Power Systems

Load Curve, Load Duration Curve and Mass Curve

1. Load Curve

A load curve is a graph showing the variation of electrical load (demand) on a power station with respect to time. The load (in kW or MW) is plotted along the Y-axis and time (hours, days, months) along the X-axis.

• Daily load curve — variation of load over 24 hours of a day.
• Monthly load curve — obtained by averaging the daily load curves of a month.
• Annual load curve — obtained by combining the monthly load curves.

The area under the load curve gives the energy consumed (kWh) in that period. The load curve identifies the peak load and the base load, and hence forms the basis for deciding plant capacity and the operating schedule of generating units.

2. Load Duration Curve

A load duration curve is obtained by rearranging the load values of a load curve in descending order of magnitude against the duration for which each load occurs (or is exceeded). The X-axis represents the percentage of time (0–100%) or the number of hours in the period, and the Y-axis represents the load.

• The area under the load duration curve also equals the total energy consumed in the period.
• The highest load appears at the extreme left (duration = 0) and the lowest load at the extreme right (duration = 100%).

It is more useful than the load curve for generation planning because it directly shows for how many hours a given load level must be supplied. It helps in:

• deciding the capacity of base-load, intermediate and peak-load plants,
• economic scheduling (loading) of generating units,
• determining the reserve capacity requirement.

3. Mass Curve

A mass curve (or integrated load curve) is a plot of the cumulative energy (kWh) consumed against time. It is obtained by integrating the area under the load curve up to successive instants and plotting this cumulative energy versus time.

• The slope of the mass curve at any point gives the load (demand) at that instant.
• A steeper slope indicates a higher load; a flatter slope indicates a lower load.
• Mass curves are widely used in hydroelectric plants to compare the cumulative inflow (from the flow duration curve) with the cumulative demand, and thus to determine the storage capacity of the reservoir required to meet the load throughout the year.

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Significance of Load Factor, Plant Factor and Diversity Factor in Generation Planning

Load Factor

$\text{Load Factor} = \frac{\text{Average Load}}{\text{Maximum (Peak) Load}} = \frac{\text{Energy consumed in a period}}{\text{Peak load} \times \text{hours in the period}}$

Significance:

• A high load factor means the plant capacity is utilised for a greater part of the time, so the fixed cost per unit (kWh) generated is reduced.
• It indicates how uniformly the load is distributed; a low load factor means a large idle capacity during off-peak hours.
• It guides the planner in fixing tariffs (higher tariff for low load factor consumers) and in deciding the size of the plant so that the plant is neither oversized nor overloaded.

Plant Factor (Capacity Factor)

$\text{Plant Factor} = \frac{\text{Average Load on the plant}}{\text{Rated (installed) capacity of the plant}} = \frac{\text{Actual energy generated}}{\text{Maximum possible energy}}$

Significance:

• It measures the degree of utilisation of the installed plant capacity over a period.
• A high plant factor means the plant is being used efficiently and the capital invested is well recovered; a low plant factor indicates excess or idle capacity.
• It is used to decide which plant should be used as a base-load plant (high plant factor) and which as a peak-load plant (low plant factor), and to select the most economic plant capacity during expansion planning.

Diversity Factor

$\text{Diversity Factor} = \frac{\text{Sum of individual maximum demands}}{\text{Maximum demand of the whole system}}$

Significance:

• Since the maximum demands of different consumers occur at different times, the system peak is less than the sum of individual peaks. The diversity factor (always > 1) quantifies this.
• A high diversity factor means the installed capacity and the size of distribution equipment (transformers, feeders, conductors) can be reduced, thereby lowering the capital cost of the system.
• It is used in load forecasting and system planning to estimate the coincident system maximum demand from the connected load, and hence to decide the generating capacity to be installed.

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In summary: The load curve, load duration curve and mass curve describe the time-variation of demand and energy, while load factor, plant factor and diversity factor quantify the utilisation and coincidence of that demand — together they form the basic data for deciding plant capacity, type of plant (base/peak), reserve margin, equipment rating and economical operation in generation planning.

