Simulated Annealing Algorithm for Dynamic Economic Dispatch Problem in the Electricity Market Incorporating Wind Energy

The formulation of the BBDED mathematical model is presented as a bilevel problem of optimization. The higher level is the market-clearing, and the lowest level is the maximization of social profit, considering the minimization of the generation cost of generators. The objective function, including conventional power generators and wind energy generators, can be represented by the formula below.

$\text{MaxPF}\sum\limits_{t=1}^{T}{\left[ \sum\limits_{j=1}^{Nd}{B{{c}_{j}}({{d}_{j,t}})-\sum\limits_{i=1}^{Ng}{C{{g}_{i}}({{p}_{i,t}})+\sum\limits_{k=1}^{Nw}{C{{w}_{k}}({{p}_{k,t}})}}} \right]}$              (1)

$B{{c}_{j}}\left( {{d}_{j,t}} \right)={{a}_{dj}}{{d}_{j,t}}^{2}+{{b}_{dj}}{{d}_{j,t}}+{{c}_{dj}}$         (2)

$C{{g}_{i}}\left( {{p}_{i,t}} \right)={{a}_{gi}}{{p}_{i,t}}^{2}+{{b}_{gi}}{{p}_{i,t}}+{{c}_{gi}}$          (3)

$C{{w}_{k}}\left( {{p}_{k,t}} \right)={{d}_{k}}{{p}_{k,t}}$          (4)

where, Bc_j(d_j,t) and Cg_i(p_i,t) are the demand and supply bid functions of customers j and generator i, respectively; Cw_k(p_k,t) is the bid function of the wind generator k. d_j,t is the demand bid amount of power for customer j at period t; p_i,t is the supply bid amount of power for thermal generator i at period t; p_w,t is the bid amount of wind generator w at period t. N_d is the customer's number; N_g is the number of thermal generators; N_w is the wind generator number. a_dj, b_dj, and c_dj are the bid demand function coefficients. a_gi, b_gi, and c_gi are the bid supply function coefficients; d_k is the bid function coefficient of wind generator; t is the periods' number.

2.1 Bidding strategies in the electricity market

Bidding strategies can be implemented by setting the power capacity and price to be submitted as bids for each generator. The bidding strategies are either static or dynamic and generally depend on generation supply and the customer's demand. Each GENCO submits a supply bid in the electricity market, and each customer submits a demand bid to ISO. Usually, these bids are linked to an offered price and matched by the ISO. There are two components of the objective function. The first one is the customer benefit function. To achieve maximum benefit, the customer must implement a bidding strategy using the price bid coefficients (a_dj) and (b_dj). The customers bidding strategy is implemented as High (H), Medium (M), and Low (L) bidding strategy according to the bid price coefficient. The authors have demonstrated through experiments and based on the literature, for high bidding strategy, (a_dj) ≥ 0.09, for medium bidding strategy, (a_dj) can be within the range of 0.05, and for low bidding strategy (a_dj) ≤ 0.01. For the bid coefficient (b_dj), it is formed using the equation 0 < (b_dj) < λm, where λm is the market-clearing energy price, which can be determined by intersecting the GENCOs and customers bid curves [20]. The second one is the cost function of generators, which represents the supply side. Each GENCO should develop a bidding strategy according to the marginal cost to maximize the profit [21].

2.2 Problem constraints

These constraints are particularly equality, inequality, and ramp-rate limits.

2.2.1 Equality constraints

It is known as the balance of active power flow between supply and demand in the power system, and it is given below.

$\sum\limits_{i=1}^{Ng}{{{p}_{i,t}}=}\sum\limits_{j=1}^{Nd}{{{d}_{j,t}}}+{{p}_{L}}$         (5)

The calculation of the exact real power loss is complex. When developing optimization models that include power loss, approximating formulas are often sufficient. One such approximation technique is the B-matrix power loss formula shown in (6). The use of the B-matrix approximation avoids the need to calculate the loss for each transmission line, as long as the structure of the power system remains relatively uniform. This approximation is effective for large variations in the system load.

${{P}_{L}}({{p}_{i,t}})=\sum\limits_{i=1}^{Ng}{\sum\limits_{j=1}^{Ng}{{{p}_{i,t}}{{B}_{i,j}}}}{{p}_{j,t}}+\sum\limits_{I=1}^{Ng}{{{B}_{i,0}}{{p}_{i,t}}}+{{B}_{0,0}}$       (6)

where, B_i,j is the matrix loss coefficient; B_i,0 represents the vector loss coefficient; B_0,0 represents the constant of loss;

Loss coefficients formula derivation is based on the following four basic elements [22]: (1) The power factor for the generator busses remains constant. (2) The voltage angle at each voltage-controlled (PV) bus is constant. (3) The voltage amplitude at each PV bus is fixed. (4) The load current to total load current ratio remains constant. The coefficients lose significant precision when the calculations used to derive the loss formula are not correct. are violated. The coefficients of the polynomial loss equation are determined by the power flow solutions.

