Application of Interval Arithmetic in the Evaluation of Transfer Capabilities by Considering the Sources of Uncertainty

Total transfer capability TTC is an important index in a power system with large volume of inter-area power exchanges. This paper proposes a novel technique to determine the TTC and its confidence intervals in the system by considering the uncertainties in the load and line parameters. The optimal power flow OPF method is used to obtain the TTC. Variations in the load and line parameters are incorporated using the interval arithmetic IA method. The IEEE 30 bus test system is used to illustrate the proposed methodology. Various uncertainties in the line, load and both line and load are incorporated in the evaluation of total transfer capability. From the results, it is observed that the solutions obtained through the proposed method provide much wider information in terms of closed interval form which is more useful in ensuring secured operation of the interconnected system in the presence of uncertainties in load and line parameters.


Introduction
Due to the deregulation of the power industry, there has been an increasing interest in quantifying the transmission transfer capabilities of power systems.Interconnected systems are replacing isolated systems for economic and reliability reasons, but the resulting looped networks give rise to new technical concerns.Assessment of total transfer capability and available transfer capability ATC are such issues in a power system.ATC can be mathematically defined as the total transfer capability TTC minus the transmission reliability margin TRM , minus the sum of existing commitments and the capacity benefit margin CBM 1 .
In the past few years, a number of methods have been proposed for ATC determination.The literature of ATC calculation can be divided into two broad categories: deterministic methods and probabilistic methods.The deterministic methods can be further divided into the continuation power flow CPF method 2 , the repeated power flow RPF method 3 , and the optimal power flow OPF methods 4, 5 .Starting from a solved base case, the CPF method obtains a series of power flow solutions by increasing the transfer parameter by using a prediction-correction scheme.RPF method starts from a base case and repeatedly solves the power flow equations, each time increasing the power transfer by a small increment until it reaches an operation limit.OPF method formulates the TTC calculation as an optimization problem, with equality constraints of power flow and inequality constraints from operation limits.The objective function, obviously, is to maximize power flow on the specified transmission route.These methods seem to be appropriate and efficient in management of the transmission system.However, because it fails to consider the probabilistic nature of the power system, the obtained ATC may be too conservative and therefore lead to costly and inefficient use of system resources.
The quantitative uncertainty is quantifiable in nature and is represented in numerical terms by a mathematical function with deterministic parameters.This could be achieved either by probabilistic variables or by interval variables.In the former, the uncertainty is defined by probability density functions 6, 7 .The application of the interval Arithmetic method for power flow analysis of transmission networks was first proposed in 8 .This concept has been applied for radial distribution system analysis 9, 10 .The basic idea in this paper is to extend the application of interval arithmetic in order to consider the probabilistic aspects of the system input parameters in the calculation of total transfer capability.The problem is formulated as an optimization problem, where the objective function is to maximize the power transfer between a specific set of generator s and load s without violating any of the system constraints.
The remainder of this paper is organized as follows: in Section 2, the problem formulation of TTC is presented; the functional procedure of the interval arithmetic is discussed in Section 3 and calculation of base case power flow is briefly discussed in Section 4; the results for two important case studies on transfer capability are discussed in Section 5 and finally the conclusions are summarized in Section 6.

Problem Formulation for Total Transfer Capabilty
The mathematical formulation of TTC can be expressed as follows.
The objective function for the OPF reflects the maximum power transfer from one bus/area to another bus/area.
Objective function is The equality constraints of the OPF reflect the power system.The power system is enforced through the power flow equations which require that the net injection of real and reactive power at each bus is equal to zero.
Equality constraints are

2.2
The inequality constraints of the OPF reflect the limits created to ensure system security.The limiting constraints considered in this paper consist of three types: generation capacity, transmission line capacity, and voltage level.
The generators' maximum and minimum outputs for real and reactive powers are as follows: For the maintenance of system security, the transmission line MVA ratings are taken into account To maintain the quality of the electrical service and system security, bus voltage limits are considered where Δ P Di : is active power increment of load bus; P Gi , Q Gi : are real and reactive power generation at bus i; P Di , Q Di : are real and reactive load demand at bus i; n: is bus number of the system; V i , V j : are voltage magnitudes at bus i, j; S ij : is line MVA limit.

