CC BY
6
Gayibov Tulkin Shernazarovich, Doctor of technical science, professor, Head of chair "Power Plants, Systems and Networks" Tashkent State Technical University, E-mail: tulgayibov@gmail.com Reymov Kamal Mambetkarimovich, Assistant teacher of chair "Power Plants, Systems and Networks"
Tashkent State Technical University, E-mail: kamal-tstu@mail.ru
OPTIMAL PLANNING OF SHORT-TERM MODES OF POWER SYSTEMS WITH CONTROL OF LOADS OF ELECTRIC CONSUMERS TAKING INTO ACCOUNT OF NETWORK FACTOR
Abstract: In the article the problem of optimal planning of power system's short-term modes with load control of regulated electric consumers and the network factor are considered. A mathematical model and an algorithm for solving of the problem are given. The results of research of efficiency of the proposed algorithm are presented.
Keywords: power system, short-term planning, network factor, power losses, derivative oflosses, load schedule, fuel consumption, optimization, control, optimal mode.
I. Introduction power consumption during the planning period under
Functioning of many modern power systems is consideration. In such conditions, it become possible to characterized by a sharp non-uniform loading sched- obtain an additional economic effect due to the optimal
ules, often changing of electrical network schemes and corresponding complication of electrical power system's (EPS) mode control.
Sharp non-uniformity of loading schedules leads to increase of expenses for production of electricity and corresponding growth of its costs for consumers. It, in particular, is connected with operation in power plants at maximum loadings the less economic equipment, which use power resources not enough effectively, frequent change in the composition of operating units in thermal power plants.
One of the rational ways to ensure the economical operation of such power systems in the short-term planning is alignment of the load schedule through control ofpower consumption. The alignment of the load schedule can be achieved by involving consumers in optimization process on basis of administrative and economic measures [1, 83-86]. At short-term planning of power system mode for some large consumers non-rigid loads for each time interval are given, but their minimum and maximum possible values while maintaining a constant
planning of the power system modes taking into account non-rigid load schedules of consumers. Part of the additional generated economic profit, at the same time, can be spent to stimulate consumers for their participation in optimization process.
A mathematical description and an algorithm for solving of the problem under consideration without taking into account the network factor were proposed by the authors of this paper in [1, 83-84]. Here we discuss mathematical modeling and algorithm of solving of the problem taking into account of network factor.
Taking into account of network factor in planning of power system's short-term modes with optimal load control of electric consumers, generally, can lead to significantly changing of the load schedules of regulated electric consumers and redistribution of the total load of power system between power plants. Here, optimization is reduced to reducing the total fuel costs in power system by reducing losses as a result of choosing rational schedules for loads of regulated electric consumers and power plants.
II. Theoretical Part
The problem under consideration is formulated mathematically as follows:
to minimize the total fuel cost at the thermal power plants (TPPs) for control cycle T
b = zz bt p) (1)
tsT isN
subject to restrictions
- on balance of power in power system in each time interval of the control cycle T
w =xp-xp-n=0, tet, (2)
ieN jeM
- on electricity balance for control cycle for each of the regulated electric consumers
4=!^ - ET = 0, je M, (3)
teT
- on capacities of power plants and electric consumers
p >min < p < p i e n , t eT;
p,,min < p < p' m , t eT, (4)
- on power flow on controlled power transmission lines (PTL)
p; < p, l e Lp, t eT, (5)
where N, M are sets ofTPPs and consumers participated in optimization process, accordingly; Lp is set of PTL in which power flows are supervised; p', P' are loadings of TPP i and consumer j in time interval t of the regulation cycle T; Bt (p') is fuel cost in TPP i at loading pf in time interval t of the regulation cycle; Wt, fy are nonbalance functions ofcapacity in time interval t and electricity for consumer j for regulation cycle, accordingly.
Decision of the described problem can be found through minimization of function
l=bzp;-zp;-n
\ ieN
j^M
+
jsM
+ XAj (z P - E[]+ZZ 5;,
(6)
where Ht, are undefined Lagrange multipliers entered for taking into account the condition of balance of capacity in time interval t and electricity for consumer j; S' is penalty function considering limitations on power flow in supervised PTL l in time interval t .
In general, in optimization process participate also Hydro Power Plants (HPPs) which have water reservoirs [2, 34-56; 5, 1975-1976; 6, 125-127; 7, 405-412]. In the described model HPPs are taken into account in basis of choosing of corresponding Lagrange multipliers and
their reduction, in calculation sense, to equivalent TPPs as in [2, 35-38, 6, 126-127].
