eee 103 2nd exam review
DESCRIPTION
EEE 103 2nd Exam ReviewTRANSCRIPT
Load Flow Studies
EEE 103 2nd Exam Review
Outline
• Bus Admittance Matrix
• Gauss-Seidel Iterative Method
• Tips for Reliable Solutions
The Bus Admittance Matrix
𝑌𝐵𝑈𝑆 =𝑌11 ⋯ 𝑌1𝑛⋮ ⋱ ⋮𝑌𝑛1 ⋯ 𝑌𝑛𝑛
• YBUS building rules:
– Diagonal elements Yii = sum of all admittances connected to bus i
– Off-diagonal elements Yij = negative of net admittance connected
between bus i and bus j
• Yii is the self-admittance or driving-point admittance
• Yij is the mutual admittance or transfer admittance
Properties of the YBUS
• Complex
• Symmetric
• Sparse (can have zero elements)
• Singular if there are no admittances between
any of the buses and the reference bus
Example 1: 3-bus system
Example 1: 3-bus system
Form the YBUS:
• Y12 = Y21 = -(1 / (0.02 + j0.04 Ω))= -10 + j20
• Y13 = Y31 = -(1 / (0.01 + j0.03 Ω)) = -10 + j30
• Y23 = Y32 = -(1 / (0.0125 + j0.025 Ω)) = -16 + j32
• Y11 = Y12 + Y13 = (10 – j20) + (10 – j30) = 20 – j50
• Y22 = Y21 + Y23 = (10 – j20) + (16 – j32) = 26 – j52
• Y33 = Y31 + Y32 = (10 – j30) + (16 – j32) = 26 – j62
Example 1: 3-bus system
𝑌𝐵𝑈𝑆 =
20 − j50 –10 + j20 −10 + j30−10 + j20 26 − j52 −16 + j32−10 + j30 −16 + j32 26 − j62
Source Transformation
• Is = Vs / Zs
• Ys = 1 / Zs
Injected Current
• The current that flows through a bus
• Current from sources flow into the bus
• Current to admittances flow out of the bus
Injected Current
KCL equation at bus i:
• IS = Iij + Iik + Iil
• IS = Yij(Vi – Vj) + Yik(Vi – Vk) + Yil(Vi – Vl)
• IS = YijVi – YijVj + YikVi – YikVk + YilVi – YilVl
• IS = (Yij + Yik + Yil)Vi – YijVj – YikVk – YilVl
Injected Current
𝐼1⋮𝐼𝑛
=𝑌11 ⋯ 𝑌1𝑛⋮ ⋱ ⋮𝑌𝑛1 ⋯ 𝑌𝑛𝑛
𝑉1⋮𝑉𝑛
𝐼𝐵𝑈𝑆 = 𝑌𝐵𝑈𝑆 𝑉𝐵𝑈𝑆
• IBUS: Bus current vector (i.e. current sources)
• YBUS: Bus admittance matrix
• VBUS: Bus voltage vector
Injected Power
𝑆1⋮𝑆𝑛
=𝑉1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑉𝑛
𝑌11 ⋯ 𝑌1𝑛⋮ ⋱ ⋮𝑌𝑛1 ⋯ 𝑌𝑛𝑛
𝑉1⋮𝑉𝑛
∗
𝑆𝑗 = 𝑉𝑗 𝑌𝑗𝑖𝑉𝑗∗
𝑛
𝑖=1
• The complex power that flows through a bus
Example 2: 4-bus system
Example 2: 4-bus system
Example 2: 4-bus system
The Gauss-Seidel (G/S) Iterative Method
• A solution to a non-linear equation with the form f(x) = 0
• Steps:
1. Rewrite function into x = g(x)
2. Initialize x(k), k = 0
3. Approximate x(k+1) with g(x(k))
