Multiagent coevolutionary genetic fuzzy system to develop bidding strategies in electricity markets: computational economics to assess mechanism design

Abstract

This paper suggests a genetic fuzzy system approach to develop bidding strategies for agents in online auction environments. Assessing efficient bidding strategies is a key to evaluate auction models and verify if the underlying mechanism design achieves its intended goals. Due to its relevance in current energy markets worldwide, we use day-ahead electricity auctions as an experimental and application instance of the approach developed in this paper. Successful fuzzy bidding strategies have been developed by genetic fuzzy systems using coevolutionary algorithms. In this paper we address a coevolutionary fuzzy system algorithm and present recent results concerning bidding strategies behavior. Coevolutionary approaches developed by coevolutionary agents interact through their fuzzy bidding strategies in a multiagent environment and allow realistic and transparent representations of agents behavior in auction-based markets. They also improve market representation and evaluation mechanisms. In particular, we study how the coevolutionary fuzzy bidding strategies perform against each other during hourly electric energy auctions. Experimental results show that coevolutionary agents may enhance their profits at the cost of increasing system hourly price paid by demand.

Appendices

Appendix A: Chromosome representation

The chromosome representation adopted in this paper follows the approach devised in [3].

1.1 Granularity

GFRBS granularity, that is, the number of linguistic terms, defines the component Cr ₁ encoded by a variable length chain of integers(N, M, n, m),with n = (n ₁, ..., n _i , ..., n _N ) and m = (m ₁, ..., m _j , ..., m _M ), where N is the number of input variables, M the number of output variables, n _i the number of linguistic terms of input variable i and m _j the number of terms of output variable j, with i = 1, ..., N; j = 1, ..., M.

The total number of linguistic terms is given by expression (5) where L _a is the number of linguistic terms of the antecedents and L _c the number of linguistic terms of the consequents.

$$ L = L_{a} + L_{c}, \quad L_{a} = \sum_{i=1}^{N} n_{i}, \quad L_{c} = \sum_{j=1}^{M} m_{j} $$

(5)

1.2 Membership functions

Following the approach of [54, 55], we consider strong fuzzy partitions. The number of fuzzy sets is kept the same, but the change of only one parameter results in a new fuzzy partition of the input or output spaces, which allows global adjustment of membership functions. The FRBS obtained are likely to be more transparent because usually it results in cognition frames.

The form of the membership functions is not evolved; we assume trapezoidal membership functions. Since strong partitions are adopted, each linguistic input variable i, 2 × (n _i  − 1) requires real numbers to define the partition of the corresponding universe. Similarly, for output variable j, 2 × (m _j  − 1) real numbers define a partition of the universe. These real numbers are encoded in the component Cr ₂ of the chromosome whose length is given by expression (6). Since no normalization is performed, each value must lie within the corresponding interval [v ^X _min , v ^X _max ].

$$ L_{mf} = \sum_{i=1}^{N} 2 \times (n_{i} - 1) + \sum_{j=1}^{M} 2 \times (m_{j} - 1) $$

(6)

For single input single output (SISO) system, we have N = 1 and M = 1. In this case Cr ₁ is (1, 1, (n ₁), (m ₁)). In expression (7) the parameters of the membership functions encoded in Cr ₂ corresponding to this Cr ₁ component are shown. $l_k^{X_1}$ denotes the parameter on the left of the k-th trapezium top that defines the k-th fuzzy set of variable X ₁ and r _k the parameter on its right.

$$ \begin{array}{ll} &((r_{1}^{X_{1}}, l_2^{X_1}, r_2^{X_1}, \ldots, l_{k}^{X_{1}}, r_{k}^{X_{1}}, \ldots, l_{n_{1}}^{X_{1}}),\\&(r_{1}^{Y_{1}}, l_2^{Y_1}, r_2^{Y_1}, \ldots, l_{k}^{Y_{1}}, r_{k}^{Y_{1}}, \ldots, l_{m_{1}}^{Y_{1}})) \end{array} $$

