CLIAMS:We Claim:
1. A method to manage dynamic economic dispatch of power comprising:
receiving a predetermined schedule load demand for power dispatch;
receiving ramp rate constraints, power balance constraints and generator operating limit constraints of plurality of connected power generating units (101) in an electrical power network;
receiving predetermined power load data from plurality of buses, plurality of transmission lines (105) and plurality of power generating units (101) for obtaining power flow solution using at least one numerical iterative method;
employing at least one numerical technique for calculating one of at least two transmission loss coefficients;
determining slack bus power using the power flow solution;
determining slack bus power using the economic dispatch;
obtaining slack bus power mismatch; the determined economic dispatch result being updated in bus system provided the obtained slack power mismatch being lesser than tolerance value;
determining the power load for a consecutive defined time interval for at least 24 hours;
obtaining the economic dispatch by selecting at least one of the transmission loss coefficients satisfying dispatch constraints;
updating generator limits at consecutive defined time interval for at least 24 hours in generator characteristics database (119);
a remote control unit for transmitting data from dynamic economic dispatch processor (113)to a remote terminal unit controlling plurality of generating units (101);
regulating plurality of the power generating units (101) by updating the load demand data in at least one processor (113) for scheduling and dispatching.
2. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the operation for determining the transmission loss coefficients shows a percentage error in the range of 1 to 1.9, preferably 1.4 to 1.7.
3. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the at least one numerical iterative approach for determining power flow solution is selected from Guass Siedel approach, Newton- Raphson approach, and/or fast decouple approach, preferably Newton-Raphson approach.
4. The method to manage dynamic economic dispatch of power as claimed
in claim 1, wherein the at least one numerical technique to determine
one of the at least two transmission loss coefficients, preferably A
transmission loss coefficient, is selected from iteration technique,
perturbation technique, curve fitting method and/or slack bus power
correction technique, preferably perturbation technique and slack bus
power correction.
5. The method to manage dynamic economic dispatch of power as claimed
in claim 1, wherein the at least one numerical technique to determine
one of the at least two transmission loss coefficients, preferably B
transmission loss coefficient, is selected from iteration technique,
perturbation technique, curve fitting method and/or slack bus power
correction technique, preferably slack bus power correction technique.
6. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the at least one processor (113) obtains said slack bus power correction by iteratively solving economic dispatch and finding loss coefficients.
7. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the plurality of numerical techniques to determine the economic dispatch is selected from gradient method, lambda-iteration method, newton’s method, evolutionary algorithms, genetic algorithm, neural networks, simulated annealing, particle swarm optimization, bacterial foraging algorithm, artificial honey bee algorithm, and firefly algorithm, preferably gradient method.
8. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein determining the slack power mismatch is by finding the difference between slack bus power predetermined by power flow solution and slack bus power predetermined by economic dispatch.
9. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the predetermined schedule load demand is obtained from forecasted load data.
10. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the dispatch is obtained by selectively using A transmission loss coefficient for allocating economic dispatch if load is in the range of 0.9 to 1.4 per unit.
11. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the dispatch is obtained by selectively using B transmission loss coefficient for allocating economic dispatch if load is in the range of 0 to 0.9 per unit.
12. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the ramp rate constraints, power balance constraints, generator operating limit constraints are obtained from predetermined data.
13. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the tolerance value is in the range of 0 to ±10 -8per unit.
14. The method to manage dynamic economic dispatch of power as claimed in claim 1, wherein the costs associated with each of the plurality of generating units is calculated according to the formula (1) and formula (2).
15. A dynamic economic dispatch system to manage optimal power dispatch and power generation comprising:
at least one processor (113) for determining at least two transmission loss coefficients using at least one numerical technique and regulating plurality of power generating units by updating load demand data for minimizing the costs associated with each of the plurality of generating units (101);
wherein the at least one processor (113) automatically and selectively use either of the at least two transmission loss coefficients for allocating economic dispatch employing a dispatch constraint for a defined consecutive time interval for at least 24 hours;
wherein the at least one processor (113) determines ramp rate constraints, power balance constraints and generator operating limit constraints of plurality of connected power generating units in an electrical power network;
wherein the at least one processor (113) receives predetermined load data from plurality of buses, plurality of transmission lines (105) and plurality of power generating units (101) for determining power flow solution using at least one numerical iterative approach;
wherein the at least one processor (113) is operative for obtaining economic dispatch data by adopting from plurality of numerical techniques;
wherein the at least one processor (113) determines slack bus power mismatch and the determined economic dispatch data being updated in bus system when the obtained slack power mismatch is lesser than tolerance value;
wherein the at least one processor (113) updates the generator limits at defined time interval for at least 24 hours to a remote control unit for allocating output to one of the plurality of generating units (101).
16. The dynamic economic dispatch system as claimed in claim 15, wherein the remote terminal unit being any compatible programming module measuring electrical and mechanical parameters.
17. The dynamic economic dispatch system as claimed in claim 15, wherein the plurality of buses is selected from any electrical power network, preferably IEEE 6 bus system, IEEE 14 bus system, IEEE 30 bus system, and IEEE 118 bus system.
18. The dynamic economic dispatch system as claimed in claim 15, wherein the power generating units are selected from an Alternating Current and/or Direct Current power generating unit.
19. The dynamic economic dispatch system as claimed in claim 18, wherein the Alternating Current power generating units (101) are selected from thermal power generation unit, nuclear power generation unit, hydroelectric power generating unit, wind power generating unit, biogas power generating unit, tidal power generating unit and/or solar power generating unit.
20. The dynamic economic dispatch system as claimed in claim 18, wherein the Direct Current power generating units are selected from lead acid battery, fuel cells.
,TagSPECI:A SYSTEM WITH MULTIPLE TRANSMISSION LOSS COEFFICIENTS FOR DYNAMIC ECONOMIC GENERATOR DISPATCH

Field of invention

The disclosure relates to power management and load scheduling of plurality of power generation units (101).Particularly, the disclosure is directed to a method and a system for economically allocating the output among plurality of power generation units (101), to a given load in the most economical manner. More particularly, this invention relates to a new and novel means for computing optimal generation of power among plurality of power generation units(101) through the determination of transmission loss in transmission lines (105) in addition to load demand, so as to establish a minimum incremental cost of power delivered from each unit to the load center (104)and maximum economical operation of the plurality of power generation units (101).

Background of invention

A power system can be primarily divided into four systems namely generation, transmission, distribution, and load. A schematic of a typical power system is given in fig1.