4. Stop approximations when |x(k+1) – x(k)|< ε, where ε is the desired accuracy
Example 3: G/S solution
Find a root of f(x) = x3 – 6x2 + 9x – 4 = 0
Desired accuracy is 0.01
1. x = -(1/9)x3 + (6/9)x2 + (4/9) = g(x)
2. Let x(0) = 3.5
Example 3: G/S solution
Find a root of f(x) = x3 – 6x2 + 9x – 4 = 0
3. First iteration, x(1) = g(3.5) = -(1/9)(3.5)3 + (6/9)(3.5)2 + (4/9) = 3.8472
4. Check convergence |x(0) – x(1)| = 0.3472
5. Second iteration, x(2) = g(3.8472) = 3.9848
6. Check convergence |x(1) – x(2)| = 0.1376
7. Third iteration, x(3) = g(3.9848) = 3.9998
Example 3: G/S solution
Find a root of f(x) = x3 – 6x2 + 9x – 4 = 0
8. Check convergence |x(3) – x(2)| = 0.015
9. Fourth iteration, x(4) = g(3.9998) = 4.0
10. Check convergence |x(4) – x(3)| = 0.0002
11. Stop iterations, x = 4.0
The Load Flow Problem
• The ff. information are available
– Source (Generator) • Voltage and real power
delivered
– Line (Transmission) • Impedance
– Load • Complex power
consumed V, P
Z
P, Q
• The ff. are unknown
– Bus voltages
– Bus currents
– Line power flows
The Load Flow Problem
• We need the equation f(x) to solve for x
• f(x) is the complex injected power
• x is the bus voltage
• G/S will be extended to a linear system
The Load Flow Problem
For a bus i, 𝑓 𝑉𝑖 = 𝑆∗
𝑃𝑖 − j𝑄𝑖 = 𝑉𝑖∗ 𝑌𝑖1𝑉1 +⋯𝑌𝑖𝑖𝑉𝑖 +⋯𝑌𝑖𝑛𝑉𝑛
𝑌𝑖𝑖𝑉𝑖 =𝑃𝑖 − j𝑄𝑖𝑉𝑖∗ − 𝑌𝑖1𝑉1 +⋯𝑌𝑖 𝑖−1 𝑉𝑖 𝑖−1 + 𝑌𝑖(𝑖+1)𝑉𝑖(𝑖+1) +⋯𝑌𝑖𝑛𝑉𝑛
The (k+1)th approximation to the voltage Vi is
𝑉𝑖(𝑘+1)=1
𝑌𝑖𝑖
𝑃𝑖 − j𝑄𝑖
𝑉𝑖(𝑘)∗− 𝑌𝑖1𝑉1
(𝑘)+⋯+ 𝑌𝑖(𝑖−1)𝑉(𝑖−1)
(𝑘)+ 𝑌𝑖(𝑖+1)𝑉(𝑖+1)
(𝑘)+⋯+ 𝑌𝑖𝑛𝑉𝑛
(𝑘)
System Representation
Step-by-step G/S Formulation
1. Build the [YBUS]
2. Initialize voltages (Vi∠δ) and power (Psch,Qsch)
A. PV bus: use given V, set δ = 0.0°, and Psch = Pg – Pl
B. PQ bus: set V = 1.0, δ = 0.0°, Psch = Pg – Pl, and Qsch = Qg – Ql
3. Approximate voltages (except slack bus). Use the latest approximations.
A. PQ bus: 𝑉𝑖(𝑘+1)=1
𝑌𝑖𝑖
𝑃𝑖𝑠𝑐ℎ−j𝑄𝑖
𝑠𝑐ℎ
𝑉𝑖(𝑘)∗ − 𝑌𝑖1𝑉1
(𝑘)+⋯+ 𝑌𝑖(𝑖−1)𝑉(𝑖−1)
(𝑘)+ 𝑌𝑖(𝑖+1)𝑉(𝑖+1)
(𝑘)+⋯+ 𝑌𝑖𝑛𝑉𝑛
(𝑘)
Step-by-step G/S Formulation
3. Approximate voltages (except slack bus). Use the latest approximations.
B. PV bus: i. Approximate Qsch
using 𝑄 = −Im 𝑉𝑖
∗ 𝑌𝑖1𝑉1 +⋯𝑌𝑖𝑖𝑉𝑖 +⋯𝑌𝑖𝑛𝑉𝑛
ii. If there are MVAR limits, Qsch = Qmin if Q ≤ Qmin and Qsch = Qmax if Q ≥ Qmax; otherwise Qsch = Q
iii. Approximate V’ as if it were a PQ bus
iv. V = |V(0)|∠(angle of V’)
