2NjRGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjA3RidGQEZDRkZGQC1GPTYmUTMwLjA2ODQzNTc2NDkyNjI0MTBGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE0MC4wMDI5NDUyMjI3NTM0Njg5NkYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRmRzLUY1NistRjg2JC1GLDYkLUY9NiZRJTAuMDFGJ0ZARkNGRkZALUY9NiZRNDAuMDA2MjU4NDkxMjcyOTQxMDhGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE0MC4wMDI5ODAzNDMwNTI3MjQzNEYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRkxGT0ZSRlVGWEZaLUYyNjAtRjU2Ky1GODYkRltwLUY9NiZRMzAuMDYwNTkyMTM2NzM4ODg3NkYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZpdi1GPTYmUTIwLjEwNTk1Nzk2MjcxMjUzN0YnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JC1GLDYkLUY9NiZRJTAuMjVGJ0ZARkNGRkZALUY9NiZRMjAuMjUzMzA5ODM5MjY0OTczRidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkRl9vLUY9NiZRMzAuMDg2MDUzMjE2MjkzNDMyOUYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZocy1GPTYmUTQwLjAwMjAyMjQwODMwMDQ1MzAwRidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkRmhzRl90RkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JC1GLDYlRmVxRmlwRkAtRiw2J0ZlcS1GPTYmUTMwLjA0MjQ2MTU1OTk4NDU2NzhGJ0ZARkNGRkZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiQtRiw2JUZlcUZqeEZALUYsNidGZXEtRj02JlE0MC4wMDgyMzMzMDA1MzYyOTYxNkYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRkxGT0ZSRlVGWEZaLUYyNjAtRjU2Ky1GODYkRmdwLUY9NiZRMzAuMDM4NzQxMjMyMzg1NzAwNkYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZldy1GPTYmUTMwLjA2Nzc0NzEwODQxMTk2MTBGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGX28tRj02JlEzMC4wODYyNTc0MDUwMzU1OTc5RidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkLUYsNiQtRj02JlElMC4xMEYnRkBGQ0ZGRkAtRj02JlEyMC4xMDAxMzU1OTcyNDUyMzhGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE0MC4wMDEyOTMwODE4MTE2NTkxNUYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRmFcbC1GNTYrLUY4NiRGZF1sLUYsNidGZXEtRj02JlE0MC4wMDc1MTIyNDgwMjU3OTgxMUYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZjcS1GLDYnRmVxLUY9NiZRMzAuMDE2NDUwMjgyMjk0MzQ4OUYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRkxGT0ZSRlVGWEZaLUYyNjAtRjU2Ky1GODYkLUYsNiRGXHNGQC1GPTYmUTMwLjAyMDE2MTQwNTYxNjY5ODFGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE1MC4wMDAxNTc1MjE3NTk4Njk1NTVGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE1MC4wMDAxMDk3MzAwOTIxOTc1MjRGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE2MC4wMDAwNjk4NjE5NjEyODUzMDA3RidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkLUYsNiQtRj02JlElMC4wOEYnRkBGQ0ZGRkAtRj02JlEzMC4wODEwMTc2MDE0NTI3NTcyRidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW5GZHMtRjU2Ky1GODYkRmhzLUY9NiZRNzAuMDAwMDA2Mzg4OTE3Nzc1NDYyMDRGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHMtRj02JlE3MC4wMDAwMDMwNDI0NTMxODQ5NzkzOUYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRkxGT0ZSRlVGWEZaLUYyNjBGYVxsRmFcbEZhXGxGYVxsRmFcbC1GNTYrLUY4NiQtRiw2JC1GPTYmUSUxLjAwRidGQEZDRkZGQC1GPTYmUSMxLkYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRmRzRmRzRkxGT0ZSRlVGWEZaLUYyNjAtRjU2Ky1GODYkLUYsNiVGZXFGanNGQC1GLDYnRmVxLUY9NiZRNjAuMDAwMDgzMTg2OTY4NzQwMjIwOEYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZlZmwtRiw2J0ZlcS1GPTYmUTUwLjAwMDE0NTQ2OTcyOTM0NjcxOEYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JC1GLDYkLUY9NiZRJTAuMDVGJ0ZARkNGRkZALUY9NiZRMzAuMDUxOTE0MDExOTM3ODM1MUYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZoeC1GPTYmUTMwLjAxMjI1MzUzMDgxNDQ4OTBGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGZWZsLUYsNidGZXEtRj02JlE3MC4wMDAwMDI3NzY1NjUxNjM4MDU3NEYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRmFcbC1GNTYrLUY4NiQtRiw2JC1GPTYmUSUwLjIxRidGQEZDRkZGQC1GPTYmUTIwLjIwODIxNTI5NDU3NjcxN0YnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZpZ2wtRj02JlEzMC4wNDk1NTU0ODQ0MjI2NTU4RidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW5GTEZPRlJGVUZYRlotRjI2MC1GNTYrLUY4NiRGZWZsLUYsNidGZXEtRj02JlE1MC4wMDAxMzAyNjI4NDMyNjgzNDdGJ0ZARkNGRkZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGZWZsLUYsNidGZXEtRj02JlE1MC4wMDAyMjc3OTE2OTQzMTM5OTZGJ0ZARkNGRkZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGaHgtRj02JlEzMC4wMTI4MTQxMTYwODc4MDIwRidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW4tRjU2Ky1GODYkRl1ibC1GPTYmUTMwLjAxOTE4Nzg1NzE2NzkyODZGJ0ZARkNGRkZMRk9GUkZVRlhGWkZmbkZpbi1GNTYrLUY4NiRGZWZsLUYsNidGZXEtRj02JlE3MC4wMDAwMDQzNDc4MzU3MDM1MzAxMEYnRkBGQ0ZGRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluRmFcbC1GNTYrLUY4NiRGaWdsLUY9NiZRMzAuMDUyMTMyMTkxNjIxNjkyOEYnRkBGQ0ZGRkxGT0ZSRlVGWEZaRmZuRmluLUY1NistRjg2JEZbZGwtRj02JlEzMC4wNzc1OTkxNDg1MjQ3ODU4RidGQEZDRkZGTEZPRlJGVUZYRlpGZm5GaW5GTEZPRlJGVUZYRlpGTEZPRlIvJSZhbGlnbkdRJWF4aXNGJy9GVlEpYmFzZWxpbmVGJy9GWVEnY2VudGVyRicvRmVuUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdRJXRydWVGJy8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRltfbS8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZmX20vJSZmcmFtZUdGZl9tLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGVC8lLWVxdWFsY29sdW1uc0dGVC8lLWRpc3BsYXlzdHlsZUdGVC8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZjX21GTEZPRlJGQEZAL0krbXNlbWFudGljc0dGJFEnTWF0cml4RidGQ0ZGLyUlb3BlbkdRJyZsc3FiO0YnLyUmY2xvc2VHUScmcnNxYjtGJ0ZpYG0=

