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NEW INFORMATION TECHNOLOGIES IN OPERATIVE CONTROL OF MODES IN REGIONAL ELECTRICAL POWER SYSTEMS
ANDREY D. TEVJASHEV,
TATJANA B. TIMOFEEVA
Kharkov National University ofRadioelectronics, Ukraine, 61166, Kharkov, Lenin Ave, 14, [email protected]; [email protected]
Abstract. The problem of development of software for management of modes in electrical power systems in connection with casual character of a load in network is considered. The stochastic mathematical model of a system for operating control of modes in regional electrical power systems is offered. The methods for problem solving of operating control and operating planning of operational modes in regional electrical power systems are developed. The application of the developed models and methods will allow to increase efficiency of management in regional electrical power systems.
The problems of electric energy spare on all stages its life cycle are one from central problems facing by modern electrical power ingineering. The complex decision of these problems is connected with transition on new information ecological safe technologies of production, transmissions and consumptions of electric energy. Development of computer facilities and equipping with it regional electrical power systems (EPS) dispatching services opens new possibilities for increase of management efficiency of modes of operation of electrical systems and networks. In this work the stochastic approach to solution of a problem of operating control of modes of regional EPS is offered. It allows to ensure reliable supply of all consumers with electrical energy of specific quality with minimum losses of power in EPS.
The rich experience in modeling and optimization of EPS modes, i.e. in planning and management of modes oftransfer of electrical energy has been accumulated. The results of optimization of structure and parameters of electrical networks have opened significant reserves of economy of losses of electrical energy up to 15-20 % from the total volume of the transmitted electric power. However, models and methods of optimization, whichrealized inthe software packages (RASTR, ANARES, OPT, Dispatcher, Space), are determined and do not take into account the actual conditions of operation of electrical networks and electric power market. The optimum solutions, received with these methods, meet only concrete boundary conditions and are, as a rule, on the boundary of allowable area. It is natural, that such “optimum” solutions appear unacceptable in practice. From a formal point ofview the problem of the account of actual conditions of EPS operation is that in models of mathematical programming, by which the problems of planning and management of modes are reduced, some parameters of criterion function and restrictions are casual variables.
Therefore it is necessary for increasing the management efficiency ofEPS to develop models and methods for operating control of modes of EPS operation, taking into account a casual character ofprocesses in EPS. The application ofthese models and methods will allow to reduce actual losses of power in EPS and to increase efficiency of operation of regional EPS.
1. Stochastic model of a system for operating control of modes in regional EPS
Basic data for a problem of operating control by normal modes of EPS operation is the information about EPS structure, about values of characteristics of network elements and operative information about network state in preceding time moments ....t0 -2, t0 -1, t0 . The calculation of parameters of a planned mode is carried out in the moment t0 , with anticipation 1,2,3,...,T, incorrespondence with a criterion of management J and requests on admissibility of a mode.
It is possible to change a mode of EPS operation influencing on controlled parameters of a system. The vector of management of a EPS is:
U(t) = {R(t),KTr(t),SComp(t),NCond(t)}, (1)
where R(t) - vector of the Boolean variables describing a position of switches in switching equipment, defining the scheme of feed of the EPS consumers; KTr(t) - vector of coefficients of transformation; SComp(t) - vector of power of synchronous equalizers; Ncond(t)- vector of the amount of blocks of static condensers.
Set of load knots in each considered moment t is characterized by stochastic functions of the amount of an active and jet power consumed in this knot ?H,i(t,®),QH,i(t,®), fflefl, (Q,B,P), where Q - space of elementary
outcomes, B- borel’s sigma - algebra, P- probability measure.
Let’s designate
Zi (t, m) = {PH,i (t, m), QH,i (t,«)} (2)
- amount of an active and jet power consumed in the knot i in the moment t.
The stochastic mathematical model of EPS mode in the moment t is:
P{Pr,i(f,t) - PHi(Ui,f,t, a) - PSh,i(t) -ZPi,j.(t) = 0}=1,
j
(3)
P{Qr,i (f,t) - QH,i(Ui,f,t, rn) - Qm,i(t) Qy(t) = 0} = 1,
j
(4)
where Pr>i (f ,t), Qr^ (f ,t) - active andjet power are generated in the knot i in the moment t; PHi, QHi - active and jet power are consumed in the knot i in the moment t; Pm>i (t), Qm,i (t) -loss of a power in shunts in the moment t; PiJ, QiJ - active and j et power going out from knot i to knot j in the moment t; f -frequency in EPS.
