Considering the turbine backpressure effect of thermal units to optimize the PQ _ power generation in power system

A new algorithm of PQ_power optimization is mentioned and some typical numerical examples are presented in this article. The fuel cost characteristics being obtained in form of superposition of some high order polynomial and sinusoidal functions can approximately simulate the turbine back-pressure effect of the generator units at the electrical thermal stations and solve the problem of economic active power dispatch. A new loss factor formula expressing the network transmission power losses is a second order polynomial function of generator powers containing a square matrix. This loss factor formula is proposed for optimum solution of generator reactive powers in multi-machine power system.


INTRODUCTION
Optimal pq_power generation (OPQG) containing economic p_power generation (EPG) and optimum q_power generation (OQG) is important problem to be solved in the operation and planning of a power system.The objective of the OPQG problem is to determine the optimal combination of pq_power output of all generator unit so as to meet the required load demand at minimum operating cost while satisfying system technical constraints.
The main objective of an economic power dispatch strategy is to determine the optimal operating state of a power system by optimizing a particular objective while satisfying certain specified physical and operating constraints.In its most general formulation, the economic power dispatch is a nonlinear, nonconvex, large-scale, static optimization problem.
In reality, the turbine fuel cost characteristics can be obtained in form of some convex fracture, this is notably arised from the back-pressure turbine characteristic, in addition, the turbine regulation method also raises some effect to the appearance of turbine fuel cost characteristics, and it may be briefly called the turbine backpressure effect, practically showing as follows [1],[2] In most cases, the fuel cost characteristics of the units rating under 25MW may be determined experimentally and fit for the 2 nd order polynomial function.The fuel cost characteristics of unit rating upper 25MW may be fit for n th order polynomial superposing some sinusoidal function to simulate the turbine back-pressure effect of generator on the thermal electrical stations, approximately

STATEMENT OF EPG PROBLEM
Using the fuel costs (1) to solve the economic p_power dispatch (EPD) problem minimizing the total fuel cost in the whole of electrical power system which consists of many stations.The target function is G(P , V ) P (P , V ) P P ; (5 ) N g is number of generator unit; (i,m,r=1..N g ); P D is total MW_power load demand;  P P (P g i ,V) is total transmission MW loss which is taken form in where V m is m th bus voltage modul;  m is argument of current output from the m th generator; cos m is power factor of m th generator; J h is h th bus current; J  is sum of all of bus currents; C jm is current distribution factor; N is number of bus (h=1..N); N b is number of branch (j=1..N b ).
The solution of the problem (2), ( 3) is iterative calculation of hessian matrix [1], [3], as follows The iteration process of generator p_power optimization will be converged on condition of (F (t) +f (t) )0.

STATEMENT OF OQG PROBLEM
The OQG problem is to minimize the transmission active power losses, taking into account the steady-state stability margin of every generator in electric power system.The target function is S i is MVA output from i th generator; Q D is total MVAR load in power system; Q L is total transmission MVAR loss in the inductive elements of network; Q C is total capacitive reactive power charging of transmission lines.
The solution of the problem (10),( 11), ( 12),( 13) is iterative calculation of gradient [1], as follows  is gradient step value; The t th iterative gradient of target function [1] may be determined as follows:  q is penalty factor; The iteration process of generator q_power optimization will be converged on condition of (F (t)  q 10 -6 ).