1. Factors Influencing the Choice of Transmission Voltage

The choice of transmission voltage for a new line is a techno-economic decision — a higher voltage reduces current (and hence I²R loss, conductor size and line drop) but increases the cost of insulators, towers, transformers and switchgear. The main factors are:

(a) Amount of power to be transmitted Power transmitted is roughly proportional to the square of the voltage for a given conductor size and distance: $P \propto V^2$ So a large block of power demands a high voltage. For example, a 500 MW hydro project evacuating power over 300 km needs 400 kV, whereas 20 MW over 20 km is handled by 66 kV or 132 kV.

(b) Distance of transmission Line reactance drop and losses grow with distance. Empirically, the economic voltage is given by: $V = 5.5\sqrt{0.6L + \frac{P}{100}} \text{ kV}$ where L is in km and P in kW. Longer lines therefore require higher voltages — 132 kV for ~100 km, 220 kV for ~200 km, 400 kV for ~400 km.

(c) Economic voltage (cost of conductor vs cost of insulation) The most economical voltage is where the total annual cost — interest + depreciation on conductor, insulators, towers and switchgear plus the cost of I²R loss — is minimum. This "economic voltage" rises with load and distance.

(d) Voltage regulation and power transfer limit The steady-state power limit of a line is: $P_{max} = \frac{V_1 V_2}{X} \sin\delta$ Higher voltage raises this limit and improves regulation, which matters for long lines and interconnected grids.

(e) Corona loss and radio interference Corona loss and interference increase with voltage but decrease with conductor size. At very high voltages, bundled conductors are used to keep the surface gradient below the corona limit. This sets a practical lower bound on conductor size for a given voltage.

(f) Standardisation and existing system voltage It is cheaper and operationally simpler to adopt the voltage already used in the neighbouring network. Nepal uses 66 kV, 132 kV and 220 kV; 400 kV lines (e.g. Dhalkebar–Muzaffarpur cross-border) are being introduced for bulk exchange with India.

(g) Insulation and right-of-way Higher voltage needs larger clearance, taller towers and wider right-of-way, which is costly in hilly terrain — a key constraint in Nepal.

Examples of voltage selection

For instance, the 456 MW Upper Tamakoshi evacuates power over ~100 km at 220 kV, while a rural 5 MW mini-hydro supplies a nearby load centre at 33 kV. This shows that voltage rises with both power and distance.

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2. ABCD Parameters for Transmission Line Modelling

A transmission line is a two-port network with sending-end quantities (Vₛ, Iₛ) and receiving-end quantities (Vᵣ, Iᵣ). It is described by the generalised circuit (transmission) equations:

$\begin{aligned} V_s &= A V_r + B I_r \\ I_s &= C V_r + D I_r \end{aligned}$

or in matrix form:

$\begin{bmatrix} V_s \\ I_s \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_r \\ I_r \end{bmatrix}$

where A, B, C, D are the generalised circuit constants (ABCD parameters).

Nature of the parameters:

• A and D are dimensionless (voltage ratio and current ratio).
• B has the unit of impedance (Ω).
• C has the unit of admittance (S).
• For a linear, passive, bilateral network: AD − BC = 1.
• For a symmetrical line: A = D.

Values for different line models:

where Z = series impedance of the whole line, Y = shunt admittance of the whole line, γ = √(ZY) = propagation constant, Z₀ = √(Z/Y) = surge impedance.

Advantages of ABCD parameters over other models

1. Cascading is simple. For two lines in series, the overall parameters are obtained by multiplying the individual matrices: $\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix} \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}$ This makes analysis of a multi-section line or a line with a transformer very easy — no need to re-derive from Kirchhoff's laws each time.

2. Uniform treatment of all line lengths. The same two-port framework covers short, medium and long lines; only the values of A, B, C, D change. This gives a single, general method.

3. Direct computation of performance indices. Once A, B, C, D are known, the sending-end voltage and current for any receiving-end load can be found directly. Regulation, efficiency and the receiving-end voltage for a given load are easily computed.

4. Useful for network reduction. A transmission line, a transformer or a combination can be represented as a single two-port block, which simplifies load-flow and fault studies on large networks.

5. Physical interpretation. A = Vₛ/Vᵣ at no load (Iᵣ = 0), B = Vₛ/Iᵣ at short circuit (Vᵣ = 0), C = Iₛ/Vᵣ at no load, D = Iₛ/Iᵣ at short circuit. These open-circuit and short-circuit tests allow the parameters to be measured experimentally, just like the h-parameters of a two-port network.

