# A multi-area approach to the economic optimization of electric power system

## Abstract

In this paper, the problem of short-term economic dispatch of active power in a combined hydro-thermal electric power system is considered. The problem studied is a 24 hours optimization, with operational cost as associated criterion. It is assumed that consumer's demand, defined as a number of time functions, as well as technical constraints concerning power production units, and transmission line capabilities are satisfied within the period of optimization. The deterministic time discrete mathematical model of the above stated problem consists of a set of nonlinear algebraic equations and a set of nonequalities. The power system is decomposed into a number of interconnected power areas. Consequently, a number of local less dimensional area optimization subproblems are defined. Each of them is a typical nonlinear programming problem. Coordination among subproblem solutions is performed by a higher level decision making effort. It is done by specially derrived coordination algorithm.

It is shown that the proposed multi-area approach is computationally effective and fast. This property has a particular importance since in practice the system under control is influenced by frequent structural disturbances (failures of production units and/or transmission lines) causing power generation rescheduling. Using the presented algorithm a short term dynamic optimization problem of 220 kV network of Serbia has been solved. The results of this solution are also discussed in the paper.

## Keywords

Electric Power System Hydro Power Station Hydro Power Generation Generation Schedule Dynamic Optimization Problem## List of symbols

- t
current index denoting time (hour of a day), t=1,...,24,

- N
number of interconnected areas composing power system,

- i
current index denoting area number, i=1,...,N,

- D
_{i}^{t} i-th area power demand at time t,

- h
_{i} number of hydro power plants in i-th area,

- s
_{i} number of thermal power plants in i-th area,

- r
_{i} number of transmission lines connecting i-th area with other ones,

- j
current index denoting plant number within area; for hydro power plants j=1,...,h

_{i}, for thermal power plants j=1,...,s_{i},- r
current index denoting number of a transmission line connecting area with other ones, r=1,...,r

_{i},- H
_{ij}^{t} active power output of (i,j) hydro power plant at time t,

- H
_{i}^{t} h

_{i}-dimensional vector of active power outputs of hydro power plants in the i-th area at time t,- q
_{ij}^{t} flow of water through turbines of (i,j) hydro power plant at time t, q

_{ijmax}being maximum allowed flow of water,- Q
_{ij} a volume of water available for hydro power production of (i,j) hydro power plant,

- s
_{ij}^{t} active power output of (i,j) thermal power plant at time t; S

_{ijmin}and S_{ijmax}being technical minimal and maximal output powers, respectively,- s
_{i}^{t} s

_{i}-dimensional vector of active power outputs of thermal power plants in the i-th area at time t,- T
_{ir}^{t} active power of (i,r) transmission line at time t, T

_{irmax}being maximal allowed active power,- T
_{i}^{t} r

_{i}-dimensional vector of powers in transmission lines connecting i-th area with all others at time t,- P
_{i}^{t} (s

_{i}+h_{i}+r_{i})-dimensional vector of power production and interchange in the i-th area at time t, P_{i}^{t}=(s_{i}^{t},H_{i}^{t},T_{i}^{t}),- E
_{i}^{t} active power exchange of i-th area at time t,

- L
_{i}^{t} transmission losses in the i-th area at time t,

- L
^{t} transmission losses in the whole system at time t,

- B
_{i} (h

_{i}+s_{i}+r_{i})×(h_{i}+s_{i}+r_{i}) — dimensional matrix of B-coefficients of area i,- F
_{ij}(S_{ij}^{t}) operational cost of (i,j) thermal power plant at time t, if active oputput power is s

_{ij}^{t},- F
_{i}(S_{i}^{t}) s

_{i}-dimensional column vector function of area production costs,- λ
_{i}^{t}, μ_{ij}^{t}, ν_{ir}^{t} Lagrange multipliers,

- μ
_{i}^{t} s

_{i}-dimensional row vector multiplier,- ν
_{i}^{t} r

_{i}-dimensional row vector multiplier,- k
current index denoting the number of iteration,

- C
_{ij}^{t} worth of water of (i,j) hydro power plant at time t.

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