STATEMENT OF REACTIVE POWER OPTIMIZATION FOR VAR SUPPORTING DEVICES
Let's refer to [4], [5].In this case, the operation conditions requiring a specific steady state at each time interval of load changing in 24 hours, and the test algorithm can reach a purpose which is to solve optimization commitment of device supporting MVAR power, such as TSC, TCR, SVC or synchronous condensers...This proposed mathematical model can be applied to schedule the operation charts of VAR optimization in a power system with multiple voltages level.Here, the problem of optimizing the voltage and q_power in a power system is solved separately for p_power, i.e. the number of bus of p_power generation in the target network may be generally different from the number of bus of q_power supporting device, and assuming that the bus p_power does not change, it may have been optimized before.The target of VAR optimization problem is established to minimize the total cost value [4] consisting of following cost components: -electrical energy and power losses in the power transmission network; -installation and operation for var supporting devices; -depreciation and operation of transfo-LTC; -q_power generation of power plant; -optimization of voltage level of power system; A general form of the objective function of the total cost calculation is written as c(x j )=c qb (q bj )+c dp (x j )+c mba (q kr )+c qg (x j )+c du (x j ); (16) where j=1,2,…number of independent bus in power system; q b j is the controlled MVAR capacity at the bus (j) to meet objective; c qb (q bj ) is the cost of installation and operation for VAR supporting devices; c dp (x j ) is the cost of electrical energy and power losses in the power transmission network; c mba (q kr ) is the cost of depreciation and operation of transfo-LTC; c qg (x j ) is the cost of VAR generation of power plant; c du (x j ) is the costs optimizing the average voltage level in the power system; x j is the controlled variables corresponding to bus (j) to meet objective.The controlled variable (x) is a collective set of the numeric value of voltage module of the power plant busbars and of the transformer station busbar with LTC; or a set of numeric value of VAR capacity of compensation devices located at the bus (i) in power system; or a set of numeric value simulating the transfo LTC at the bus (i) in a power system.

The problem of VAR and voltage optimization is written as follows:
Determine the condition w(x,y)0, such that c(x)min and satisfy the constraints : (v - j v j (q c j ,a j )v + j ); j=1,2 ,.., total bus number; ( q c j 0); (k - min m ≤ k m (v r ,a r ) ≤ k + max m ) ; m=1,2,…., total transformer number; (q - g i q g i (v i, a i )q + g i ); i=1,2 ,.., total PV bus number; where: q bj is the VAR capacity to be supported at the bus (j); (q b , a i ) is the voltage at the j th bus; q gi (v i , a i ) is the generated VAR capacity of the i th power plant; a i is the i_th element of the eigenimage vector A simulating a certain steady state structure of power system; k m is the numeric value corresponds to one simulated ratio of the LTC of m_th transformer; w(x,y) is the vector balance indicating a certain technical condition of steady state of power system; x is the vector controlled to meet objectives (the v g , q b and the k mba ); y is the non-controlled vector.
In reality, we can apply some specific factors or choose some parameters depending on concrete conditions of q_power optimization to take account of total cost function c(x j ) or of just one component function.
In this article the component function c dp (x j ) is used to make the target function of q_power optimization problem.The statement of q_power optimization of VAR supporting device is to determine c dp (q bj )min.Then, this pq_power optimization problem may be solved with multitarget function by applying the method of optimum co-ordination, i.e. the main function (2) must be satisfied (10) under condition of c dp (q bj )min.

NUMERICAL EXAMPLE
Let's survey the optimum condition operartion of a 68-bus power system consisting of 4 thermal stations with 15 generation units and of 5 SVC stations.Basic power is 100MVA.The linedata is given in p.u. in table 1 as follows: The pilot-slack bus is 68 th and voltage of which is 1.08p.u.Let's investigate the case of slight load.

Typical results are shown in table 5 by comparing the initial powers with the optimum power as follows
In this case, the SVC-data is also refered to the table 2a.The busdata of initial operation condition of power system is given in table 7a and 7b as follows: Trang 70 The comparison of voltage levels may be graphically shown as refering to the figures 3 and 4.

CONCLUSION
The new algorithm of optimal pq_power flow problem is proved with good convergence.
The application of the speccific type fuel cost functions for the optimal pq_power generation problem allows to simulate the back-pressure effect of turbine regulation.
The process of calculation obtains a good results of voltage optimum levels according to the solution of optimal pq_power flow problem.


Some typical unit fuel cost characteristics are approximately rated for simulation of turbine back-pressure effect as follows

Fig. 2
Fig.2 Typical Unit rating fuel cost functions.
factors B mr can be found on condition of the t th solution of LPF problem [1].

Fig. 3
Fig.3 Voltage levels in case of loud load

Table 4 .
Generation limit data

Table 5 .
Comparison of generation

Table 8 .
Comparison of generation