6. Reciprocity check. The condition AD − BC = 1 provides a ready check on the correctness of computed or measured parameters.

Because of these features, ABCD parameters are the standard tool for representing transmission lines in power system analysis.

Importance of Load Flow Analysis in Power System Studies

Load flow (power flow) analysis determines the steady-state voltage magnitudes and phase angles at every bus, and the active and reactive power flows on every branch, for a given set of generation and load conditions. Its importance:

1. System planning: It verifies whether proposed lines, transformers and generators can supply future loads within permissible voltage limits (typically ±5%) and without overloading any equipment. It is the basic tool for deciding the size, location and timing of new transmission and generation additions.
2. System operation and control: The load dispatch centre uses it to check that the present or forecast operating state is secure, to schedule generation (economic dispatch), to set transformer taps and generator excitation, and to decide preventive or corrective actions.
3. Contingency and security analysis: By repeating the load flow for each outage (line, transformer or generator), the system is tested for N−1 security, and weak lines or buses are identified.
4. Voltage and reactive power management: It gives the voltage profile and the reactive power balance at each bus, which is the basis for placing capacitors, reactors and SVCs and for voltage stability studies.
5. Loss evaluation and tariff/accounting: It quantifies I²R losses and inter-area power exchanges, needed for loss allocation, wheeling charges and interchange settlement.
6. Input to other studies: The converged load flow solution is the starting point (base case) for short-circuit, transient stability, small-signal stability and optimal power flow studies.

Basis of Bus Classification

In a load flow problem each bus has four quantities: P, Q, |V| and δ, of which two must be specified and the other two are computed. Since the losses are unknown beforehand, the slack (reference) bus is needed to supply the difference. Buses are therefore classified by which two quantities are specified:

Notes:

• The slack bus is the only bus where generation is not scheduled in advance; it absorbs the system losses and any mismatch. Its voltage phasor is fixed so that the angular reference for the whole network is established.
• A PV bus is converted to a PQ bus when its computed reactive generation exceeds Qmax or falls below Qmin; the Q is then fixed at the limit and the voltage is allowed to fall.
• The number of unknowns equals the number of equations: 2(N−1) for a system of N buses, since the slack bus angle and magnitude are known.

Common Algorithms for Load Flow Analysis

1. Gauss–Seidel Method

The bus voltage equation derived from nodal analysis is solved iteratively:

$V_i^{(k+1)} = \frac{1}{Y_{ii}}\left[\frac{P_i - jQ_i}{V_i^{(k)*}} - \sum_{j \neq i} Y_{ij} V_j^{(k)}\right]$

for PQ buses, and the reactive power is updated for PV buses at each iteration.

Advantages:

• Very simple to program; needs little memory.
• Each bus is updated using the latest available values, so convergence is usually reached in a moderate number of iterations for small systems.
• No matrix inversion or factorization is required.

Limitations:

• Convergence is linear and slow; the number of iterations grows roughly in proportion to the number of buses.
• May diverge or converge very slowly for ill-conditioned or heavily loaded systems.
• The slack bus must supply all losses, and convergence is sensitive to the choice of slack bus.
• Practically unsuitable for large networks (more than a few hundred buses).

2. Newton–Raphson Method

The power mismatch equations are expanded in a Taylor series and the second and higher order terms are neglected, giving the linearised system:

$\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} H & N \\ J & L \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}$

where the coefficient matrix is the Jacobian. The Jacobian is rebuilt and factorised at every iteration, and the correction vector is added to the previous estimate.

Advantages:

• Quadratic convergence: the number of significant digits roughly doubles each iteration, so 3–5 iterations are usually enough irrespective of system size.
• The number of iterations is almost independent of the number of buses, making it the standard method for large systems.
• Handles ill-conditioned cases and PV buses efficiently; the Jacobian gives useful sensitivity information.

Limitations:

• Each iteration is computationally heavy because the Jacobian must be formed and factorised afresh.
• Requires good initial estimates; poor starting values can cause divergence.
• Programming is more involved, and memory requirement is larger.
• Does not easily accommodate the slack bus losses without an outer iteration.