In a power system, the power is generated at generating station and consumed at load centres. The transmission lines connect the generating stations and load. These transmission lines run through several thousand kilometres. In substations, various activities like stepping up/down the voltages will be carried out.

The generating stations are situated in various places in the power system and the load centres are also widely distributed in a power system network (fig 2). The load on a power system varies with time, as we can operate our electrical equipment whenever we need it.

The amount of power generated by the power generating companies should match the load requirement at any point of time. If power generation is less than the load, then some parts of load need to be disconnected. If the generating power is more than required power, then load will get power. However, the surplus energy generated will be wasted. So, the amount of generation should be equal to the load and losses, which occur when power is transferred through transmission lines. The block diagram of a typical power system is given in fig1.

The cost of generation of electric power in various generating units will be different. So, all the generators in the power system network cannot share the load equally. In a power system network, the power output of various generating units are decided in such a way that the total cost of power generation is minimized and is called as Economic dispatch.Economic dispatch (ED) problem involves scheduling optimal generation of committed units to minimize the total operating cost of the power system, subject to various operating constraints like ramp rate constraints, power balance constraint, generator operating limit constraint, etc.Economic dispatch is carried out for a standard time interval(Say 1 Hour). If, economic dispatch is carried out for a period of more than one time interval and ramp rate limits of the generators are taken into account then, it is termed as Dynamic Economic Dispatch(DED). Thus, DED is an extension of ED in which ramp rate limits of generating units are taken into account. The change of power output of each generating unit from one hour to the next hour is limited by ramp rate limits.

DED schedule is carried out in Dispatch centers(fig3). The load dispatch center carry out various activities like DED, Unit commitment, Contingency analysis, Load forecasting. Day ahead load forecasting is carried out in load dispatch centers. According to the forecasted load for the next day, the DED is carried out and the schedule is sent to the generating stations or the sub dispatch center. DED schedule has the amount of power generated by each generating station at each period. One period of schedule will range from 15 minutes to 1 hour. The schedule undergoes various revisions as per the response from the generating stations (fig 4).

The dispatch is executed on the next day of scheduling. There will be load following action which checks the load at each minute and changes the generation. Thus, Load Frequency Control (LFC) unit regulates generators. The LFC unit will monitor the change in frequency at the generator end and vary the output of generator accordingly. Various functions like voltage regulations will be carried out in the substations. The present invention is implemented in the dispatch center in solving DED.

According to the schedule the commands to regulate the generating units are sent to generating stations. While solving a DED problem, transmission network losses are also taken into account. If the schedule is carried out using existing loss evaluation methods, then the schedule will be inaccurate. This will force the generating units to change the amount of power generated, which will add extra cost to the generating units. Different modes of transmission loss calculation are known in literature.

M.Y. Aktaret al., Proceedings of the IEE, Jan. 2010, pp.533-538, presented a method of economic dispatch by calculating losses with precise incremental loss calculation. In their paper transmission loss is modelled by employing principle of movements and removal of loops in a power system network. Imran ziariet al., Proceedings of the IUPEC, Sep. 2006,pp.393-397, presented a method of modelling transmission losses as a second order polynomial equation. B. H. Choudary, Proceedings, IEEE Region 5 Conference, Spanning the Peaks of Electrotechnology, 1988, pp.50-54, proposed a rule based DED in which 16 rules to raise or lower each generating unit is implemented. In US 7454270 and US7321810, DED problem is solved using ANN based Very Short Term Load Prediction module by Lagrange dual multiplier problem solving approach.

The literature shows that B loss coefficients are widely used to evaluate losses in a DED problem. However, evaluation of B loss coefficients is based on various assumptions. Losses calculated by using B loss coefficients are inexact. So, a DED involving B-loss coefficients may not be valid.

In a DED problem, loss can be calculated using load flow solution at each iteration. However, running load flow solution at each iteration is a tedious and time-consuming process. This method will not be suitable for estimation of loss coefficients and the existing method of evaluating losses using DED is not accurate. Thus, there is a need for a method, which calculates the loss accurately with respect to the load in a simple manner within short time.

Summary

The present invention relates to method and system of power management and load scheduling of plurality of power generation units. Specifically, a method and a system is disclosed for economically allocating the output among plurality of power generation units, to a given load so that over a set of predetermined time intervals the integrated cost associated with power generation and output is minimized. A method and system for scheduling optimal power generation of committed power units and to minimize the cost of generation of power, through the determination of transmission loss using multiple transmission loss coefficients and taking into consideration the power balance constraint, generator operating limit constraint and ramp rate limit constraints is disclosed.

Brief description of the drawing

The drawings are necessarily not to scale. The emphasis is placed upon illustrating the principles of the invention. A wide variety of potential embodiments will be more readily understood through the following detailed description, in which like reference numerals generally refer to the same parts throughout the detailed description and accompanying drawings in which,

Figure 1: Schematic block diagram of a power system
Figure 2: Single line diagram of a power system
Figure 3: Block diagram of a power system showing dispatch center
Figure 4: Implementation of Dynamic Economic Dispatch in Power system
Figure 5: Selective use of A/B loss coefficients
Figure 6: Estimation of A loss coefficients using perturbation technique with
slack bus power correction.
Figure 7: Forecasted load patterns
Figure 8: (a) Error in losses
(b) Classification of regions of loss coefficients
Figure9: Comparison of DED with A,B loss coefficients and proposed method
(AB).

Definitions

The terms “including” and “in which” are used as the plain-English equivalents of the respective terms “comprising” and “wherein”, respectively. Moreover, the terms “first”, “second” and “third”, etc. are used merely as labels, and are not intended to impose numerical requirements on their objects.

When the following terms are used herein, the accompanying definitions apply:

A loss coefficients:The A loss coefficients are constants, which are used to express the transmission power loss in terms of power generated. This is mathematically expressed as follows
P_L=[?_(i=1)^ng¦?A_i P_i ?]^2
ng= number of generating units
PL=Transmission loss
Pi= Power generated by ith generating unit
Ai=loss coefficient

B loss coefficients:The B loss coefficients are constants, which are used to express the transmission power loss in terms of power generated.
This is mathematically expressed as follows,

P_L=?_(i=1)^ng ?_(j=1)^ng¦P_i B_ij P_j+?_(j=1)^ng¦P_i B_i0+B_00
ng= number of generating units
PL= Power loss
Pi= Power generated by ith generating unit
Pj= Power generated by jth generating unit
Bij, Bi0, B00= B loss coefficients

Base load:
This is the amount of total load in the power system for which the real load and reactive load in all the power system network nodes(buses) are available.