Thus, the mathematical model of the problem of power system's short-term mode optimal planning with load control of electric consumers and the network factor consideration is characterized by entering into active power balance equation for each t-th time interval of control cycle the total active power losses in electrical networks. These losses are complex functions of all the operating parameters and, therefore, can be determined on the basis of calculation of steady-state modes of electrical networks for all the time intervals of the control cycle. The proposed algorithm for solving of this problem is described below.
At optimum values of variables - active capacities of power plants, loads ofadjustable electric consumers and indefinite Lagrange multipliers, take place next condition:
= b; + ßt (l-a; ) + Z si = 0, i sN, ; eT;
OP leL,
dL
dP;
:t (-l-a' ) + VZ sj = 0, j e M, ; eT;
dL
SL = w = 0, ; eT;
dL
— = ^ = 0, j e M,
OA.
(7)
where b; =
dBt ÖP7
is the relative increase in fuel costs in
the i-th TPP at the load of the t-th time interval p; si, sj are derivatives of the penalty function for taking into account of restriction on power flow in l -th supervised line on power of the i-th TPP and the j-th node with load in the t-th interval of the planning period T;
is derivative of total losses of active power in electrical networks in t-th time interval of the power regulation cycle on the power i-th TPP; g'j is derivative of the total losses of active power in electrical networks in the t-th time interval of power regulation cycle on the j-th load node with adjustable electrical consumers in the t-th interval of the control cycle.
Dividing the first equation in (7) by 1 - and the second equation by 1 + , and also denoting the network coefficients for i-th TPP in t-th time interval of control cycle as n' = 1/ (l ,for j-th load in t-th time interval of the control cycle as n] = 1/(l,the system of equations (7) can be presented in following form:
%
A. «4 if^T(iM);
£i>'-Er=0, jeM.
jeM
The last system, together with simple limitations (4), is a prerequisite for the optimality of covering the load schedule of power system by thermal power plants, taking into account the network factor, while optimally controlled of load of the regulated electric consumers.
Solution of the problem under consideration on the basis of solution of system of equations obtained from (8) is connected with non-reliability of convergence of the iterative calculation process. Therefore, this system can be used only to represent the essence of the optimization problem under consideration taking into account the network factor.
In the algorithm proposed here, the problem of optimal planning of power system's short-term mode with optimization of loads of electric consumers and taking into account the network factor is solved by minimizing the function
F = B + XZ m! (9)
tsT lsLp
with gradient method taking into account all the given constraints. In order to take into account constraints (2) and (3), the optimized capacities of the plants and loads of nodes with regulated electric consumers are divided into two groups - independent and dependent variables. The constraints imposed on independent variables are taken into account by fixing at each iteration variables that went beyond the allowed limits on the violated limit values, verifying the necessity of their detachment in subsequent iterations. Restrictions on the limiting values of dependent variables are taken into account by the penalty function as in [2, 24-27; 3, 2156-2157; 6, 106; 7, 421-422].
In the case of adoption as a dependent variables the active power of the first TPP in all time intervals of control cycle and the power of the loads of nodes with adjustable electrical consumers in the first interval of the control cycle, calculation of the power ofplants and loads of nodes taken as independent variables in the regular k - th iteration is performed according to the following formulas:
dF(k-1)
— , i e N (i * 1), t e 7,(10)
dP
t<=T
(8)
P'(k) = p
t(k-i)
- h
t (k -i)
p;
m _ pf(fc-D _ jjtGfc-D d-F J ' ' DP'
(*-D
jeM, teT(t* 1),(10 a)
where the components of gradient of the function are determined as
dF_(k-1) dPt
= b(k-1) + sj™ +(
t (k-1) + st(k-1)
dF
(k-1)
= St(k-1) + &lj "I"
a
t (k-1) + St(k-1) l1
M-'+«
t (k-1)
L + CT
t (k-1)
), (11) ), (11 a)
St(k-1) St(k-1)
are derivatives from the penalty function introduced in violation of the restriction on the flow of the l- th power transmission line on capacities of the i-th TPP and loads of j-th node in the i-th interval of the control cycle and in k-1 -th iteration; hti (k-1), ht(k-1) - steps in the direction of the antigradient for the independent capacities of the i-th plant and the load of j-th node in the k-1 -th iteration, determined by the conditions given in [2, 154-155; 3, 351-352].