3. Fast Decoupled Load Flow

Using the practical observations that in EHV networks the resistance is much smaller than the reactance (so that ∂P/∂|V| and ∂Q/∂δ are small) and that voltage magnitudes are close to 1.0 p.u., the Jacobian is reduced to two constant matrices:

$[B'][\Delta \delta] = [\Delta P/|V|]$ $[B''][\Delta |V|] = [\Delta Q/|V|]$

Advantages:

• The two matrices are constant and are factorised only once, so each iteration is very fast.
• Memory requirement is low and convergence is reliable for normal transmission systems.
• Widely used in real-time energy management systems and contingency analysis.

Limitations:

• Based on approximations (r ≪ x, |V| ≈ 1 p.u.), so it performs poorly for distribution networks, low-voltage networks or systems with high r/x ratio.
• Convergence is geometric rather than quadratic, so more iterations are needed than Newton–Raphson.
• Not suitable when voltage deviations are large or when the network is heavily loaded near the stability limit.

4. DC Load Flow

A further simplification in which all resistances, shunt admittances and voltage magnitude variations are neglected, so the problem becomes linear:

$[B'][\delta] = [P]$

Advantages:

• Extremely fast and non-iterative; used for screening a very large number of contingencies.
• Gives active power flows and angles accurately enough for security ranking.

Limitations:

• Gives no information on voltages or reactive power.
• Losses are completely ignored, so results are approximate.

Summary of comparison

In practice, the Newton–Raphson method (or its decoupled version) is the standard choice for transmission system studies, while Gauss–Seidel is retained only for small networks or as an initialiser.

Selection of Transmission Voltage and Conductor Size for an Urban Supply System

1. Factors Affecting the Selection of Transmission Voltage

The transmission voltage is the single most important design decision, since it fixes insulation levels, tower heights, right of way width and the cost of switchgear. The main factors are:

(a) Amount of power to be transmitted. For a three-phase line, power transferred is approximately

$P = \sqrt{3}\, V_L I_L \cos\phi$

For a given power and permissible current, the voltage and current are inversely related. A growing urban area with a large connected load (say 100–200 MW) therefore demands a high voltage so that the current, and hence the conductor size and $I^2R$ loss, remain within limits.

(b) Length of the transmission line. The voltage drop and the line losses depend on the line length. As a rough guide, the economic voltage rises with distance:

$V \approx 5.5\sqrt{0.62\,L + \frac{P}{100}}\ \text{kV}$

(Still's formula, with $L$ the line length in km and $P$ the power in kW)

Short lines (10–20 km) can be served at 33 kV or 66 kV, whereas a line of 100 km or more justifies 132 kV, 220 kV or 400 kV.

(c) Power loss and voltage regulation. Losses vary inversely as the square of the voltage ($P_{loss} \propto 1/V^2$). A higher voltage reduces losses and improves regulation, which matters greatly for an urban load containing motors and electronic equipment sensitive to voltage variation.

(d) Cost economics. The total annual cost is the sum of the capital cost (which rises with voltage because of larger insulators, taller towers, wider right of way and costlier transformers and switchgear) and the cost of losses (which falls with voltage). The economic voltage is the one at which the total annual cost is minimum. Kelvin's law is applied to the conductor, and a similar cost comparison is made for voltage levels.

(e) Standardisation. Only standard voltages (11, 33, 66, 132, 220, 400 kV in Nepal) should be chosen, so that transformers, insulators and protective equipment are interchangeable and spares are available. Nepal's urban distribution is largely at 11 kV and 33 kV, with 132 kV and 220 kV used for bulk transmission from hydropower stations to load centres such as Kathmandu.

(f) Future expansion. The urban area is "growing", so the voltage must be chosen with a view to the load 15–20 years ahead; uprating a line later is far costlier than building it at the higher voltage initially.

(g) Other factors. Availability of a right of way corridor, environmental and safety requirements, short-circuit capacity of the system, and the voltage of the existing network to which the line must be interconnected.

2. Factors Affecting the Selection of Conductor Size

(a) Current-carrying capacity (thermal limit). The conductor must carry the maximum continuous current without exceeding the permissible temperature (usually 75 °C for ACSR), otherwise annealing and loss of tensile strength occur. This fixes the minimum cross-section.

(b) Corona and radio interference. For high-voltage lines, the conductor diameter must be large enough to keep the surface voltage gradient below the disruptive value; otherwise corona loss and radio interference become excessive. This often makes the corona limit, not the thermal limit, the governing criterion at 220 kV and above.