Base generation:
The total power generated by the power system to supply base load.

Contingency analysis:
Contingency is the loss or failure of a small part of the power system (e.g. a transmission line), or the loss/failure of individual equipment such as a generator or transformer. This is also called an unplanned "outage".
Contingency analysis is a computer application that uses a simulated model of the power system, to evaluate the effects, and calculate any overloads, resulting from each outage event.
DBSCAN
DBSCAN is a clustering algorithm used to discover clusters with arbitrary shape. It classifies clusters as dense region of objects in data space that are separated by a region of low density. The algorithm grows "regions with sufficiently high density" into clusters and discovers "clusters of arbitrary shape".

Dispatch center (112)
Dispatch center is the apex body to ensure integrated operation of a power system. The functions of dispatch center (112) includes, optimum scheduling and dispatching of electricity, grid monitoring, maintain power system database (119),supervision and control of power system.

Dispatch constraint:
The dynamic economic dispatch problem is solved subject to various constraints. They are,
Power balance constraint.
Generator operation limit constraints.
Ramp rate constraints.
Economic dispatch
Economic dispatch will give us the amount of power generated by each generator in the system. The amount of power generated by each generator in the power system is the given by economic dispatch result.
An example economic dispatch result of three unit system with a load of 175.2 MW is given below.
PG1 PG2 PG3 PL
50.00 74.30 55.40 4.50

Electrical power generating unit: a device for generating electrical power.

Electrical power system: an interconnected collection of electrical power generating units, transmission equipment, and transmission lines (105).

Generator data
The characteristics of each generating unit are defined in generator data. The generator data is in following format.

Unit No Pmin Pmax a b C Ramp up limit (MW) Ramp down limit (MW)

Here,
Pmin- Minimum power generated by the power plant.
Pmax- Maximum power generated by the power plant.
a,b,c- Generator cost constants.
Ramp up limit: The maximum increase in output power the power plant can provide in a particular period(1 Hour) is called ramp up limit.
Ramp down limit: The maximum decrease in output power the power plant can provide in a particular period(1 Hour) is called ramp up limit.

Generator limits or Generator operating limit constraint
At any point of time the power output of a generator will be within a specified range (From Pmin to Pmax). This is Generator operating limit constraint.

Generator output:
Amount of power generated by the generator.

Gradient method: In optimization techniques, Gradient method is a series of steps to solve a minimization or maximization problem with the search direction defined by the gradient of a function at current solution point. This method is used to solve economic dispatch problem in our invention.

Load Frequency Control (LFC): A tool for controlling the alternating current frequency of power output by an electrical system and/or one or more components thereof.

Line data
Line data contains details of various parameters of transmission lines (105) in a power system. The format of line data is given below.

Start bus End bus Resistance
R (Pu) Inductance
X (Pu) Line charging Admittance
B (Pu) Transformation Ratio

Each transmission lines (105) connect two buses (Start bus and End bus).
Each transmission line (105) has its own resistance, Inductance and charging admittance. If the transmission lines (105) have a transformer connected to it, then transformation ratio of the transformer is also given in line data.

Predetermined power load data/ Predetermined schedule load demand:
A representation of expected demand for system power output over a predetermined future time period.

Forecasted load
The expected power system load, which may occur the next day (In future, next hour, next day, next week, next month) is called forecasted load.

Forecasted load Pattern:
The expected power system load, which may occur over a range of time is called forecasted load pattern.
For Example

Time 1 2 3 4 5
Load (MW) 100 110 120 90 80

{100,110,120,90,80}-Forecasted Load Pattern
100-Forecasted load, 110 Forecasted Load

Load
In a power system, the power is consumed by some equipments (All equipments using power) these equipments are collectively termed as load.

Loss coefficients/ Transmission loss coefficients:
The loss coefficients are constants, which are used to express the transmission power loss in terms of power generated.

Newton-Raphson method:
In numerical analysis, Newton's method (also known as the Newton–Raphson method) is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. Newton Raphson (NR) method is used for solving optimization problems, by using iterative successive approximation technique. NR method is used for finding load flow solutions (ie Parameters like Real Power, Reactive power, Voltage, voltage angle, Currents and power losses) in a power system network.

Operating constraint:
A limitation on power output for an electrical system and/or one or more components thereof.

Perturbation technique: A small change in a physical system.

Power balance constraint
The amount of power generated in the power system should be equal to the sum of load and losses. This is called power balance constraint.

Power flow solution/ Load flow/ Load flow analysis:
Load flow studies involves the solution of the power system network under steady state conditions. From the load flow solution, various power system parameters such voltage and angle at each bus, power flow, and the power system loss can be achieved.

Power generating units(Power plants)
Electrical power is produced in a power generating unit. Various types of power generating units are Hydel power plant, Thermal power plant, Atomic power plant, Solar power plant, wind power plant etc.,

Power loss:
The power which is dissipated without performing any useful work. Power loss in a power system is expressed in terms of power generated in every generating unit.

PL= f(Pg1,Pg2,Pg3.......)

where, Pgi= Power generated by ith generating unit.

Power output: Amount of power generated by the generator.

Ramp rate limits: Maximum rate at which power output from an electrical power-generating unit can be increased and/or decreased.

Ramping constraints/ Ramp rate limit constraint: A limit on a rate at which power output from an electrical power generating unit can be increased and/or decreased.

Ramping: Changing the load level of a generator in a constant manner over a fixed time (eg. Ramping up or down)

Remote control unit:
A computer or a processor (113) in power plant, which controls the generators in the power plant.

Slack Bus Power Correction:
The process of minimizing the slack power mismatch up to a tolerance value is called Slack Bus Power Correction. The slack power mismatch (SPM) is the absolute difference between slack power generation obtained by economic dispatch and also from load flow solution.

Slack bus (110):
The power system consists of two types of buses namely, Load bus(PQ), and Generator bus(PV) (The bus where generator are present). One generator bus is assumed as slack bus (110). The slack bus (110) makes up the difference between scheduled load and generated power caused by the losses in the power system.

Scheduling/schedule
Scheduling is the process of allocation of electrical output of each generator in the system over the time. An example schedule is shown below. This schedule is prepared by solving the DED problem for a time horizon of 3 Hour.