Realization of the conditions of balances of an active and jet power (3), (4) with probability 1 completely describes the EPS
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state in the moment t, accounts a stochastic character of consumed in knot i active and jet power and work of local systems of automatic regulation of EPS modes. From a system (3), (4) information about significances of modules U and phases 8 ofvoltages in each knot in the moment t is received.
The various methods of solution of system equations for mathematical model of a steady mode in regional EPS are considered. It is shown, that for using during operating control by a load flow in EPS is expedient to decide equations (3-4) by the method of Newton. The solution of this system of mathematical model of the steady mode by the method of Newton [2] is conducted.
The criterion function of a problem of operating control by a load flow EPS in the interval [t0,T] is:
T
M | J(X(t);Y(t);Z(t, ra))dt ^ min (5)
® t0 X.(t)eQ , (5)
where J(X(t);Y(t);Z(t, ra)) - optimum criterion that was selected, X(t) - vector of management of a power system, Y(t) - vector of determined parameters of a mode; Z(t, ra) -vector of casual parameters of a mode.
As criterion of optimality in the moment t it is possible to consider:
- total losses of an active and jet power in EPS:
Jl(X(t);Y(t);Z(t, ra)) = P2 (X(t);Y(t);Z(t, ra)) +
+Q Z (X(t);Y(t);Z(t, ra))
transformation. They are determined depending on a type of the transformer and taking into consideration technical state of the equipment and parameters of reliability. The graduated character of regulation of coefficients of transformation is taken into account
2. On a power of synchronous equalizers
SK ^ SK(t) ^ SK , (7)
where K - number of synchronous equalizers; SK ,SK -lower and upper limits of a modification of a power of synchronous equalizers. It is determined depending on a type SE and taking into consideration technical state of the equipment and parameters of reliability.
3. On number of blocks of static condensers
N l < Nl(t) < N l', Nl. = {Nl.,N? ....}, (8)
j lj lj
where l - number of static condensers; N l - amount ofb locks
of static condensers; N'pNl - limits of a modification of an amount of blocks of the static condenser. It is determined depending on a type BSC and taking into consideration technical state of the equipment and parameters of reliability. The graduated character of regulation of the given parameter is taken into account.
For holding conditions of reliability of power supply and the qualities of the electric power the following restrictions are considered:
P{Uk < Uk(c®)< uk} = 1, (9)
- functional of cost of active power transmitted to the consumers:
J 2 (X(t);Y(t); Z(t, ra)) = Pj (X(t);Y (t); Z (t, ta)) c(t)
where Pj(X(t);Y(t);Z(t,ra)),Q2(X(t);Y(t);Z(t,ra)) -total losses of an active and jet power in EPS in the moment t; c(t) - factor of costs calculated according to the acting tariff for a current consumption and factor of daily nonuniformity: c(t) = kj (t) pi, where pi- acting tarifffor a current consumption; kj (t) - coefficient of daily nonuniformity (ki- night, k2- peak, k3 - halfpeak):
kj
fk
(t) Hk
1 -2 =
lk3 =
0.25, t e [0,6] '
1.8, t e [8,10] u [l7,20] \
1.02, t e[6,7] u [11,16] [21,23J
In this problem the part of restrictions (area Q) is the equations of balance of an active and jet power in knots EPS (3), (4) and they ensure connection between adjustable EPS parameters and calculated parameters of a mode - modules and phases of voltages in network nodes. Besides the next restrictions are considered:
Restrictions on adjustable parameters of a mode:
1. On coefficients of transformation of transformers
K ^ < Kjt) < K tij, Ktrj = {K^K^ ....}, (6)
where j - number of adjusting transformers; Ktrj, Ktrj -lower and upper limits of modification of coefficients of
where k - number of knots in EPS, U k, U K - limiting significances of modules of nodal voltages.
P{IZ <Iz(i,®)<IZ} = 1, (10)
where z - number of elements in EPS, IZ, IZ - allowable significances of currents on elements.
Thus, the mathematical model of a system of operating control by modes of operation in regional EPS is generated. It represents a problem of digital - continuous nonlinear stochastic programming of a M-type with line probability restrictions and differs from known models by a possibility of the account of a casual character of power consumption by the consumers.
2. General approach to a solution of a problem of operating control of modes in regional EPS.
The complicated character of influence of various groups of the factors on processes of power consumption has shown that these processes canbe described as stochastic, containing determined (polyharmonic and polynomial) and casual components. This made it necessary to choose such structure of optimum control, which also contains two components: determined and stochastic. The determined vector of
management is determined in the moment t0, in an outcome ofa solution ofa problem of operating planning ofEPS mode. The solution of a problem of operating planning of a EPS mode is carried out on the basic of determined data both about the structure parameters of EPS and prognosis of future significances of stochastic processes which are calculated with specific anticipation {1,2,...,T} (conditional expectations
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of future significances of a current consumption). The casual vector of management is determined for each moment to + 1,t0 + 2,...,t0 + T . It is outcome of a solution of a problem of stabilization ofEPS mode.