(c) Mechanical strength and sag. The conductor must withstand wind and ice loading and its own weight without excessive sag. A minimum size (e.g. not below 35 mm² or so) is fixed by mechanical considerations, and the tension is limited so that sag does not violate ground clearance.

(d) Voltage drop and regulation. The conductor resistance and reactance determine the voltage drop; a longer or more heavily loaded line needs a larger cross-section to keep regulation within limits (say ±6 %).

(e) Economic current density. Kelvin's law states that the most economical size is that for which the annual cost of the conductor (interest + depreciation) equals the annual cost of the energy lost. In practice the size is chosen from standard sizes nearest to this economic value.

(f) Mechanical and climatic conditions. Wind pressure, ice loading, temperature range and the terrain (hilly terrain in Nepal needs shorter spans and higher tensions) all influence the choice.

3. Role of Sag and Tension

Sag is the vertical distance between the lowest point of the conductor and the line joining the two supports. For a level span,

$S = \frac{wL^2}{8T}$

where $w$ = weight per unit length (kg/m), $L$ = span (m), $T$ = tension (kg).

• Sag is inversely proportional to tension and directly proportional to the square of the span. If the tension is too high, the conductor may fatigue or fail; if too low, the sag is large and ground clearance is lost.
• The conductor is strung at a tension such that the sag at the worst condition (maximum wind or maximum ice with minimum temperature) does not exceed the permissible value. The stringing tension is therefore determined by the "governing condition" among (i) maximum temperature, (ii) minimum temperature with wind, and (iii) maximum wind/ice load.
• Sag also affects the height of towers: a larger sag needs taller towers, which increases cost. Hence span, tension and tower height are optimised together.

4. Clearances and Right of Way

• Ground clearance is the minimum vertical distance between the lowest conductor and the ground, roads, railways, rivers, buildings and other lines. It depends on the voltage level (e.g. about 6.1 m for 33 kV, 6.7 m for 66 kV, 7.6 m for 132 kV, 8.8 m for 220 kV over open ground, with larger values over roads and inhabited areas). It must be satisfied at maximum sag and maximum temperature.
• Right of way (ROW) is the strip of land along the line kept free of obstructions. Its width increases with voltage (roughly 15–30 m for 132 kV and 30–50 m for 220 kV). In an urban area, acquiring ROW is difficult and expensive, so the route must be selected carefully, and taller towers or underground cables may be preferred near dense settlements.
• Safety clearances between conductors and from conductors to structures are governed by the standard clearance formulae, e.g. $a = 0.4 + 0.01V$ (metres) for phase-to-phase spacing in some codes.

5. Corona Phenomenon — Views

Yes, corona is definitely expected on a high-voltage urban transmission line, and it must be considered in design.

Cause. When the surface voltage gradient of the conductor exceeds the disruptive value of air (about 30 kV/cm peak, or 21.1 kV/cm rms), the air surrounding the conductor ionises, producing a faint violet glow, a hissing sound, ozone and radio interference. This is corona.

Disruptive voltage for a conductor of radius $r$ and spacing $D$ is

$V_d = 21.1\, m_0\, \delta\, r \ln\frac{D}{r}\ \text{(kV, rms)}$

where $m_0$ = surface irregularity factor and $\delta$ = air density factor.

Effects.

1. Power loss — corona loss increases with voltage and frequency and decreases with conductor diameter. It is significant above 220 kV.
2. Radio and television interference — the ionised air radiates noise, disturbing communication near the urban area.
3. Ozone and corrosion — ozone attacks insulators and fittings.
4. Reduced effective capacitance and increased apparent diameter — corona acts as if the conductor diameter is larger, which slightly reduces the voltage gradient (a self-limiting effect).

Mitigation. Corona can be reduced by:

• using larger conductor diameter (e.g. ACSR with larger outer strands),
• using bundled conductors (two, three or four sub-conductors per phase) which increase the effective diameter and reduce the gradient,
• increasing conductor spacing,
• using smooth, clean conductors and avoiding sharp edges on fittings,
• corona rings at insulator strings.