Hour Load PG1 PG2 PG3 PL
1.00 175 50.00 74.30 55.40 4.50
2.00 165 50.00 69.51 49.65 3.97
3.00 159 50.00 66.43 45.96 3.66

PG1- Power generated by generator no 1. PL- Power losses.

Slack bus power :
The amount of power generated by slack bus (110).

Slack bus power mismatch(SPM): The slack power mismatch is the absolute difference between slack power generation obtained by economic dispatch (Gradient method) and also from load flow solution (NR method)

Tolerance value(Tol)
The amount of power generated in the power system should be equal to the sum of load and losses. This is called power balance constraint. The difference between the amount of power generation in the power system and the sum of load and losses should be less than a small value. This small value is termed as tolerance.

Transmission lines (105):
Transmission lines (105) are wires, which carry electrical current from one place to other.

Transmission loss:
When electrical power is transmitted from one place to other, power loss occurs. This is called transmission loss.

Transmission network:
A transmission network transmits power from one(or more) place to the other. Here, more number of transmission lines (105)is present.

Unit commitment:
The Unit Commitment (UC) problem in a power system involves determining the start-up and shut down schedules of thermal units to be used to meet forecasted demand over a future short term (24-168 hour) period. The objective is to minimize total production costs while observing a large set of operating constraints.

White Gaussian Noise (WGN):
White Gaussian Noise is a basic noise model to mimic the effect of random processes that occur in nature. In this invention, WGN are added to the forecasted load pattern to make the power system more realistic.

Objectives of the invention:

The objective of the disclosure is to provide a method and system for scheduling optimal power generation of committed power units and to minimize the cost of generation of power, through the determination of transmission loss, subject to power balance constraint, generator operating limit constraint and ramp rate limit constraints.

One of the objectives of the disclosure is to provide a system for scheduling optimal power generation of committed power units, subject to power balance constraint, generator operating limit constraint and ramp rate limit constraints.

Yet another objective of the disclosure relates to a new and novel means for computing and controlling the generation of power among plurality of power generation units (101), so as to establish a minimum incremental cost of power delivered from each unit to the load center (104)and maximum economical operation of the plurality of power generation units (101).

Another objective is to formulate a Dynamic Economic Dispatch to schedule optimal generation of committed units to minimize the cost of generation, subject to power balance constraint, generator operating limit constraint and ramp rate limit constraints, where, the fuel cost for each unit at each hour is given by,
f_i (P_(i,t) )=a_i+b_i P_(i,t)+c_i ?P_(i,t)?^2 (1)
And objective function of the DED problem is formulated as:
Minimize??F_T ?=?_(t=1)^T ?_(i=1)^ng¦?f_i (P_(i,t) ) ? (2)
Subject to
1) Power balance constraint.
?_(i=1)^ng¦?P_(i,t)=P_Dt+P_Lt ? (3)
2) Generator operation limit constraint.
P_(min,i)=P_i=P_(max,i)for i=1,2....n. (4)
3) Ramp rate constraints.
P_(min,i,t)=max?{P_(min,i),P_(i,t-1)-Ud_i } (5)
P_(max,i,t)=min?{P_(max,i),P_(i,t-1)-Up_i } (6)

Ramp rate constraint limits the change in power generated by generating unit within permissible values, which will ensure safety of the equipments. The ramp rate constraints are taken care by updating the values of generator limit constraints in each hour by using equation (5) and equation (6).
Yet another objective is to evaluate transmission losses in a power system network using equation (7) or equation (8).
B transmission loss coefficients
P_L=?_(i=1)^ng ?_(j=1)^ng¦P_i B_ij P_j+?_(j=1)^ng¦P_i B_i0+B_00 (7)
A transmission loss coefficients
P_L=[?_(i=1)^ng¦?A_i P_i ?]^2 (8)

One another objective is to reduce the average percentage error in computation of loss when A and B transmission loss coefficients are selectively used.

Yet another objective, the dynamic variation of the generator output will be reduced.

One another objective is to satisfy both the losses and the electricity load ofthe generator, so that the generators will operate as per schedule.

Another objective is to reduce the cost of power generation.

One of the objectives is to apply the DED solution to transmission loss coefficient calculation irrespective of any distance or area.

Yet another objective is to do Performance analysis by employing DBSCAN on DED results of various test systems(IEEE 6 bus system, IEEE 14 bus system, IEEE 30 bus system, and IEEE 118 bus system) and load patterns.

One more objective is the DED solution can be applied to DC power generation and transmission sectors apart from AC power generation.

Still another objective is to apply the DED solution to power grids or industries with in-house power generation units(101) if they are injecting power to the transmission grid or have more than one generating unit and, the distance between load and the generating unit is higher.

Detailed description

In the following detailed description, reference is made to the accompanying drawings that show, by way of illustration, specific embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. It is to be understood that the various embodiments of the invention, although different, are not necessarily mutually exclusive. Furthermore, a particular feature, structure or characteristic described herein in connection with one embodiment may be implemented within other embodiments without departing from the scope of the invention. In addition, it is to be understood that the location or arrangement of individual elements within each disclosed embodiment may be modified without departing from the scope of the invention. The following description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims, appropriately interpreted, along with the full range of equivalents to which the claims are entitled.

Embodiments of the invention include features, methods or process embodied within machine-executable instructions provided by a machine-readable medium. A machine-readable medium includes any mechanism which provides (ie., stores and/or transmits) information in a form accessible by a machine (e.g., a computer, a network device, a personal digital assistant, manufacturing tool , any devise with a set of one or more processors, etc.). In an exemplary embodiments, a machine-readable medium includes volatile and/or non-volatile media (eg., for example but not limited to, an electronic, magnetic, optical, electromagnetic, a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), a portable computer diskette (magnetic), an erasable programmable read-only memory (EPROM or Flash memory) (electronic), and a portable compact disc read-only memory (CDROM) (optical), or semiconductor system, apparatus, or device as well as electrical, optical, acoustical or other form of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.)

Such instructions are utilized to cause a general or special purpose processor, programmed with the instructions, to perform methods or processes of the embodiments of the invention. Alternatively, the features or operations of embodiments of the invention are performed by specific hardware components. Embodiments of the invention include digital/analog signal processing systems, software, data processing hardware, data processing system-implemented methods, and various processing operations, further described herein.