2.2.2 Inequality constraints

The conventional or wind generator is designed to produce electricity between a maximal and minimal secured range to prevent over-or under-generation during load satisfaction.

${{p}_{i,,t\min }}\le {{p}_{i,t}}\le {{p}_{i,t\max }}$       (7)

${{p}_{k,t\min }}\le {{p}_{k,t}}\le {{p}_{k,t\max }}$         (8)

The demand side is also limited between a maximal and a minimal power bid to balance with the supply side.

${{d}_{j,t\min }}\le {{d}_{j,t}}\le {{d}_{j,t\max }}$        (9)

2.2.3 Ramp-rate limit constraints

The output of a generator is limited by the ramp-rate limit. It represents the generated power variation, whether it increases (ramp-up) or decreases (ramp-down). It is therefore expressed in the unit of power output by the minute.

$R_{i}^{down}\le {{p}_{i,t}}-{{p}_{i,(t-1)}}\le R_{i}^{up}$          (10)

where, R_i^down is the generator's maximal power decreasing capacity over a period; R_i^up is the generator's maximal power increasing capacity over a period;

2.3 Wind energy

Wind energy is one of the world's renewable energy source that has been largely used in the last decade. It has many benefits, like low investment cost as well as the absence of pollution. However, wind energy varies according to location and wind speed availability, leading to great uncertainty concerning the power produced. Wind speed has been calculated for the desired height according to the equation of power law [23].

$\frac{W{{s}_{h}}}{W{{s}_{10}}}={{\left( \frac{h}{10} \right)}^{N}}$        (11)

Wind energy generation varies mainly according to wind speeds, the air density, the area swept, and the generators' efficiency. The wind energy equation can be stated as follows:where, Ws_h is the wind speed at a specific height h; Ws₁₀ is the standard wind speed at the height of 10m; N is the hellman coefficient (0.143).

$WP=\frac{1}{2}.AD.a.e.Ws_{h}^{3}$      (12)

where, AD is the density of air in kg/m³ (1.225); a is the wind generator swept area in m²; e is the efficiency constant of the wind generator (0.49); Ws_h³ is the wind speeds in m/s.

Wind speed is a value that is variable in this equation by ignoring minor non-linearities. The function between a given wind speed and power output can be represented in the curve shown in Figure 1.

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Figure 1. Simple power curve for wind generator

The figure above shows that no power is produced at wind speeds lower than (V_in) or higher than (V_out) the power produced at wind speeds between (V_r) and (V_out) is equivalent at the wind turbine's rated output, for wind speed from (V_in) to (V_r), the power produced is a linear function.

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Figure 3. Comparison of results under bidding strategies

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Figure 4. Social profit convergence characteristics

The wind generator's cost coefficient is calculated from the equivalent coefficient of the conventional unit, with 37.55% [31, 32]. The bid-based dispatch was carried out for each wind speed, considering the discussed constraints in Eq. (5)-(10). In this case, two outcomes are simultaneously optimized as objective functions, minimizing the total generation cost and maximizing the total social profit with the wind speed/energy variation. The customers' load demand is kept as in the previous case. The simulation results are illustrated in Table 7, Figures 5 and 6 after the trading periods.

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Figure 5. Total generation cost with wind power penetration

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Figure 6. Total social profit with wind energy penetration

From Table 7, Figure 5, and Figure 6, it is noticed that, when the injection of wind energy generation is increased, the thermal generators' power is continuously reduced, which leads to the minimization of total generation cost. If the wind generator works with 25% of its power injection, we can save 1270.1 \$ of total generation cost with a 14869.13 \$ of total social profit. If the wind generator works with 100%, we can save 1117.59 \$ of the total generation cost with 15022.39 \$ of the total social profit. The customer benefit is kept higher and not affected by wind energy injection than system losses, which are decreased in all cases. Therefore, social profit is significantly maximized due to the minimization in the generation cost of thermal generators when wind energy is increased, as shown in Table 8, which compares the two study cases outcomes.

Table 8. Results comparison of two study cases

Table 8 shows that the BBDED solution using the SA algorithm, including wind energy, reduces thermal generators' output, transmission system losses, and the total cost of generation, which leads to a maximum social profit for the generating companies and an improved benefit for the customers. Figure 7 depicts the convergence curves of the proposed SA algorithm solution for both study cases.

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Figure 7. Social profit convergence curve comparison with and without wind energy