Conclusion. For an urban area in Nepal, a 132 kV or 220 kV line would be typical. At 132 kV, corona loss is small but radio interference near settlements must still be checked; at 220 kV and above, corona becomes a governing design factor and bundled conductors are adopted. Thus the selection of voltage and conductor size, the control of sag and tension, the provision of adequate clearances and right of way, and the suppression of corona are all interlinked aspects of the same design problem.

Short Note on Economic Load Dispatch in Interconnected Power System

Meaning. Economic Load Dispatch (ELD) is the process of allocating the total real power demand among the committed generating units of an interconnected power system so that the total operating (fuel) cost is minimised, while satisfying the load demand and all unit and network constraints. It is a real-time operational problem solved at the control centre, typically every few minutes, after unit commitment has decided which units are ON.

Objective function. For $N$ generating units with output $P_i$ (MW) and fuel cost $C_i(P_i)$ (Rs/h),

$$\min C_T=\sum_{i=1}^{N} C_i(P_i)$$

The cost of each unit is normally approximated by a quadratic polynomial:

$$C_i(P_i)=a_iP_i^{2}+b_iP_i+c_i \quad \text{(Rs/h)}$$

Constraints.

1. Power balance (equality):

$$\sum_{i=1}^{N} P_i = P_D + P_L$$

where $P_D$ is total demand and $P_L$ is transmission loss.

2. Generation limits (inequality):

$$P_i^{\min}\le P_i\le P_i^{\max},\qquad i=1,2,\dots,N$$

Solution by Lagrange multiplier. Forming the Lagrangian with multiplier $\lambda$ (the incremental cost of received power, Rs/MWh),

$$L=\sum_{i=1}^{N}C_i(P_i)+\lambda\left(P_D+P_L-\sum_{i=1}^{N}P_i\right)$$

For a minimum, $\partial L/\partial P_i=0$:

$$\frac{dC_i}{dP_i}+\lambda\frac{\partial P_L}{\partial P_i}-\lambda=0$$ $$\Rightarrow \frac{dC_i}{dP_i}=\lambda\left(1-\frac{\partial P_L}{\partial P_i}\right)=\frac{\lambda}{PF_i}\quad\Longleftrightarrow\quad PF_i\,\frac{dC_i}{dP_i}=\lambda$$

where $PF_i=\dfrac{1}{1-\partial P_L/\partial P_i}$ is the penalty factor of unit $i$. This is the coordination equation: the incremental cost of every unit multiplied by its penalty factor must be equal to the system $\lambda$.

Lossless (classical) case. When transmission losses are neglected ($P_L=0$, $PF_i=1$), the condition reduces to the well-known equal incremental cost criterion:

$$\frac{dC_1}{dP_1}=\frac{dC_2}{dP_2}=\dots=\frac{dC_N}{dP_N}=\lambda$$

For the quadratic cost model, $dC_i/dP_i=2a_iP_i+b_i$, hence

$$P_i=\frac{\lambda-b_i}{2a_i}$$

and $\lambda$ is found from $\sum P_i=P_D$, respecting the limits. Units violating limits are fixed at $P_i^{\min}$ or $P_i^{\max}$ and the remaining units are dispatched again (water-filling / $\lambda$-iteration method).

Figure.

        Rs/MWh
          |            Unit 1 (cheaper)   Unit 2 (costlier)
          |              /                  /
          |             /                  /
     λ  --|------------*------------------*------  equal incremental cost line
          |           /                  /
          |          /                  /
          +--------------------------------------------> P (MW)
              P1        P2
        Total demand  P1 + P2 = PD  (+ PL)

   [ λ-iteration: increase λ until Σ Pi = PD + PL ]

Effect of losses. Because $PF_i>1$ for units whose output increases losses ($\partial P_L/\partial P_i>0$, typically far from the load centre), the condition $dC_i/dP_i=\lambda/PF_i$ forces them to run at a lower incremental cost, i.e. they generate less; units near the load centre generate more. This reduces total loss and hence total cost, though the incremental costs are no longer equal.

Methods of solution. (i) Analytical $\lambda$-iteration method, (ii) gradient method, (iii) Newton–Raphson method, (iv) linear programming, (v) dynamic programming, (vi) modern soft-computing methods (GA, PSO, etc.) for non-convex cost curves.

Significance. ELD gives minimum operating cost, better economy, reduced losses, optimal reserve utilisation and reliable operation of the interconnected grid. In Nepal, ELD is applied in the Integrated Nepal Power System (INPS) to schedule hydro and thermal/import generation economically.