A number of figures show block diagrams of systems and apparatus of embodiments of the invention. A number of figures show flow diagrams illustrating systems and apparatus for such embodiments, the operations of the flow diagrams will be described with references to the systems/apparatuses shown in the block diagrams. However, it should be understood that the operations of the flow diagrams could be performed by embodiments of systems and apparatus other than those discussed with reference to the block diagrams, and embodiments discussed with reference to the systems/ apparatus could perform operations different than those discussed with reference to the flow diagrams.

A schematic diagram of a typical power system is given in fig1. In a power system, power generation (101) is carried in different modes like thermal, fossil, hydro, nuclear, gas, etc., in power generating stations (101), which are situated in various places. The generated power is then transmitted (102) through the transmission lines (105) connecting the generating stations (101). In substations (103), various activities like stepping up/down the voltages will be carried out and distributed (103) and consumed at load centres (104). In the power system, the load centres (104) are also widely distributed and consist of very large, large, medium and residential customers.

The Single line diagram of a power system (fig 2) represents a typical power system network consisting of power generating units (101) which are connected by power generating bus systems (106) which in turn are connected to step up transformers (107) for transmission (102) of the generated power. The transformers (107) are connected to EVH bus (109) and breaker (109a) through EVH transmission lines (105) and again stepped down using transformers (107) for transmission to different load centres through distribution bus (108). The distribution feeders (108a) finally distribute the generated power to the load (104).

Block diagram of a power system showing Dispatch center (112) is given in fig 3. DED schedule is carried out in Dispatch centers (112). The power generated in power generating units (101) is transmitted through slack bus (110) or a generating bus (106). They are then stepped up or down using transformers (107) and transmitted through load bus (111) to load centres (104).

In another embodiment, the Implementation of Dynamic Economic Dispatch in Power system is as given in fig 4, where the load dispatch center carry out various activities like DED, Unit commitment, Contingency analysis, Load forecasting (115). Day ahead load forecasting (115) is carried out in load dispatch centers (112) using the historic data from the database (119), different processors (113) and dynamic economic dispatch program (114). These data are then transmitted to remote generation unit (101) through Internet via different communication ports (116) like USB, Ethernet, ports and router (117) or to SCADA (118).
In yet another embodiment, the Dynamic Economic Dispatch is elaborated in fig 5 through selective use of transmission loss coefficients. The load forecasting (115) is done on hourly load basis for the next day by reading the data (120) from system, generator, line and buses. Then A loss coefficient is calculated (121) using perturbation and slack bus power correction methods. Then B loss coefficient (122) is calculated using slack bus power. Ramp rate limits (123) are updated and the load for the tth hour (124) is obtained. Using the constraints (126) decision is made and the iteration is continued by increasing the hour till dispatch horizon (125) is satisfied. The dispatch schedule (127) is created based on the decision using the above constraints using A or B loss coefficients.
The values of A loss coefficients are calculated from power flow solution (128) obtained for initial operating state by using the perturbation technique. The A loss coefficients of the power system is calculated using the perturbation technique with slack bus power correction (129) as given in fig 6. As show in figure, the load flow analysis is carried out using which the transmission loss is determined at base load flow conditions (130) followed by varying generation of power in the corresponding generating unit (131, 101). This process of analysing the load flow solution and determining the power generated in the generating unit taking into account the transmission loss is repeated enabling economic dispatch of power (132). Based on this data the transmission loss coefficient is calculated using the equation (14). Economic dispatch is carried out by the calculated A-loss coefficients using gradient method (133).The absolute difference between scheduled slack bus power generation obtained by economic dispatch and slack bus power generation obtained from power flow solution is termed as slack power mismatch (134). The economic dispatch result is updated in bus data by replacing the real power generated by the PV busses with the real power calculated by economic dispatch. The above process is iterated until slack power mismatch is within a specified tolerance value (135).
In one of the embodiments of the present disclosure, a method to manage dynamic economic dispatch of power comprising, receiving a predetermined schedule load demand for power dispatch; receiving ramp rate constraints, power balance constraints and generator operating limit constraints of plurality of connected power generating units in an electrical power network; receiving predetermined power load data from plurality of buses, plurality of transmission lines (105) and plurality of power generating units for obtaining power flow solution using at least one numerical iterative approach; obtaining data for economic dispatch adopting from plurality of numerical techniques; calculating at least two transmission loss coefficients using at least one numerical technique; determining slack bus power using the power flow solution; obtaining slack bus power mismatch; determining the updated economic dispatch data when the obtained slack power mismatch is lesser than tolerance value; determining the power load for a consecutive defined time interval for at least 24 hour; obtaining the economic dispatch by selecting at least one of the transmission loss coefficients employing a dispatch constraint; updating generator limits at consecutive defined time interval for at least 24 hour to a remote control unit for allocating output to one of the plurality of generating units wherein the respective allocation are configured for minimizing the costs associated with each of the plurality of generating units when producing power according to the respective allocations of output power; and regulating plurality of the power generating units by updating the load demand data in at least one processor (113) for scheduling and dispatching.

In another embodiment, a dynamic economic dispatch system to manage optimal power dispatch and power generation comprising at least one processor (113) for determining at least two transmission loss coefficients using at least one numerical technique and regulating plurality of power generating units by updating load demand data for minimizing the costs associated with each of the plurality of generating units.

In one more embodiment, the remote terminal unit is a compatible programming module measuring electrical and mechanical parameters.

Another embodiment of the present disclosure is a remote terminal unit operative for allocating the output power to one of the plurality of generating units.

In yet another embodiment, the at least one processor (113) operative for determining the transmission loss coefficients shows a percentage error in the range of 1 to 1.9, preferably 1.4 to 1.7.

In another embodiment of the disclosure at least one numerical iterative approach for determining power flow solution is selected from Guass Siedel method, Newton- Raphson method, and/or fast decouple method, preferably Newton-Raphson approach.

In yet another embodiment, the at least one numerical technique to determine one of the at least two transmission loss coefficients is selected by using standard formula and perturbation technique and/or slack bus power correction technique, preferably perturbation technique.

Still one more embodiment of the present disclosure is, the at least one numerical technique to determine at least one of the two transmission loss coefficients is selected from loss coefficients with correction techniques such as penalty factor method and or slack bus power correction technique, preferably slack bus power correction technique.

Yet another embodiment, the plurality of numerical techniques to determine the economic dispatch is selected from gradient method, lambda iteration method, newtons method, evolutionary algorithms, genetic algorithm, neural networks, simulated annealing, particle swarm optimization, bacterial foraging algorithm, artificial honey bee algorithm, and firefly algorithm, preferably gradient method.