For realization of the approach it is necessary to select an interval of planning At1and interval of management At2, (At1 > At2). The size of an interval of management is selected proceeding from conditions of physical realization of management in EPS as well as from the realized system for information acquisition and processing, which is the most important when deriving operative data on the values of the component vector Z(t, ra). The choice of the interval of planning At1, is defined by the required accuracy of values of parameters of a stochastic process Z(ra,t). For simplification of an entry hereinafter we shall consider only one interval of planning [t0,T], where T -10 = At1.
Let’s make digitization of the interval of management At1 on K steps with the interval At2 . The discrete analog of a problem (5), (3-4), (6-10) is:
K
M{ Z J(X(k);Y(k);Z(k,ra))> ^ min (11)
ra k=0 XeQ’
where the area Q is determined by the equations (3), (4), (6)-( 10).
It is offered to search for an optimum solution X*(k* as two
vectors: X (k) = X0(k) + 8X (k), where X0(k) -
determined vector calculated in the initial time moment, *
8X (k) - casual vector calculated in each moment k = 0,1,2, — K -1.
It is shown, that the considered problem can be presented as:
I(K) * EJ(X0(k);Y(k);Z0(k)) +
k=1
1 K N „ „
+ X Z EJ^i(5X(k);Y(k);Z0(k))DZi(k) +
2 k=1 i=1
K
- Z ZJ^Z,(SX(k);Y(k);Z0(k))KZiJ(k)-
k=1 i<J
► min
(X0(k),SX(k))eQ
(12)
where Z0(k) - conditional expectation calculated in the moment t = 0 with the anticipation k; DZ (k) - variance of
magnitude of a load in a knot i, KZij(k) - coefficient of correlation between magnitudes of loads in knots.
Thus, it is offered to consider a problem of operating control by modes of operation in regional EPS as a two-steps stochastic programming problem. The solution ofa problem (5), (3), (4), (6)-(10) can be obtained in two stages. At the first stage the determined problem (planning of a mode EPS) is decided.
K
I0(K) =Z [J(X0(k);Y(k);Z0(k))] ^ min
k=0 X0(k)eQ’
it is a problem ofprograms management ofa mode inEPS. The solution of this problem allows to receive determined component of vector of management - x0 (k).
Casual component of vector of management SX*(k) is received in an outcome of a solution of a problem of the second stage (stabilization of a mode EPS)
1 K N
AI(K) = -1 IjZi (5X(k);Y(k);Z0(k)) D^ (k) +
2 k=1 i =1 K
I ZJZiZj(8X(k);Y(k);Z0(k)) KZij(^)^ nkn,
k=1 i<j 6Xlk)eU
where the area q is determined by the equations (3), (4), (610).
3. The solution of a problem of operating planning of modes in regional EPS
The problem of operating planning of modes in EPS allows to define determined component of vector of management X0(t),t e [t0,T]. It is a central task in a problem of operating control by a mode in EPS. The correctly planned mode allows to avoid origin of a lot of situations requiring decision making on management of a mode in EPS in an actual time scale. In an outcome of such planning during operating control the psychological load on EPS (dispatcher ofa system) is sharply reduced and his role in control procedure is changed. Moreover, the maintenance of stability ofwork in EPS on all an interval of planning increases the role of local automation and opens a perspective of transition from automated control systems EPS to completely automatic systems.
The problem of operating planning of operational modes in regional EPS is aproblemof discrete mathematical programming with nonlinear additive criterion of optimization and nonlinear restrictions. Restrictions are the equations of balance EPS (34), restrictions on a range of a modification of parameters of vector of management (6), (8) and regime’s restrictions (9), (10).
Structurization and research of properties of solutions of a problem of planning of a mode in regional EPS is conducted.