Protection of transmission lines against lightning

A transmission line is protected against lightning by a combination of shielding, low tower footing resistance, and co-ordination of insulation:

1. Overhead earth (shield) wires — One or two earth wires are strung at the apex of the towers above the phase conductors. They intercept the lightning stroke and divert it to earth, so the phase conductors are shielded within the "protective angle" (usually about 20°–30°). This is the primary protection.

2. Low tower footing resistance — The surge current from the earth wire must be quickly discharged to ground. Tower footing resistance is kept low (generally below 10 Ω) by counterpoise wires, driven rods, or chemical earthing, so that the tower top potential does not rise enough to flash over the insulator strings (back-flashover).

3. Co-ordination of insulation (basic impulse insulation level) — The insulator strings and air gaps are designed so that their impulse withstand level is higher than the voltage developed across them during a stroke. A protective gap or surge arrester may be provided at substation entries to limit the surge.

4. Overhead ground wires with shield angle and continuous earth continuity — Earth wires are bonded to each tower and to the tower footing, giving a continuous low-impedance path to earth.

5. Reduction of surge impedance / travelling wave effects — The earth wire also reduces the induced voltage on phase conductors by electrostatic and electromagnetic coupling, lowering the induced overvoltage.

6. Additional measures — Arcing horns, grading rings, and, in some designs, line surge arresters on exposed spans; also, a lower tower height, since a taller tower increases the line's exposure to direct strokes, and a lower footing resistance, which reduces the chance of back-flashover.

Optical Ground Wire (OPGW) at the apex of towers

The OPGW is a composite cable in which optical fibres are housed inside a metallic (usually aluminium or steel) tubular structure that also serves as the earth (ground) wire. It is run along the apex of the towers, replacing or supplementing the conventional earth wire.

Uses:

• Lightning shielding — It performs the normal function of an earth wire, protecting the phase conductors from direct strokes.
• Communication channel — The embedded optical fibres carry data, telemetry, SCADA signals, teleprotection, telephony, and internet traffic between substations and control centres.
• Earth continuity — It provides a continuous earth path along the line.

Importance:

• Dual function — One cable does the work of both earth wire and communication link, saving the cost and tower loading of a separate optical cable.
• High bandwidth and immunity — Optical fibres are immune to electromagnetic interference (EMI) and induced voltages from the power line, giving reliable, high-capacity communication.
• Reliability and security — Data cannot be tapped easily; the cable is protected mechanically by the metallic armour.
• Grid management — Enables real-time monitoring, protection signalling, and remote control of the power system, which is essential for modern smart-grid operation.
• Cost-effective — Utilises the existing tower apex and right-of-way, avoiding separate communication infrastructure.

Thus, the earth wire at the apex gives the basic lightning protection, while the OPGW adds a high-capacity, interference-free communication backbone to the same mechanical path.

Three Part Tariff

A three part tariff is a tariff in which the total charge for electrical energy is divided into three components:

1. Fixed charge (or standing charge): A fixed amount payable per billing period irrespective of the energy consumed. It depends on the maximum demand (contract demand) of the consumer and covers the interest and depreciation on the capital cost of the plant, transmission lines and distribution network.

2. Semi-fixed charge (or demand charge): A charge proportional to the maximum demand recorded during the billing period (in kW or kVA). It covers the cost of the plant capacity that must be kept ready to meet the consumer's peak demand.

3. Running charge (or energy charge): A charge proportional to the actual energy consumed (in kWh). It covers the running cost of the plant, i.e. the cost of fuel, labour and maintenance.

Thus, if Rs a is the fixed charge per month, Rs b per kW of maximum demand and Rs c per kWh consumed, the monthly bill for a consumer having maximum demand MD (kW) and energy consumption E (kWh) is:

$\text{Bill} = a + b \times MD + c \times E$

Advantages: It is the most fair and equitable tariff, since the fixed cost, capacity cost and running cost of supply are recovered separately from the consumers who are responsible for each. It gives correct economic signals to the consumer to reduce peak demand as well as total energy consumption.