In one more embodiment, the at least one processor (113) determines the slack power mismatch by finding the difference between slack bus power predetermined by power flow solution and slack bus power predetermined by economic dispatch.

In another embodiment of the present disclosure, the predetermined schedule load demand is obtained from historical data or forecasted data.

In one of the embodiments, the power generating units are selected from an Alternating Current generating units (101) such as thermal power generation unit, nuclear power generation units, hydroelectric power generating unit, solar power generating unit and/or Direct Current power generating unit such as lead acid battery, fuel cells.
In one more embodiment, one of the at least two transmission loss coefficient is determined according to a formula

P_L=[?_(i=1)^ng¦?A_i P_i ?]^2

where
P_Lis transmission loss
A_iis one of the at least two transmission loss coefficients for ith generating
unit (A transmission loss coefficient)
P_iis the power generated by ith generating unit in MW
ngis number of generating units

In yet another embodiment, one of the at least two transmission loss coefficient is determined according to a formula
P_L=?_(i=1)^ng ?_(j=1)^ng¦P_i B_ij P_j+?_(j=1)^ng¦P_i B_i0+B_00
where,
P_Lis transmission loss
ngis number of generating units
P_iis the power generated by ith generating unit in MW
P_jis the power generated by jth generating unit in MW
B_ij,B_(i0,) B_00areB transmission loss coefficient.

In one of the embodiments, the dispatch constraint is to selectively use B transmission loss coefficient for allocating economic dispatch if load is in the range 0 to 0.9 per unit and use A transmission loss coefficient for allocating economic dispatch if load is in the range 0.9 to 1.4 per unit.

Yet another embodiment, the plurality of buses is selected from any electrical power network such as IEEE 6 bus system, IEEE 14 bus system, IEEE 30 bus system, and IEEE 118 bus system.

In still one more embodiment, the tolerance value is in the range of 10-3.

In another embodiment, the cost associated with each of the plurality of generating units is calculated according to the formula (2).

Example 1:Calculation of one of the two transmission loss coefficient (A transmission loss coefficient) using perturbation and slack bus power correction
The A transmission loss coefficient of a power system is calculated by using perturbation technique. In this procedure, the load flow analysis is carried out repeatedly by varying the generation in each generator(101). The load flow analysis is carried out for the base generation. The value of generation in each generator (101) is set as given in the bus-data and the load flow is carried out. This is called base load flow solution. The corresponding losses for the base load flow solution are noted. Then, the value of power generated by each generator (from generator 1 to ng) is varied and corresponding transmission loss is noted by running the load flow.
The A coefficients are evaluated from base load flow solution using perturbation technique. The number of A coefficients needed to evaluate a system with 'ng' generating units is 'ng'.
The method used to calculate A loss coefficients using perturbation technique is explained briefly below.
Let there be three generating units in the system and bus1 be the slack bus (110). From base load flow solution, find power in all generator buses P10, P20,P30 and transmission loss PL0. The superscript '0' represents base load flow solution. Equation (8) can be expanded as follows,
v(?P_L?^0 )=?[A_1 ?P_1?^0+A_2 ?P_2?^0+A_3 ?P_3?^0 ] (9)?^
If a small perturbation is applied at second generating bus(106) and load flow study is carried out, then the power at buses 1, 2, 3 and transmission losses in the power system are P12, P22,P32 and PL2 respectively. The superscript '2' represents load flow solution is achieved by applying perturbation at second generating bus (106).
v(?P_L?^2 )=?[A_1 ?P_1?^2+A_2 ?P_2?^2+A_3 ?P_3?^2 ] (10)?^
The change in power generated in bus number two result in change in power generated in slack bus (110) and change in transmission losses. The power at all buses except slack bus (110) and bus at which perturbation is applied will remain same as that of base load flow solution. If the amount of perturbation applied at second generating bus (106) is ?P_2 and the resulting change in slack bus generation is ?P_12, then the equation (10) can be modified as follows,
v(?P_L?^2 )=?[A_1 ??(P?_1?^0-?P_12)+A_2 (?P_2?^0+?P_2)+A_3 ?P_3?^0 ] (11)?^
By applying perturbation at third generating bus (106), the following equation is achieved.
v(?P_L?^3 )=?[A_1 ??(P?_1?^0-?P_13)+A_2 ?P_2?^0+A_3 (?P_3?^0+?P_3)] (12)?^
The equations (10), (11), and(12) can be represented in matrix form as follows,
[¦(v(?P_L?^0 )@@v(?P_L?^2 )@@v(?P_L?^3 ))]=[¦(?P_1?^0&?P_2?^0&?P_3?^0@&&@&&@??(P?_1?^0-?P_12)&(?P_2?^0+?P_2 )&?P_3?^0@&&@&&@??(P?_1?^0-?P_13)&?P_2?^0&(?P_3?^0+?P_3 ) )][¦(A_1@@@A_2@@@A_3 )] (13)
The above equation can be generalised for 'ng' number of generating buses as follows,
[¦(v(?P_L?^0 )@.@.@v(?P_L?^i )@ .@.@v(?P_L?^n ))]=[¦(?P_1?^0&?P_i?^0&?P_3?^0@.&.&.@.& .&.@&&@??(P?_1?^0-?P_1i)&(?P_i?^0+?P_i)&?P_n?^0@.&.& .@.&.&.@??(P?_1?^0-?P_1n)&?P_i?^0&(?P_n?^0+?P_n))][¦(A_1@.@.@@A_i@.@.@A_ng )](14)
The values of A loss coefficients can be obtained by solving the above equation.

Example 2:Determination of slack power mismatch

The process of estimation of A loss coefficients with slack bus power correction is illustrated in fig 6. The values of A loss coefficients are calculated from power flow solution obtained for initial operating state by using the perturbation technique. Economic dispatch is carried out by the calculated A-loss coefficients using gradient method. The absolute difference between scheduled slack bus power generation obtained by economic dispatch and slack bus power generation obtained from power flow solution is termed as slack power mismatch. The economic dispatch result is updated in bus-data by replacing the real power generated by the PV busses with the real power calculated by economic dispatch. The above process is iterated until slack power mismatch is within a specified tolerance value of 0 to±10-3 per unit.