It is shown, that the problem of operating planning of operational modes in regional EPS on an interval of planning [0,T] is:
K N N „
I0(K) = Z [ EE (Pij (Xo (k),Y(k),Z0 (k)) +
k=1 i=1j=1
i* J
+ Qij(Xo(k),Y(k),Z 0(k)))] ^ min
J X0(k)eQ,
where
Pij (X0(k),Y(k),Z 0(k)) = V3Ui(k) cos 8i (k) Iay (k) + W3u i (k) sin 5 i (k) Iry (k),
Qij(X0 (k), Y(k), Z0 (k)) = V3Ui (k) sin Si (k) Iaij (k) --«s/3Ui(k)cos 8i(k)Iry(k),
where Iaij(k),Irij(k) - active and jet component of current in connection ij look like:
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Iaij(k)
-=(Ui(k)cos5i(k) - Uj (k) cos5 j(k))gij(k) -
-J3 (Ui (k) sinSi (k) - Uj (k) sin5 j (k)) bij (k),
Irij (k) = -3 (Ui(k) cosSi (k) - Uj (k) cos5 j (k)) bij (k) --33 (Ui (k) sin Si (k) - Uj (k) sin 5 j (k))gy (k),
where k - O..K ,K
T
----, At? - selected interval of
At 2 2
management.
The system of restrictions on allowable management for moment k = 0..K look like:
Pi(k) - gu(k)Ui(k)2
n+1
Ui(k) £ Ui(k) *
j=l
j*i
* (g ij (k) cos 5ij(k) - bij (k) sin 5 j (k)) = 0,
Z Qs(k) + £ Sk(k) + £ Nt (k), - bii (k)Ui (k)2 -
S k t
n+1
- Ui(k) £ Ui(k) * (bij (k) cosSj (k) + gij (k) sin Sj (k))
j=l
j*i
Sij(k) = Si(k) -5 j(k), i = 1,n, j = 1,n, where n - number of knots in a network.
Conductance and susceptance are possible to express as:
gij(k)
bij (k)
Z rn n ktr
n m m
(k)
(Z xn)2 + (Z rn)?’
nn
Z xn n
•nktr (k)
m
m
(Z Xn)2 + (Z rn)2
nn
The restrictions on adjustable parameters of a mode look like (6)-(10).
The considered problem is a problem of digital - continuous programming. It is necessary to take into account that a part of components of vector of management has a discrete character of modification. The conducted researches have shown, that application of methods of discrete mathematical programming to a solution ofthis problem, which allow to find a global minimum requires the large temporary costs. It is unacceptable in the work of complex of operating control by modes in EPS. The algorithm, which allows to reduce a solution of an initial digital - continuous problem of mathematical programming to a solution of a problem of continuous mathematical programming is offered. This algorithm ensures hit of a solution in e - neighborhood of a point of a global optimum with rigid restrictions during the time of a solution of a problem (fig. 1).
The analysis ofpossible methods of a solution of a continuous problem of planning of a mode in EPS is conducted. It is shown, that mo st expedient is to use the differential algorithm for solution of this problem.
Properties of a continuous problem of operating planning of operational mode in EPS:
- A criterion of optimization function (12) is nonlinear and convex on a set of allowable solutions q .
- The set of allowable solutions q is nonlinear and convex.
- The necessary conditions for a point of a local minimum are simultaneously and sufficient conditions.
The analysis of a kind both properties of criterion function and restrictions has allowed to modify differential algorithm in connection with features and properties of a considered problem. The offered modification of differential algorithm is more effective and less power consuming than traditional method.
The solution of a problem of planning of a mode EPS with the following algorithm was conducted:
1. The initial approximation X(0) and precision of calculations s are set. X - vector of restrictions on adjustable parameters of a mode (6-8).
02. The necessary conditions of a optimum (analysis of conditional partial derivatives) are checked up:
- if these conditions are executed on all components of a vector X - pass to a stage 5;
- differently k = k +1. The set J'c J (J = {1,2,...,n} )is formed. It consists of indexes ofthose variables, on whichthe violation of necessary conditions happens, and we pass to the following stage.
3. The analysis of partial derivatives of criterion function is carried out, with the purpose of clearing up of a character of violations of necessary conditions on Xr , for everything r e J'. Proceeding from a kind of violations, AXr is calculated:
if(.SL)<k) > 0, x« 8 Xr r
then AXrk) = max [X'r -Xr , <_m.r
X£' * X'r
' -X(k)- AX(k)
AX« <0
8I0
=0
max
6V(k)
8Xr
Y"i _Yi
(k)
<0
SYi vSXr y
x (k) ’
max
Y'i -Yi
Xk)
8Y
8Xr
(k) /
>0
SYi vSXr y
x (k)
];(13)
if (|Xr)(k) < 0, Xrk) * X"r
oXr
then AX® = min [X"r -X®; AX(rk)
AX« >0
min
Y'i -Yi
(k)
8Y;
8Xr
(k)
<0
SYi
x(k) ’
min
Y"i -Yi
8I0 _ 8Xr (k)
vSXr y
8Y;
8Xr
(k)
>0
SYi vSXr y
x (k)
]
. (14)
0
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4. For all r e J', Xr = X^0) + AXr is calculated; transition to a stage 2.