Power Factor Tariff

Power factor tariff is a tariff in which the electricity charges are made to depend upon the power factor of the consumer's load, so as to encourage consumers to maintain a high (near unity) power factor. Low power factor increases the current for the same power, thereby increasing line losses, voltage drops and the required plant and conductor ratings; hence the consumer causing it must bear the extra cost. The common forms are:

1. kW and kVA maximum demand tariff: The consumer is charged on both his maximum kW demand and maximum kVA demand. Since kVA = kW / cos φ, a consumer with poor power factor has a higher kVA demand for the same kW demand and therefore pays more.

2. Sliding scale (average power factor) tariff: A specified reference power factor (say 0.85 lagging) is fixed. Consumers with power factor above this are given a discount (penalty is negative), and those below it are charged a penalty, the rate being proportional to the deviation from the reference value.

3. kVA maximum demand tariff: The consumer is charged only on the maximum demand in kVA. This automatically penalises low power factor, since kVA demand rises as power factor falls.

4. Three part tariff with power factor clause: The demand charge is based on kVA instead of kW, so that the consumer pays for the apparent power actually drawn from the supply.

Thus power factor tariff encourages the installation of capacitors or synchronous condensers by the consumer to improve the power factor and reduce his bill.

1. Introduction

A bus-bar is the common conductor to which incoming and outgoing circuits of a substation are connected. The bus-bar scheme fixes reliability, flexibility and cost. Two switching devices are used at 132 kV:

• Circuit breaker (CB): makes and breaks load and fault current.
• Isolator: no arc-quenching capacity; operated only with the circuit de-energised (after the CB is open); gives a visible break for safe maintenance.

2. Single bus-bar

  Line-1        Line-2        Line-3
    |             |             |
  ISO-L         ISO-L         ISO-L     line-side isolators
    |             |             |
   CB1           CB2           CB3
    |             |             |
  ISO-B         ISO-B         ISO-B     bus-side isolators
    |             |             |
 ===+=============+=============+=====  132 kV BUS

All circuits are connected to one bus; each has a CB with an isolator on both sides. It is the simplest and cheapest scheme, but a bus fault, or bus maintenance, shuts down the whole substation, and CB maintenance needs that circuit shut down. It is used in small, non-critical substations.

3. Main and transfer bus

  Line-1        Line-2        Line-3
    |             |             |
  ISO-L         ISO-L         ISO-L
    |             |             |
    +----ISO-T----+----ISO-T----+----ISO-T--- TRANSFER BUS ---+
    |             |             |                              |
   CB1           CB2           CB3                         ISO  CB-T  ISO
    |             |             |                              |
  ISO-B         ISO-B         ISO-B                            |
    |             |             |                              |
 ===+=============+=============+===== MAIN BUS ===============+

Normally all circuits feed from the main bus through their CB; the transfer bus is idle. A transfer (bus-coupler) CB with isolators connects the main and transfer buses. To maintain the CB of a circuit: close the transfer isolator of that circuit, close the transfer CB so that the circuit is now fed from the main bus via the transfer bus, then open the circuit's own CB and isolators. The circuit stays in service, protected by the transfer CB, while its own CB is repaired.

Advantage: CB maintenance without interruption; only one extra CB for the whole station. Drawback: a main-bus fault still trips the whole station, and one circuit at a time can be transferred.

4. Double bus-bar

 ===+=============+=============+=====  MAIN BUS
    |             |             |    \
  ISO-a         ISO-a         ISO-a   ISO-C1
    |             |             |      |
    +-------------+-------------+     CB-C   bus coupler
    |             |             |      |
  ISO-b         ISO-b         ISO-b   ISO-C2
    |             |             |    /
 ===+=============+=============+=====  RESERVE BUS

In the usual form each circuit has one CB (line side, with isolator) and two bus-side isolators, one to the main bus and one to the reserve bus; the bus coupler (CB with an isolator on each side) joins the two buses. Normally all circuits work on the main bus, with the reserve bus charged through the coupler. Any circuit can be moved to the reserve bus without interruption, using the coupler to equalise the two buses first, so a bus can be maintained without shutdown, and if the main bus faults, circuits are shifted to the reserve bus after a short outage.

Advantage: high flexibility and reliability. Drawback: more isolators and a coupler CB, so higher cost and more complex protection.

5. Comparison

Interlocking rule for all schemes: isolators are opened only after the CB is open, and closed before the CB is closed.