4. If max(|V(k+1) – V(k)|) ≤ ε then stop iteration, otherwise repeat steps 3 and 4.
Example 4: 3-bus Load Flow
• Use 100 MVA base and ε = 0.01
Line R (pu) X (pu)
1-2 0.02 0.04
1-3 0.01 0.03
2-3 0.0125 0.025
Bus Type V (pu) δ (deg) PGEN
(MW)
QGEN
(MVAR) PLOAD
(MW)
QLOAD
(MVAR)
1 SLACK 1.05 0 - - - -
2 LOAD - - 0 0 400 250
3 GEN 1.04 - 200 - - -
Example 4: 3-bus Load Flow
• The [YBUS] was solved earlier in Ex. 1
𝑌𝐵𝑈𝑆 =
20 − j50 –10 + j20 −10 + j30−10 + j20 26 − j52 −16 + j32−10 + j30 −16 + j32 26 − j62
• The initial voltage vector [VBUS] is [1.0500∠0.00° 1.0000∠0.00° 1.0400∠0.00°]
• The scheduled complex powers in per-unit are P2
sch = −4.0, Q2sch = −2.5, and P3
sch = 2.0
Example 4: 3-bus Load Flow
• First iteration V2
(1) = [1/(26−j52)+ * ,*(−4.0+j2.5) / (1.0∠0)+ − [(−10+j20) * (1.05∠0) + (−16+j32) * (1.04∠0)]}]
= 0.9755∠−2.49
Q3(1) = − Im{[1.04∠0] * [(−10+j30) * (1.05∠0) +
(−16+j32) * (0.9755∠−2.49) + (26−j62) * (1.04∠0)]}
= 1.16
V3(1) = *1/(26−j62)] * {[(2.0-j 1.16) / (1.04∠0)+ −
[(−10+j30) * (1.05∠0°) + (−16+j32) * (0.9755∠−2.49)]}] = 1.04∠−0.29
Example 4: 3-bus Load Flow
• Second iteration V2
(2) = 0.9708∠−2.56
Q3(2) = 1.2911
V3(2) = 1.04∠−0.36
• Third iteration V2
(3) = 0. 9720∠−2.56
Q3(3) = 0.8605
V3(3) = 1.04∠−0.36
Max. voltage mismatch: 0.0037, stop iterations.
Example 5: 4-bus Load Flow Line R (pu) X (pu)
1-2 0.01008 0.05040
1-3 0.00744 0.03720
2-4 0.00744 0.03720
3-4 0.01272 0.06360
Bus Type V (pu) δ (deg) PGEN
(MW)
QGEN
(MVAR) PLOAD
(MW)
QLOAD
(MVAR)
1 SLACK 1.0 0 - - - -
2 LOAD - - - - 170 105.35
3 LOAD - - - - 200 123.94
4 GEN 1.02 - 318 - 80 49.58
Example 5: 4-bus Load Flow
• Build the bus impedance matrix
• Solve for the bus voltages using G/S. Use 100 MVA base and ε = 0.01
Example 5: 4-bus Load Flow
• [YBUS]
8.9852–j44.9260 –3.8156 +j19.0781 –5.1696 +j25.8478 0
–3.8156+j19.0781 8.9852–j44.9260 0 –5.1696+j25.8478
–5.1696+j25.8478 0 8.1933–j40.9663 –3.0237+j15.1185
0 –5.1696+j25.8478 –3.0237+j15.1185 8.1933–j40.9663
Example 5: 4-bus Load Flow
• First iteration
V2 = 1.0034∠−1.82°
V3 = 1.0013∠−2.34°
V4 = 1.0200∠−4.28°
Q4 = 1.0468
Max mismatch = 0.0762
Example 5: 4-bus Load Flow
• Second iteration
V2 = 1.0032∠−4.26°
V3 = 1.0001∠−3.87°
V4 = 1.0200∠6.45°
Q4 = 1.5222
Max mismatch = 0.0428
Example 5: 4-bus Load Flow
• Third iteration
V2 = 1.0016∠−5.47°
V3 = 0.9986∠−4.64°
V4 = 1.0200∠7.53°
Q4 = 1.4658
Max mismatch = 0.0211
Example 5: 4-bus Load Flow
• Fourth iteration
V2 = 1.0006∠−6.07°
V3 = 0.9978∠−5.02°
V4 = 1.0200∠8.07°
Q4 = 1.3588
Max mismatch = 0.0105
Example 5: 4-bus Load Flow
• Fifth iteration
V2 = 1.0001∠−6.37°
V3 = 0.9973∠−5.21°
V4 = 1.0200∠8.34°
Q4 = 1.2858
Max mismatch = 0.0053, stop iterations
Example 6: 3-bus Load Flow
Line R (pu) X (pu)
1-2 0.02 0.10
1-3 0.04 0.15
2-3 0.07 0.25
Bus Type V (pu) δ (deg) PGEN
(MW)
QGEN
(MVAR) PLOAD
(MW)
QLOAD
(MVAR)
1 SLACK 1.0 0 - - 0 0
2 GEN 1.0 - 44.1 - 0 0
3 LOAD - - - - 100 66.67
Tips for Reliable Solutions
1. Remember these equations: [I] = [YBUS][V] and [P-jQ] = diag([V*])[YBUS][V]
2. Report per-unit quantities in four (4.0000) decimal places and angles in two (2.00) decimal places
3. YBUS
A. Complete the upper-diagonal only. Remember that the matrix is symmetric
B. Start with off-diagonal elems.
C. The diagonal elems. have positive real terms
Tips for Reliable Solutions
4. System Modelling
A. Positive power flows from generators
B. Negative power flows into load
5. G/S Iterative Method
A. Write the voltages in tabular form
B. Setup all equations first
C. Use the latest information available
D. All quantities are in per-unit