Control of Active Power Operating Modes of Interconnected Power Systems with Long-Distance AC Transmission Lines
Abstract Generated abstract
This paper addresses the economic control of active power operating modes in interconnected power systems that include long distance high voltage AC transmission lines and intermediate systems. It formulates an optimization approach for distributing active power among sources over a load schedule, accounting for generation costs, transmission losses, reserve and load maximum coefficients, voltage and flow constraints, and nodal active and reactive power balances. The proposed procedure uses stepwise load schedule calculations, Lagrange functions with transformed inequality constraints, and equivalent energy costs to determine operating loadings that approximate the minimum calculated system costs under practical operating restrictions.
Full Text
CONTROL OF ACTIVE-POWER OPERATING MODES OF INTERCONNECTED POWER SYSTEMS WITH LONG-DISTANCE AC TRANSMISSION LINES
R. I. BORISOV
(Presented by the scientific seminar of the Department of Electric Power Stations and Electric Systems and Networks)
The full use of all the advantages provided by the interconnection of power systems presupposes their mutual participation in covering the aggregate active-load schedule. The system-forming ties in the form of long-distance AC transmission lines will have a substantial influence on the economics of the entire interconnection, while the intermediate systems located along the route of the long-distance transmission lines will have a substantial influence on the economic performance of the lines. The mutual influence of intermediate systems on the technical and economic indicators of long-distance transmission proves to be highly varied. Thus, when the values of transmitted powers decrease, the voltages on the sections will tend to rise. The action of the intermediate systems, by lowering the voltage, will reduce corona losses. Conversely, when the transmitted powers increase, the voltage will decrease, losses will grow, and the action of the intermediate systems should raise the voltage and reduce losses. Consequently, with respect to power losses one can outline, for each point of interconnection, a range of voltage values within which the losses turn out to be minimal. Naturally, this range must lie within the zone of permissible voltage-deviation values prescribed by technical regulations. The presence of intermediate electric-power systems partially or to a greater extent eliminates shunt compensation reactors, thereby reducing the specific reduced costs of transmitting electric energy. Owing to the uncertainty and error in determining the cost characteristics of interconnections, it is preferable to use other initial data for solving problems of distributing active powers among sources. Questions of regulating the active powers of 750- and 1200-kV transmission lines acquire new significance owing to the influence of the time-zone effect, meteorological conditions in different parts of the interconnections, load seasonality, and other conditions.
Let the active-power consumption schedules in each power system (or the power distributions) and certain constraints on power flow or curves of flow distribution, as well as the sale price of electric energy in each power system, be known. If, for different variants in the composition of equipment, the values of the cost of released energy are identical, then such variants are considered equivalent. This cost may differ for each source over the year or over the day, which makes it possible to take into account the influence
of the belt coefficient of time, and changes in the various operating conditions of the equipment in the interconnected power system (startup, shutdown, etc.). Then the variable part of the costs, which depends on the distribution of active powers among the systems in the interconnection and on exchanged power flows, is written as follows:
\[ Z=\sum_i \sum_{j=1}^{T} \left(P_{ij} C_{ij}+P_{\text{loss}j} C_{\text{loss}j}+P_{\text{loss}j} C_{mj} C_{pj} K_y\right)\Delta t_j \to \min , \tag{1} \]
where
\[ \begin{aligned} P_{ij} &\text{ is the delivered power of the } i\text{-th source at the } j\text{-th step}\\ &\text{of the load schedule }(j\le i);\\ C_{ij} &\text{ is the cost of a unit of generated energy of the } i\text{-th source}\\ &\text{at the } j\text{-th step of the schedule;}\\ P_{\text{loss}j} &\text{ is the total power loss at the } j\text{-th step of the schedule;}\\ C_{\text{loss}j} &\text{ is the calculated cost of active-power losses;}\\ C_{mj};\, C_{pj} &\text{ are the calculated values of the coefficients of coincidence with the load maximum}\\ &\text{and reserve coefficients for each step of load variation;}\\ K_y &\text{ is the calculated cost of installed capacity of power plants;}\\ T &\text{ is the calculation period.} \end{aligned} \]
The delivered powers and losses can be defined through the operating-mode parameters in the form:
\[ P_i=P_i(E_i\, U_i\, \delta_i); \]
\[ P_{\text{loss}j}=P_{\text{loss}j}(mP), \]
\(mP\) is the number of unknown operating-mode parameters.