The proposed algorithm for taking into account of network factor at optimal planning of power system short-term modes undertake the preliminary reconstruction of dependence of losses derivative on active power of power plants ot (P) and load nodes (P) which are used at calculation on formulas (11) and (11a). The essence of the algorithm is as follows:
1) the optimal coverage of the total load schedule of the power system by all TPPs with optimal load control of electric consumers without taking into account the network factor is carried out. As a result, optimal schedules of loads of power plants and electric consumers are obtained;
2) for each time interval of the control cycle, calculations of the steady-state modes of electrical systems are performed, total losses in electrical networks, their derivatives at the capacities ofpower plants and load nodes are determined. In view of non-uniformity of load schedules ofpower plants and consumers, the dependences of ot (P) and Oj (Pj) are obtained;
3) the optimal planning of the power system mode with the control of the load of regulated electric consumers for the period under consideration is carried out as
without taking into account the network factor. At this stage, when calculating the components of the gradient on formulas (11) and (11a the, the dependences of O (P) and O} (P) are used.
Calculation of derivative of losses in the proposed algorithm is carried out on the basis of solution of following system of linear algebraic equations obtained from conditions
dp ~dS dP dU
dQ öS dQ dU.
x
dn dS dn dU.
(12)
as in [2, 101-103; 3, 350-351; 4, 2157-2158]. In (12)
dn dn
are determined vectors of the loss
S, =Pi+jQi
derivatives on active and reactive powers of nodes;
dP dQ dP dQ
—, -, -, -are sub matrixes of partial deriva-
dS dS dU dU
tives of active and reactive powers of nodes on angles and
dn dn
modules of complex voltages of nodes; —, - are
do dU
the vectors of loss derivative on angles and modules of complex voltages of nodes. III. Experimental Part
Efficiency of the proposed algorithm was evaluated in the example of optimal coverage of the load schedule of power system by four TPP with following fuel consumption characteristics (in tones of conditional fuel in hour - t. c.f./h.):
-, oQ =-
dP Q dQ
«-) 1—
U8=242kB
3.9+j 15 J
4
3,6+j34,2
ir
=p2+jQ2 2
ocT
k
SA =P4+jQ4
^ 5 ¥
S, =P3+jQ. 5,2+jl5,3
6+j20.:
/
s 7 =P7+jQ7
^=P5+jQ5 S,= P6+jQ6
Figure 1. The circuit of the power system
Bl = 90 + 0,1P1 + 0,0007Pj2,
B6 = 70 + 0,11P6 + 0,0004P62,
B7 = 80 + 0,15P7 + 0,0005P72,
B8 = 60 + 0,12P8 + 0,00055P82. The circuit of power system is presented in Fig. 1. TPPs are located in points (nodes) of 1, 6, 7 and 8. The TPP in node 8 is a balancing plant.
Table 1. Load schedules of consumers
Load nodes 2, 4, 5, and 6 have adjustable electric consumers. Load schedules of consumers with four characteristic time intervals are given in Table 1. The limiting capacities of TPPs, loads of consumers and the amount of electricity received by consumers during the planning period are presented in Table 2.
Number of interval of time P2, MW P3, MW P , MW P5, MW Total load P , MW
1. 230 400 130 340 1100
2. 280 470 150 400 1300
3. 350 580 260 490 1680
4. 300 500 180 420 1400
Active power factors for power plants are 0.85, and for consumers are 0.9.
Table 2. Limiting capacities of TPPs, loads of consumers (in MW) and the amount of electricity received by consumers in the nodes (in MWh)
Parameters Number of nodes with TPP Number of nodes with regulated consumers
1 6 7 8 2 3 4 5 Total
prnrn, MW 200 200 200 200 230 400 130 340 1100
pmaxMW 600 600 600 600 350 580 260 490 1680
E, MWh 1160 1950 720 1650 5480
To assess the effect of taking into account the networkfactor, as well as comparing the results of optimization, in Table 3 the results of optimal planning of power system's mode with the optimal control of load of electric consumers without taking into account the networkfactor are presented (the derivative of losses on the capacities of TPPs and loads of consumers are zero).
For optimal planning of power system mode with load control of electric consumers and the network factor consideration, calculations of the steady-state modes of the electric network at the load nodes are presented in Table 3 were performed and the derivatives of losses based on the solution of the system of linear algebraic equations obtained from condition (12) were determined. As a
result, the corresponding derivatives of losses and the dependences of a (P) for all the nodes corresponding to these loads are obtained. These dependencies are used to determine the derivatives of losses when calculating the components of the objective function's gradient according to (11) and (11a) during the iterative optimization process.