Interval Arithmetic Method
Interval arithmetic is a powerful tool to determine the effects of uncertain data.It can deal with numbers that vary within a range.The basic concepts of interval arithmetic are discussed as follows.
An interval number X x 1 , x 2 is the set of real numbers x such that x 1 ≤ x ≤ x 2 .Here x 1 and x 2 are known as the lower limit and upper limit of the interval number, respectively.Let X x 1 , x 2 and Y y 1 , y 2 be the two interval numbers.The basic arithmetic operations of addition, subtraction, multiplication, and division of these two interval numbers are defined as follows 11 :

3.1
However, for the purpose of power flow analysis, calculations are based on complex numbers rather than real numbers.The basic relations involving complex interval numbers are described as follows.In general, any complex number Z X iY -where i is the complex operator-is said to be a complex interval number if both its real and imaginary parts, X and Y , respectively, are interval numbers.Therefore X can be represented as X x 1 , x 2 and Y can be represented as Y y 1 , y 2 .Let Z 1 A 1 iB 1 and Z 2 A 2 iB 2 be the two complex interval numbers.The addition, subtraction, multiplication, and division of these two complex numbers are defined as follows: 3.3

Interval Power Flow Analysis
The power flow analysis method used in this work is fast-decoupled load flow algorithm.However, to account for the uncertainty of the input load parameters, the real and reactive power loads are treated as interval numbers rather than fixed numbers.Similarly, to account for the uncertainty of the input line parameters, the resistance and reactance of the line are treated as interval numbers.Consequently, the complex arithmetic has been replaced by complex interval arithmetic where, P k is real power load of kth bus; Q k is reactive power load of kth bus; V k is voltage magnitude at kth bus; R jj is resistance of the branch-jj; X jj is reactance of the branch-jj; P k and P k u are lower and upper limits of real power load of kth bus; Q k and Q k u are lower and upper limits of reactive power load of kth bus; R jj and R jj u are lower and upper limits of resistance of a particular branchjj; X jj and X jj u are lower and upper limits of reactance of a particular branch-jj.Find the condition just before violation and calculate the lower limit and upper limit of TTC as sum of loads in the sink area For any selected option display lower and upper limits of bus voltages and lower and upper limits of real and reactive power losses End

Uncertainty Parameters in Load and Line Variation Modeling
The system is assumed to operate under normal conditions but line and load parameters vary within a certain range.In this paper, ±10% variations in load parameters and ±3% for line parameters are considered from their rated nominal value R jj 0.97R jj , R k u 1.03R jj , X jj 0.97P jj , X k u 1.03X jj .

3.5
Various case studies have been conducted and the results of the following cases are presented: i the system under normal operating conditions, ii uncertainties due to load parameters only, iii uncertainties due to line parameters only, iv uncertainties due to both load and line parameters.
The basic steps used for computing the TTC using interval arithmetic for each transaction are given in Figure 1.

Basecase Power Flow Calculation
The IEEE 30 bus system shown in Figure 2 is adopted to illustrate the proposed method.The system is divided into 3 areas.The system has six generators, with two generators in each area.We assume each area as a utility.The utility in a certain area wants to import power from another area.Hence TTC is evaluated between areas.The base case load flow for fixed values of input is given in Table 1.

Base Case Power Flow Incorporating Various Uncertainties Using Interval Arithmetic
The transfer capability is computed from a base case constructed from the system information available at a given time.Usually, there is some uncertainty or inaccuracy in this computation because the transfer capability computed at the base case does not reflect evolving system conditions.These uncertainties include inaccurate or incorrect network parameters, effects neglected in the data and approximations in transfer capability computations.As already discussed in Section 3, the uncertainties in load and line parameters have been taken into account by assuming that they vary over a range.When the load demands in the system vary within some intervals, the bus voltages and other quantities such as line flows and line losses also vary within certain intervals.The intervals of variation of bus voltages for the system under consideration have been calculated for all the three uncertainties and are presented in Table 2.In this table, the symbols V and V u denote the lower limit and upper limit of the voltage, respectively.Figures 3 and 4 show the voltage profile for load uncertainty and line uncertainty for fixed, lower, and upper limit voltages.One can see that the load uncertainty has a considerable effect on the voltage profile compared with the line uncertainty.However, when  both load and line uncertainties are taken into account, the effect is much more and is reflected in the voltage profile shown in Figure 5.