Corrections of voltage and phase angle for the new iteration step Delta_V := (l, V, delta) -> add(Jinv[l + N_bus - 2, k - 1] * (Pb[k] - P(k, V, delta)), k = 2..N_bus) + add(Jinv[l + N_bus - 2, k + N_bus - 2] * ifelse(bus_type[k] = "PV", 0, Qb[k] - Q(k, V, delta)),k = 2..N_bus)

Warning, (in Delta_V) `k` is implicitly declared local

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEZWx0YV9WR0YoZio2JUkibEdGKEkiVkdGKEkmZGVsdGFHRig2I0kia0dGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkkYWRkR0YmNiQqJiZJJUppbnZHRig2JCwoRjAiIiJJJk5fYnVzR0YoRkEiIiMhIiIsJkY0RkFGQUZERkEsJiZJI1BiR0YoRjNGQS1JIlBHRig2JUY0RjFGMkZERkEvRjQ7RkNGQkZBLUY6NiQqJiZGPjYkRkAsKEY0RkFGQkZBRkNGREZBLUknaWZlbHNlR0YmNiUvJkkpYnVzX3R5cGVHRihGM1EjUFZGKCIiISwmJkkjUWJHRihGM0ZBLUkiUUdGKEZLRkRGQUZMRkFGKEYoRig3I0Yu