The operating loading of each source (power system) over the duration of the calculation time \(T\) must be found from the costs of the actual load distribution \(Z^*\), and not from the real distribution, which is piecewise variable:
\[ Z^*=\sum \left(P_i^* C_i^*+P_{\text{loss}i}^* C_{\text{loss}i}^*+P_{\text{loss}i}^* C_{mi}^* C_{pi}^* K_y\right), \tag{2} \]
where
\(C_i^*\) is the equivalent delivered cost of electric energy of each source,
\[ C_i^*=\frac{\sum_i C_{ij} P_{ij}}{P_i^*}, \]
\(P_i^*\) is the sought operating loading of each source over the calculation time interval \(T\), including the frequency-regulating one. Since the costs of the actual distribution \(Z^*\) cannot be equal to the calculated costs \(Z\), the operating loadings of each generalized source should be determined in such a way that the costs of the actual, i.e., operating, loading correspond to the minimum of the calculated costs. The expression for the objective functional takes the following form:
\[ \Delta Z=Z^*(mP)-Z(mP') \to \min . \]
As shown in [1], the total number of independent operating-mode parameters in a three-machine scheme with three transformations and eight nodes is 25; in a four-machine scheme with two intermediate systems the total number of operating-mode parameters will be 33. With additional variables of the Lagrange function, the number of unknowns sought will be, respectively, 66 and 74.
Solution Procedure
- A calculation is performed at each step \(i\) of the electric-load schedule, and the active and reactive powers supplied to the system are determined, on the basis of relation (1), taking into account the limiting conditions for the various variables and the power balances at all nodes. In this case, the flow distribution of active powers, as well as of reactive powers, proves to be unknown, and the imbalance functions are written for conditional flows directed positively toward the nodes:
\[ \pi_i P_{nb\,i}(m\Pi)=0, \tag{5} \]
\[ \pi_i Q_{nb\,i}(m\Pi)=0, \tag{6} \]
where \(\pi\) is the enumeration symbol.
Restrictions on the operating parameters in the form of inequalities may be represented in the following generalized form:
\[ \pi_i \Pi_i \leq \pi_i(\Pi_i)\leq \pi_i \overline{\Pi_i}. \tag{7} \]
By introducing additional variables \(\pi_i \overline{x_i}\) and \(\pi_i x_i\), two-sided inequalities are transformed into doubled equalities in the usual way. The Lagrange function for each calculated load operating condition is written as follows:
\[ L = \mathcal{Z}(m\Pi)+\sum r_i\,\pi_i\bigl(P_{n i}(M\Pi_i)\bigr)+\sum \psi_i\,\pi_i\bigl(Q_{nb i}(M\Pi_i)\bigr)+ \]
\[ +\sum \vartheta_i\bigl(\pi_i(\Pi_i+x_i)\bigr)+\sum \Theta_i\bigl(\pi_i(\Pi_i+x_i)\bigr). \tag{8} \]
After the operating parameters \((\pi E_i;\ \pi U_i;\ \pi\delta_{0i})\) have been determined, the active and reactive powers supplied by each source are found. Such calculations are performed for each step of the electric-load schedule, and the values of the supplied powers are found while satisfying the required conditions and restrictions \((\pi P_{0i}\) and \(\pi Q_{0i})\).
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The variable part of the total costs and the equivalent cost of electric energy are calculated by formula (3) for each source over the calculated time \(T\).
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Next, the objective functional is formed in the form of relation (4) with respect to the new variable operating parameters \((m\Pi')\), and their values are determined with constraints and conditions of various kinds and numbers in the form of (5), (6), and (7), which must unconditionally be satisfied for the powers and the operating mode. The experimental program for these calculations is undergoing pilot testing.
References
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R. I. Borisov. Control of reactive-power operating modes of interconnected power systems with long-distance AC transmission lines. Article in this collection.
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F. Haidu, I. Peschons, W. F. Tinny and D. S. Piercy. Optimum Load-Shedding Policy for Power Systems PAS-87 pp. 784--796.