Results and Discussions
Several cases on the transfer capability are studied and from them the results of two important case studies are presented and discussed below.

Case 1: The Transfer Capability from Area 2 to 1
Using the proposed interval arithmetic method, TTC is evaluated for both fixed input parameters and for various uncertainties.The loads are modeled as constant power factor loads.The active loading in MW of area 1 before and after the transaction is shown in Table 3.The generation of area 2 increases from 56.2 to 70.0 MW and the load at area 1 from 84.5 to 98 MW and the limiting condition was the overloading of lines 6-8.The sum of the loads at the sink area is taken as the TTC.Table 4 gives the total load of area 1 and total real and reactive losses in interval parameters.One may observe that the range of the total loss is higher for load uncertainty compared to line uncertainty.Obviously, the losses increase when both load and line uncertainties are taken into account.
Table 5 shows the values of different operating parameters after the transaction has been carried out.The TTC values are presented in interval form for all the categories of uncertainties.It is observed that the interval arithmetic method indicates a wider range of intervals.

Case 2: The Transfer Capability from Area 3 to 1
The generation of area 3 is increased proportionally when the loads in area 1 are increased.The generation at area 3 is increased from 84.5 to 94 MW.The load active power vector of area 1 in bus number sequence is 0.0, 27.7, 3.06, 9.70, 0.0, 0.0, 29.1, 38.29, 0.0, 0.0 .The limiting conditions are overloading in lines 6-8 and 21-22.It is observed that the TTC lower limit and upper limit vary over a wide range when both load and line uncertainties are taken into account.Table 6 the values of TTC and other system parameters in interval form after the transaction has been carried out.

Conclusion
The need for efficiency in electrical power deregulation has increased the need for improved calculations of TTC by incorporating the uncertainties in transmission.This paper proposes a probabilistic approach using interval arithmetic to estimate the TTC.The solution obtained from the interval arithmetic method is found to be more informative in qualitative terms about the system analysis when compared to the conventional deterministic approach.It is conjectured that power planners and operators may benefit not only from the expected TTC evaluation, but also from an evaluation of confidence intervals of the calculation.The transfer capability from area 2 to 1 and 3 to 1 of the IEEE 30 bus test system is computed with fixed line and load, with line uncertainty, with load uncertainty, and with both line and load uncertainties.The total loss for the lower limit and upper limit is computed for all four types and compared.The results clearly elucidate that the total loss occurring is minimum when the bus is operated with line uncertainty.The obtained results are useful to ensure a secured

Start
Represent all the inputs in interval formChoose one of the following options a Normal operating conditions b Uncertainty due to load parameters c Uncertainty due to line parameters d Uncertainty due to load and line Select an operating condition Run optimal power flow for initial system conditions base case Specify the source bus/area and sink bus/area Increase real and reactive loads at sink bus/area and generation at source bus/area with constant power factor Power flows and voltage magnitudes are obtained by running the power flow Individual bus voltages and branch power flows are compared with their respective limits Is there any violation?

Figure 1 :
Figure 1: Flow diagram for the computation of ATC.

Figure 3 :
Figure 3: Voltage profile for load uncertainties.

Figure 4 :
Figure 4: Voltage profile for line uncertainties.

Figure 5 :
Figure 5: Voltage profile for load and line uncertainties.

Table 1 :
Base case load flow.

Table 3 :
Active loading of area 1.

Table 4 :
Base case for Cases 1 and 2.

Table 5 :
Interval values for the operating parameters in case 1.

Table 6 :
Interval values for the operating parameters in case 2.