1.3 Rule base

The fuzzy rule based system is a set of fuzzy rules combined by the union operator:

$$ R = \bigcup (R_{1}, R_{2}, \ldots, R_{k}) $$

The rule structure adopted is DNF (disjunctive normal form) Mamdani fuzzy rules [56–59]. Each rule is hence interpreted in the Mamdani sense. A rule is defined as in expression (7) where A _io , A _ip are fuzzy sets associated with the input variable X _i and C _jq , C _jr are fuzzy sets associated with the output variable Y _j , with o, p ≤ n _i , and q, r ≤ m _j .

$$ {\mathbf{If }}\; X_{i} \;{\mathbf{ is }}\; (A_{io} \;{\mathbf{ or }}\; A_{ip}) \;{ \mathbf{ and } }\; \ldots \;{ \mathbf{then } }\; Y_{j} \;{ \mathbf{ is } }\; (C_{jq} \;{ \mathbf{ or } }\; C_{jr}) \ldots $$

(7)

Each rule is encoded in a chain of bits with variable length L given by expression (5). One additional bit indicates (in)active rules. If the antecedent of a rule contains an entry like X _i is A _ij , then the corresponding bit at position $p = j + \sum_{k=1}^{i-1} n_{k}$ is 1, otherwise it is 0.

Each rule base has L _r rules, where L _r is randomly chosen in the interval

$$ \min(L_{a}, L_{c}) \leq L_{r} \leq L_{a} \times L_{c} $$

(8)

The number of rules, rule size and the rules themselves define chromosome component Cr ₃. Number of rules and rule size are encoded just for convenience, that is, they could have been computed from the rule base itself, but are encoded to be promptly accessible by the algorithm. Component Cr ₃ is composed by two integers and L _r chains of length L + 1 because the first bit indicates an (in)active rule.

Appendix B: Genetic operators

Similarly as an approach devised previously in [3], the crossover operator is as follows.

1.1 Crossover

Two different crossover operators are chosen, depending if the selected individuals (parents) have the same granularity or not.

Two individuals with Cr ₁ given by

$$ \begin{array}{ll} &(N, M, (n_1, \ldots, n_i, \ldots, n_N), (m_1, \ldots, m_j, \ldots, m_M) )\\&(N, M, (\eta_1, \ldots, \eta_i, \ldots, \eta_N), (\mu_1, \ldots, \mu_j, \ldots, \mu_M) ) \end{array} $$

have the same granularity if and only if n _i  = η _i , i = 1, ..., N and m _j  = μ _j , j = 1, ..., M.

1.1.1 Parents with same granularity

When the granularity is the same, a promising zone in the search space is found and should be appropriately exploited [51]. In this case, the granularity of the database (Cr ₁ component) is kept the same for the offspring, and the membership functions parameters (component Cr ₂) combined using the max-min-arithmetic crossover (MMA) proposed in [52].

According to the max-min-arithmetic crossover, for each pair of chromosomes, four offspring are generated through pairwise combination of the minimum, maximum, and two linear combinations of each element. The best two individuals among the offspring could be added to the next generation [51]. However, in this paper, all offspring are added. The algorithm proceeds with mutation and the population size is cut down in the next generation selection.

When parents have the same granularity, the four offspring also have the same granularity. Instead of perform rule base crossover directly, the rule base of each parent is kept with a copy of the offspring generated by the max-min-arithmetic crossover. After crossover, each pair of selected parents gives eight new individuals.

1.1.2 Parents with different granularity

When the selected pair has different granularity, a random crossover position p, 1 ≤ p ≤ (N + M − 1), is chosen. Both, granularity (component Cr ₁) and corresponding parameters of the membership functions (component Cr ₂) are recombined.

Two individuals whose components Cr ₁ are

$$\begin{array}{ll} &( (n_1, \ldots, n_i, \ldots, n_N), (m_1, \ldots, m_j, \ldots, m_M) )\\ &( (\eta_1, \ldots, \eta_i, \ldots, \eta_N), (\mu_1, \ldots, \mu_j, \ldots, \mu_M) ) \end{array}$$

crossed at position p, p ≤ N result in offspring¹⁵:

$$ \begin{array}{ll} &( (n_1, \ldots, n_{p-1}, \eta_p, \ldots, \eta_N), (\mu_1, \ldots, \mu_j, \ldots, \mu_M) )\\&( (\eta_1, \ldots, \eta_{p-1}, n_p, \ldots, n_N), (m_1, \ldots, m_j, \ldots, m_M) ) \end{array} $$

The crossover operator for rule bases with different granularity is illustrated in Fig. 5. The length of each rule and the rule base size (number of rules) may be different for the two selected rule bases. Then a specific crossover operator, denoted by × in Fig. 5, was devised. To cross over the rule bases, the same crossover position p, the position where the granularities have been crossed, is kept.