Table 3. Results of the optimal planning of power system mode without taking into account the network factor
t Loads of power consumers Power losses n, MW Loads of power plants B, t. c.f./h.
MW Py MW MW MW P1, MW MW P* MW MW
1. 269.95 479.75 159.77 399.95 58.41 321.41 267.24 455.09 324.10 706.14
2. 295.12 485.10 164.68 430.81 68.84 339.94 281.20 479.73 343.68 743.26
3. 300.00 500.00 210.00 400.00 80.33 350.52 289.70 494.51 355.59 766.13
4. 294.92 485.14 185.55 419.24 68.93 341.93 283.03 482.68 346.14 747.83
Total expense of conditional fuel for period of planning: 2963,36 t. c.f.
Table 4 shows the results of the optimal power system consumers taking into account power factor by the mode planning with optimal load control of electric algorithm described above.
Table 4. Results of the power system mode optimal planning with optimal load control of electric consumers taking into account power factor
t Loads of power consumers Power losses n, MW Loads of power plants B, t. c.f./h.
MW MW MW MW P1, MW MW MW MW
1. 269.35 479.75 159.44 399.36 58.55 322.91 457.93 285.26 300.36 708.67
2. 308.38 484.63 165.00 428.84 55.74 333.16 467.14 328.40 313.90 744.73
3. 300.66 500.00 215.31 401.98 66.10 341.16 485.71 337.07 320.11 765.53
4. 281.61 485.62 180.24 419.85 56.61 337.84 471.26 297.16 317.64 736.69
Total expense of conditional fuel for period of planning: 2955,61 t. c.f.
Comparing the results obtained as a result of optimal planning of power system mode with control of load of consumers without taking into account (Table 3) and
taking into account the network factor (Table 4), we see that the network factor is significantly cost-effective. In the example under consideration, the total consumption
of conventional fuel in the power system from taking into account the network factor is reduced by 7.75 tons of conditional fuel, which is 0.26% of the total fuel consumption in the initial mode.
IV. Conclutions
1. A mathematical model of the problem of optimal planning of short-term mode of power system with optimal control of load of electric consumers and network factor consideration is proposed.
2. The conditions of optimality for coverage of schedule of loads ofpower system with optimal control of load of regulated electric consumers taking into account the network factor are obtained.
3. The algorithm of taking into account of network factor at optimal planning of short-term modes of power system with control of loads of electric consumers is proposed. The algorithm allows solving the problem with high accuracy and reliability. Using the obtained dependences of a(P) in calculation process creates the possibility for taking into account the network factor without iteratively.
4. It has been established that taking into account the network factor in the optimal planning of short-term modes of power system with control of loads of electric consumers give a significant economic effect.
References:
1. Gayibov Tulkin., Reymov Kamal. Optimization of modes of electrical power systems with the control of power consumption / Proceedings of the 7th International Scientific Symposium on Electrical Power Engineering Ele-ktroenergetika - 2013. September 18-20, Stara Lesna, Slovak Republic. - P. 83-86.
2. Nasirov T. X., Gayibov T.Sh. Teoreticheskie osnovi optimizatsii rejimov energosystem. - T.: "Fan va texnologi-ya", - 2014. -184 p.
3. Автоматизация диспетчерского управления в электроэнергетике / Под общ. ред. Ю. Н. Руденко и В. А. Семенова. - М.: Изд-во МЭИ, - 2000. - 648 с.
4. Tulkin Gayibov, Kamal Reymov "Taking into account the Power Losses in Optimization of Modes of Power Systems", International Electrical Engineering Journal, - Vol. 7. - 2016. - No. 2. - P. 2156-2160.
5. Slama M. M., Elgazar M. M., Abdelmasoud S. M., Henry Y. A. "Solving Short Term Hydrotermal Generation Scheduling by Artifical Bee Colony Algorithm", International Electrical Engineering Journal (IEEJ), - Vol. 6. -2015. - No. 7. - P. 2078-2365.
6. Murty P. S. R., Operation and Control in Power Systems. B. S. Publications, Hyderabad, - 2008.
7. Bergen, Arthur R. Power Systems analysis / Arthur R. Bergen, Vijay Vital. Prentice Hall, Inc. Upper Saddle River, New Jersey, - 2000.