Example 3:Classification Of Region Of Loss Coefficients

In order to quantify the region of the loss coefficients, the dynamic economic dispatch is carried out for various load patterns. The load patterns are generated by the following strategies,

(i) Adding White Gaussian Noise (WGN) over the standard hourly load pattern.
(ii) Scaling the standard load pattern by a multiplication factor.
(iii) Adding a constant value to the standard load pattern.
(iv) Randomly shuffling the hourly loads in each load pattern and modifying the load
pattern to satisfy ramp rate limits.
(v) Arbitrarily generating the load patterns and modifying to satisfy ramp rate limits.
The standard load pattern of the IEEE 30 bus system is given in Table-I. DED is carried out for all the generated load patterns. The loss obtained by using the A loss coefficients(PLA) and B loss coefficient(PLB) with reference to load flow solution(PLF) at each per unit load is extracted from the results of numerous dynamic economic dispatch results. The error in calculation of losses when A loss coefficients are used is given in equation (18).

E_A=absolute(P_LF-P_LA ) (15)

The error in calculation of losses when B loss coefficients are used is given in equation (19).

E_B=absolute(P_LF-P_LB ) (16)

The relative error in calculation of losses while using A and B loss coefficients is given by equation (20)
E_R=E_B-E_A (17)

The relative error in losses is plotted in fig 8(a). The cluster analysis is carried out using the relative error and per unit load. The cluster analysis revealed that the A loss coefficients perform better than B loss coefficients when the per unit loadis between 0.9 and 1.4per unit and vice versa. The cluster analysis result showing the region of loss coefficient is given in fig8(b).

Example 4:

While solving ED, initially a value of lambda is assumed and the value of power generated by each generator is calculated by (21).
?P_(i,t)?^k=(?^((k) )-b_i-2?^((k) ) ?_(j?i)¦A_j ?P_(j,t)?^k)/(2(c_i+?^((k) ) A_i)) (18)
Where,
?P_(i,t)?^k is the value of power generated by ith generator at tth hour in kth iteration.
The power mismatch ??P?^((k)) is calculated using power generated by all generator in the current iteration and the value of transmission loss calculated by using A loss coefficients.
??P?^((k))=P_Dt+P_Lt-?_(i=1)^n¦?P_(i,t)?^k (19)
The value of change in lambda is calculated by,
??^((k) )=??P?^((k))/(?_(i=1)^ng¦[dP_(i,t)/d?]^((k)) ) (20)

?_(i=1)^ng¦?[dP_(i,t)/d?]^((k))=? ?_(i=1)^ng¦[(c_(i )+A_i b_i-2c_i ?_(j?i)¦?A_j ?P_(j,t)?^((k)) ?)/?2(c_i+?^((k) ) A_i)?^2 ] (21)

?^((k+1) )=?^((k) )+??^((k) ) (22)
The value of lambda is updated by above formula and the process is iterated till the power mismatch is less than the specified accuracy.

Example 5:

IEEE 30 bus test system is used for analysis. DED is solved for a 24 hour horizon. Table I shows the DED results obtained by using A loss coefficients. The A loss coefficients are calculated at a load of 283.4 MW, which is the base load for IEEE 30 bus system.

Example 6:

The B loss coefficients are calculated at the base load (283.4 MW). Table II shows the DED results obtained using B loss coefficients.

Example 7:

Table III represents the loss reported by the A loss coefficients(PLA), B loss coefficient(PLB), and load flow solution(PLF) at the particular operating points mentioned in the schedule in table I and II. The absolute error in calculating losses is also shown in the table III.

In order to quantify the per unit values of the load at which the A loss coefficients outperforms B loss coefficients, 13 load pattern are generated and DED is solved for all the load patterns. The analysis mentioned above is carried out in all the load patterns. Load pattern 1 is the standard load pattern which is given in Table I. Load patterns 2 to 6 are generated by shuffling load elements of load pattern1 and removing the pattern for which DED is not feasible. Load patterns 7 to 9 is obtained by adding a Gaussian noise of 5%, 10%, and 15% standard deviation to load pattern1. Load pattern 10 to 13 is achieved by adding a constant value of 100 MW to load pattern1 along with Gaussian noise of 5,10 and 15% standard deviation. The load patterns are shown in fig 7. DED is carried out for all the load patterns and losses reported by A loss coefficients, B loss coefficients and the load flow analysis is obtained for each operating point in the schedule. Each schedule is carried out for 24 hour horizon resulting in 312 operating points.

The error in losses reported by coefficients are plotted against their load in per unit value considering 283.4 MW as 1 per unit in fig 8(a). The region of loss coefficients for the corresponding per unit load is shown in fig 8(b). From fig 8(b), it is inferred that A loss coefficients perform better than B loss coefficients when the load is between 0.9 to 1.4 per unit. Therefore, in the invention the A loss coefficients are used in evaluating losses when the load is between 0.9 to 1.4 per unit. B loss coefficients are used when the load is between 0-0.9 per unit.

The average percentage error in calculation of loss with A loss coefficients, B loss coefficients, and selectively using A or B in DED is 6.2%, 2.25 % and 2.19%. The fuel cost achieved and average percentage error in DED by above stated methods is shown in fig 9. The average percentage error in computation of loss is minimal when A and B coefficients are selectively used. The fuel cost achieved by proposed method is also less than the cost achieved using conventional B coefficients. In table IV the DED result by using A/B loss coefficients selectively is given.

Example 8:

Busdata: Busdata represents the parameters of each bus in the power system. Each row represents the parameters of a bus. The parameters specified for each bus are Bus number, Bus code, Bus voltage in per unit, Bus angle in degrees, Real power demand in MW, Reactive demand in MVAR, Real power generation in MW, Reactive power generation in MVAR, minimum reactive power that should be injected at the bus and maximum reactive power that can be injected at the bus. Bus code represents the type of bus(0 represents Load(PQ) bus, 1represents Slack bus (110) and 2represents Generator(PV) bus).

Line data: The line data represents the parameters for each transmission lines (105) in the power system. Each row represents the parameter of transmission lines (105). The parameters specified for each transmission lines (105) are buses between which the transmission lines (105) is connected, resistance(R) of the transmission lines (105) in per unit, reactance(X) of the transmission lines (105) in per unit, Half line charging susceptance (1/2 B) and turns ratio. The transmission lines (105) is connected between two buses. Here, one bus is assumed as from bus and the other bus is assumed as to bus. Turns ratio represents the turns ratio of the transformer connected in the transmission lines (105). However, if transmission lines (105) don’t contain a transformer, then the turns ratio is assumed as 1.