3. Total losses and effectiveness ratio for base and optimum modes are calculated and compared each hour of the interval of planning.
5. The fulfilment of sufficient conditions is checked up: if these conditions are executed then X(k) is the solution of a problem, X* = X(k).
4. Research of the efficiency of the offered method of planning of the mode in regional EPS.
The efficiency of the offered method of planning of operational mode in EP S was carried out for a plot of the electrical power system of the Nizhnevartovsk enterprise of electrical networks. It based on actual data about parameters and condition of a system. The scheme of this EPS is represented on fig. 1.
On a plot of the network (fig. 1) there are 3 substations 220/ 110kV (Varyogan, Macht, Northern Varyogan) and 4 substations 110/35/6kV (KNS 9, KNS 1, KNS 2, West Varyogan).
For this network the optimization of a mode of operation on the criterion of minimum of total losses of active andjet power in the network having control over transformation factors of adjustable transformers is carried out.
For the analysis of efficiency of the developed method of planning of operational mode in EPS within the interval of planning [0, T] a number of computing experiment was realized. As an interval of planning the period equal to 1 day was selected. The interval of management is equal to 1 hour. As a vector of management all transformer branches of the simulation model are accepted.
Algorithm of realization of experiment is the following:
1. In the moment 0 the forecasting of loads of a considered area of EPS on the interval of planning is realized. The calculation of the prognosis of EPS loads is carried out as conditional expectations and is ensured with a minimum root-mean-square error.
2. On the basis of prognosis of the values of active and jet power of loads in EPS knots and other actual EPS parameters in the moment 0 a problem of planning of operational mode in EPS is decided. The significance of loads in EPS knots within day is accepted as constants.
For realization of the method of planning of EPS mode the significance of loads in EPS knots on the interval of planning were accepted as constants (equal to expectations of magnitudes of loads in knots). However, actually, the magnitudes of loads in EPS within a day are not identical. The replacement of such inhomogeneous behavior of a load on average significance results in an error of solution. For the analysis of influence of such replacement on an overall performance of the method the following computing experiment was conducted.
It was shown, that the application of the developed approach allows to reduce losses of a power in EPS on the average to 7 %. It is caused only increasing of efficiency operative -dispatching control in EPS.
During realization of next experiment strategies of management were realized. The strategy 1: settlement loads is constant in all the interval of planning, the issue of control actions in the interval ofplanning is carried out 1 time. (interval offorecasting of a load - 1 day). The strategy 2: significances of a load is calculated each hour and the issue of control actions is carried out on each interval of management. (interval of forecasting of a load - 1 hour).
The analysis of outcomes of this computing experiment is indicated in fig. 2.
It is shown, that the best effect from application of the developed method of management of operational mode in EPS is reached when control actions are issued each hour (strategy 2), however, this problem is now technologically not sold, therefore the offered strategy 1 of issue of control actions once on the whole interval of planning is most expedient.
5. Conclusion
In this work the new stochastic approach to problem of operating control of modes in regional electrical power systems was considered. This approach is distinguished from the known themes in that in it the casual character
of processes in EPS was taken into account. The use
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SS Zima -©
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1
Fig. 2. The analysis of the
— — parameter of efficiency for base mode
parameter of efficiency for interval of forecasting
_____ 1 day and issue of control actions 1 time on an
interval of planning // Integrated parameter of efficiency 5,69 %
parameter of efficiency for interval of forecasting 1 hour and issue of control actions on each interval of management // Integrated parameter of efficiency 6,85 %
strategies of management
of the offered stochastic approach during operating control by modes of operation in EPS allows to receive a solution ensuring the optimum, stable and reliable mode of EPS operation.
The stochastic mathematical model of a system for operating control of modes in regional electrical power systems is offered and the effective methods for problem solving of operating control and operating planning of operational modes in regional electrical power systems are developed.
The application of this method allows to reduce time of shaping ofvector of management and to increase the efficiency of operative - dispatching management in regional electrical power systems .
References: 1. Modern problems ofreliability ofpower engineering systems: models, market relations in management of reconstruction and development / Manov N., Voropay N., Sennova E., Tevjashev A. etc. M. 2000. 374p. 2. Tevjashev A., Timofeeva T., Smirnov A. Operating control by a load flow in electrical power systems under the casual character ofload // Radioelectronika i informatika. 1999. N° 3. P. 36-43 (in Russian).
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