Delta_delta := (l, V, delta) -> add(Jinv[l - 1, k - 1] * (Pb[k] - P(k, V, delta)), k = 2..N_bus) + add(Jinv[l - 1, k + N_bus - 2] * ifelse(bus_type[k] = "PV", 0, Qb[k] - Q(k, V, delta)), k = 2..N_bus)

Warning, (in Delta_delta) `k` is implicitly declared local

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxEZWx0YV9kZWx0YUdGKGYqNiVJImxHRihJIlZHRihJJmRlbHRhR0YoNiNJImtHRig2JEkpb3BlcmF0b3JHRihJJmFycm93R0YoRigsJi1JJGFkZEdGJjYkKiYmSSVKaW52R0YoNiQsJkYwIiIiRkEhIiIsJkY0RkFGQUZCRkEsJiZJI1BiR0YoRjNGQS1JIlBHRig2JUY0RjFGMkZCRkEvRjQ7IiIjSSZOX2J1c0dGKEZBLUY6NiQqJiZGPjYkRkAsKEY0RkFGTUZBRkxGQkZBLUknaWZlbHNlR0YmNiUvJkkpYnVzX3R5cGVHRihGM1EjUFZGKCIiISwmJkkjUWJHRihGM0ZBLUkiUUdGKEZJRkJGQUZKRkFGKEYoRig3I0Yu

Maximum number of iterations Max_it := 6: Acceleration coefficient lambda := 1: for Iter from 1 to Max_it do for m from 2 to N_bus do V[m] := V[m] + Delta_V(m, V, delta) * lambda: delta[m] := delta[m] + Delta_delta(m, V, delta) * lambda: end do: end do: Hence the bus voltage and phase angles are V, delta

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

These agree with the values calculated earlier with fsolve.

Power mismatch epsilon := add((Pb[m] - P(m, V, delta))^2 + ifelse(bus_type[m] = "PV", 0, (Qb[m] - Q(m, V, delta))^2), m = 2..N_bus)^0.5

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShlcHNpbG9uR0YoJCIwbzoiZVtMKjMjISNBNyNdM0U1NjZGMUVCMzg5RDlCQg==

Real and reactive power of the slack bus Ps := P(1, V, delta) + P__L[1]; Qs := Q(1, V, delta) + Q__L[1]

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Line losses for the second line are calculated as follows m := 2: i := Is[m]: j := Js[m]: ys := 1/Series[m, 3]; ysh := Series[m, 4];

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Power flow from sending to receiving terminal Pij := -V[i] * V[j] * abs(ys) * cos(argument(ys) + delta[j] - delta[i]) + V[i]^2 * abs(ys) * cos(argument(ys)) + (V[i])^2 * abs(ysh) * cos(argument(ysh))

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Power from receiving to sending terminal Pji := -V[j] * V[i] * abs(ys) * cos(argument(ys) + delta[i] - delta[j]) + V[j]^2 * abs(ys) * cos(argument(ys)) + V[j]^2 * abs(ysh) * cos(argument(ysh))

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Real power losses in the second line Ploss := Pij + Pji

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Reactive power flow from sending to receiving terminal Qij := V[i] * V[j] * abs(ys) * sin(argument(ys) + delta[j] - delta[i]) - V[i]^2 * abs(ys) * sin(argument(ys)) + (V[i])^2 * abs(ysh) * sin(argument(ysh))

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Reactive power flow from receiving to sending terminal Qji := V[i] * V[j] * abs(ys) * sin(argument(ys) + delta[i] - delta[j]) - V[j]^2 * abs(ys) * sin(argument(ys)) + (V[j])^2 * abs(ysh) * sin(argument(ysh))

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Reactive power losses in the second line Qloss := Qij + Qji

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