Load Flow Analysis of a Five-Bus Power System
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Load Flow Analysis of a Five-Bus Power System Introduction This application analyzes this five-bus power system, and calculates the voltages and real and reactive powers at each bus. Generators 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 Law where voltage magnitude, V i i real power, P i reactive power, Q i These variables are known for each bus (where the suffix G and D stand for generation and demand respectively).
Transcript of Load Flow Analysis of a Five-Bus Power System
Load Flow Analysis of a Five-Bus Power System
IntroductionThis application analyzes this five-bus power system, and calculates the voltages and real and reactive powers at each bus.
Generators 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 Law
where
voltage magnitude, Vi
ireal power, Pireactive power, Qi
These variables are known for each bus (where the suffix G and D stand for generation and demand respectively).
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Bus type Unknown variable Known variable
SlackAlways has a generator
|V|, D, QDPG, QG
PQ PG, QG, PD, QD |V|, PV |V|, PG, PD, QD , QG
Larger power systems (for example, those modeling real-life distribution systems) can consist of tens ofthousands 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:
Properties
Transmission Lines and Transformer DataEach row of the following matrix contains the properties of a transmission line
Column Data1 Sending end bus2 Receiving end bus3 Series impedance in pu4 Shunt 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 ]]):
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Shunt DataIf 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 dataBus 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 buses
delta := 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"]):
InitializationNumber 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 equipmentIs := Re(Series[.., 1]):Js := Re(Series[.., 2]):Ish := Re(Shunt [.., 1]):
Real power injection (net)
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Pb := P__G - P__L
Reactive power injection (net)Qb := -Q__L
Nodal Admittance MatrixInitialize the nodal admittance matrix.Y__bus := Matrix(N_bus, N_bus):
Add series elements to the nodal admittance matricfor 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
Add shunt elementsfor 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 and Reactive Power InjectionsReal 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 functionQ := (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
Solve the Load Flow Equation with fsolveGenerate the real power flow equations.real_power_eq := seq(Pb[i] = P(i, v, d), i in {2, 3, 4, 5})
Generate the reactive power flow equationsreactive_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 systemsys := eval({real_power_eq, reactive_power_eq}, {v[1] = 1.04, d[1]= 0, v[3] = 1.02})
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)
Check the correctness of the solutioneval([real_power_eq, reactive_power_eq],{v[1] = 1.04, d[1] = 0, v[3] = 1.02} union sys)
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Solve the Load Flow Equations with Newton-Raphson Iteration
Form the Jacobian MatrixJac := 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] =
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"PV", 1, Jac[k + N_bus - 2, k + N_bus - 2])
end do:end do:
Hence the complete Jacobian matrix isJac
Invert the Jacobian matrixJinv := Jac^(-1)
Iterative SolutionCorrections of voltage and phase angle for the new iteration stepDelta_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
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)
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Warning, (in Delta_delta) `k` is implicitly declared local
Maximum number of iterationsMax_it := 6:
Acceleration coefficientlambda := 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 areV, delta
These agree with the values calculated earlier with fsolve.
Analysis of ResultsPower mismatchepsilon := 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
Real and reactive power of the slack busPs := P(1, V, delta) + P__L[1];Qs := Q(1, V, delta) + Q__L[1]
Line losses for the second line are calculated as followsm := 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 terminalPij := -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))
Power from receiving to sending terminalPji := -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))
Real power losses in the second linePloss := Pij + Pji
Reactive power flow from sending to receiving terminalQij := 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))
Reactive power flow from receiving to sending terminalQji := 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))
Reactive power losses in the second lineQloss := Qij + Qji