Generator Data:
Generator data represents the parameters of each generating unit in the system. Each row represents the parameter of a generating unit. The parameters specified for each generator are generator number, minimum real power that can be generated by the generating unit, maximum real power that can be generated, minimum reactive power that can be generated, maximum reactive power that can be generated, the cost coefficients(a, b, c), ramp up and ramp down limit of the generating unit.

Table Va: Bus Data

Bus No Bus
Code Bus
voltage
(p.u.) Angle Load Generation Injected Capacitance
(degrees) MW Mvar MW Mvar Qmin Qmax (p.u.)
(Mvar) (Mvar)
1 1 1.06 0 0 0 0 0 0 0 0
2 2 1.043 0 21.7 12.7 50 0 -20 60 0
3 0 1 0 2.4 1.2 0 0 0 0 0
4 0 1.06 0 7.6 1.6 0 0 0 0 0
5 2 1.01 0 94.2 19 33 0 -15 62.5 0
6 0 1 0 0 0 0 0 0 0 0
7 0 1 0 22.8 10.9 0 0 0 0 0
8 2 1.01 0 30 30 22 0 -15 48.7 0
9 0 1 0 0 0 0 0 0 0 0
10 0 1 0 5.8 2 0 0 0 0 0.19
11 2 1.082 0 0 0 20 0 -10 40 0
12 0 1 0 11.2 7.5 0 0 0 0 0
13 2 1.071 0 0 0 26 0 -15 44.7 0
14 0 1 0 6.2 1.6 0 0 0 0 0
15 0 1 0 8.2 2.5 0 0 0 0 0
16 0 1 0 3.5 1.8 0 0 0 0 0
17 0 1 0 9 5.8 0 0 0 0 0
18 0 1 0 3.2 0.9 0 0 0 0 0
19 0 1 0 9.5 3.4 0 0 0 0 0
20 0 1 0 2.2 0.7 0 0 0 0 0
21 0 1 0 17.5 11.2 0 0 0 0 0
22 0 1 0 0 0 0 0 0 0 0
23 0 1 0 3.2 1.6 0 0 0 0 0
24 0 1 0 8.7 6.7 0 0 0 0 0.04
25 0 1 0 0 0 0 0 0 0 0
26 0 1 0 3.5 2.3 0 0 0 0 0
27 0 1 0 0 0 0 0 0 0 0
28 0 1 0 0 0 0 0 0 0 0
29 0 1 0 2.4 0.9 0 0 0 0 0
30 0 1 0 10.6 1.9 0 0 0 0 0

Table Vb: Line Data

From Bus To Bus R (p.u) X (p.u) 1/2 B (p.u) Tap Ratio
1 2 0.0192 0.0575 0.0264 1.0000
1 3 0.0452 0.1852 0.0204 1.0000
2 4 0.0570 0.1737 0.0184 1.0000
3 4 0.0132 0.0379 0.0042 1.0000
2 5 0.0472 0.1983 0.0209 1.0000
2 6 0.0581 0.1763 0.0187 1.0000
4 6 0.0119 0.0414 0.0045 1.0000
5 7 0.0460 0.1160 0.0102 1.0000
6 7 0.0267 0.0820 0.0085 1.0000
6 8 0.0120 0.0420 0.0045 1.0000
6 9 0.0000 0.2080 0.0000 0.9780
6 10 0.0000 0.5560 0.0000 0.9690
9 11 0.0000 0.2080 0.0000 1.0000
9 10 0.0000 0.1100 0.0000 1.0000
4 12 0.0000 0.2560 0.0000 0.9320
12 13 0.0000 0.1400 0.0000 1.0000
12 14 0.1231 0.2559 0.0000 1.0000
12 15 0.0662 0.1304 0.0000 1.0000
12 16 0.0945 0.1987 0.0000 1.0000
14 15 0.2210 0.1997 0.0000 1.0000
16 17 0.0824 0.1923 0.0000 1.0000
15 18 0.1073 0.2185 0.0000 1.0000
18 19 0.0639 0.1292 0.0000 1.0000
19 20 0.0340 0.0680 0.0000 1.0000
10 20 0.0936 0.2090 0.0000 1.0000
10 17 0.0324 0.0845 0.0000 1.0000
10 21 0.0348 0.0749 0.0000 1.0000
10 22 0.0727 0.1499 0.0000 1.0000
21 22 0.0116 0.0236 0.0000 1.0000
15 23 0.1000 0.2020 0.0000 1.0000
22 24 0.1150 0.1790 0.0000 1.0000
23 24 0.1320 0.2700 0.0000 1.0000
24 25 0.1885 0.3292 0.0000 1.0000
25 26 0.2544 0.3800 0.0000 1.0000
25 27 0.1093 0.2087 0.0000 1.0000
28 27 0.0000 0.3960 0.0000 0.9680
27 29 0.2198 0.4153 0.0000 1.0000
27 30 0.3202 0.6027 0.0000 1.0000
29 30 0.2399 0.4533 0.0000 1.0000
8 28 0.0636 0.2000 0.0214 1.0000
6 28 0.0169 0.0599 0.0650 1.0000

Table Vc: Generator Data

Gen No Pmin (MW) Pmax
(MW) Qmin
(MVAR) Qmax
(MVAR) a
($) B
($/MW) C
($/MW2) Ramp up limit (MW) Ramp down limit (MW)
1 50 200 - - 0 0.00375 2.00 50 50
2 20 80 -20 100 0 0.01750 1.75 16 19
3 15 50 -15 80 0 0.06250 1.00 10 10
4 10 35 -10 60 0 0.00834 3.25 7 7
5 10 30 -10 50 0 0.02500 3.00 6 6
6 12 40 -15 60 0 0.02500 3.00 8 8

Still other embodiments will become readily apparent to those skilled in this art from reading the above-recited detailed description and drawings of certain exemplary embodiments. It should be understood that numerous variations, modifications, and additional embodiments are possible, and accordingly, all such variations, modifications, and embodiments are to be regarded as being within the spirit and scope of the appended claims. Moreover, any activity can be repeated, any activity can be performed by multiple entities, and/or the interrelationship of elements can vary.

Accordingly, the descriptions and drawings are to be regarded as illustrative in nature, and not as restrictive. Moreover, when any number or range is described herein, unless clearly stated otherwise, that number or range is approximate. When any range is described herein, unless clearly stated otherwise, that range includes all values therein and all sub-ranges therein. Any information in any material that has been incorporated by reference herein is only incorporated by reference to the extent that no conflict exists between such information and the other statements and drawings set forth herein.