Fig. 5

Rule base crossover

Rule antecedents of one rule base are combined with rule consequents of other rule base to form new rules. The size difference, the area I of the offspring in Fig. 5, is filled randomly choosing a corresponding portion of rules from the mating individual to complete the remaining part. The area II is discarded.

For instance, consider two individuals with granularity

$$ (1, 1, (3), (5)) $$

and

$$ (1, 1, (4), (3)) $$

and the rule bases¹⁶ shown in the top of Fig. 6. Crossover of these two individuals results in two offspring with granularity

$$ (1, 1, (3), (3)) $$

and

$$ (1, 1, (4), (5)) $$

whose rule bases are shown in the bottom Fig. 6. The last rule of offspring

$$ (1, 1, (4), (5)) $$

is a recombination of the last rule I 1011 ⇒ 010 of the second parent with the second rule I 110 ⇒ 00110, randomly chosen from its mate.

Fig. 6

Rule base crossover example

1.2 Mutation

Different mutation operators are used to produce new components of the chromosome. Each of the N + M granularities (component Cr ₁) are mutated with a probability p _Mutation . When the granularity, an integer in the interval [3,9], suffers mutation, a local variation is introduced either adding or subtracting 1 with equal probability. This database mutation scheme was suggested in [51]. When the granularity increases, a new pair of membership functions parameters randomly chosen is added to Cr ₂ and a set of linguistic terms is added to the rule base component Cr ₃. When the granularity decreases, a pair of membership functions parameters from a position chosen at random from Cr ₂ and the set of corresponding bits for the linguistic term at the same position of the rule base are deleted .

Mutation of membership function parameters uses the non-uniform Michalewicz mutation operator [60]. Let an individual with membership function be given by Eq. (7). The parameter $l_{k}^{X_{1}}$ whose valid interval is given by $(v_{min}^{X_{1}},v_{max}^{X_{1}})$ becomes $\lambda_{k}^{X_{1}},$ as in Eq. (9).

$$ \lambda_{k}^{X_{1}} = \left\{ \begin{array}{ll} l_{k}^{X_{1}} + \Updelta (t, v_{max}^{X_{1}} - l_{k}^{X_{1}})& {if\; c = 0}\\ l_{k}^{X_{1}} - \Updelta (t, l_{k}^{X_{1}} - v_{min}^{X_{1}} )& {if\; c = 1} \end{array}\right. $$

(9)

In expression (9), t is the current generation, c is a random number in {0, 1}, and the function Δ(t, y), expression (10), returns a value in the interval [0, y] such that the probability of Δ(t, y) being closer to 0 increases with t.

$$ \Updelta(t, y) = y (1 - r^{(1 - \frac{t}{T})^{b}}) $$

(10)

The value r is chosen randomly using a uniform distribution in the interval [0, 1], T is the maximum number of generations and b is a parameter that establishes the degree of dependency with the number of generations. For the experiments in this paper we set b = 5 as suggested in [60]. This property causes the operator to perform an uniform search in the initial space when t is small and a local one in the the later stages of the evolutionary process [51].

Rule bases are mutated using standard, bitwise reversing operation. Whenever an allele is 0 it becomes 1 after mutation and vice versa.

Appendix C: Knowledge bases

This section reports the knowledge bases developed by the coevolutionary GFS. The universes do not evolve. Input (demand) universe is fixed between 0 and 11,000, and output (bid price) is fixed between 0 and 5 times the agent marginal cost at full capacity (as given by the last column of Table 1). The linguistic terms are presented in terms of membership parameters needed to define the strong fuzzy partition, using the notation used in Appendix A. In rule syntax a 1 means a term that is used in rule inference, and 0 otherwise. The ⇒ sign separates the consequent from the antecedent. The position of terms in the rule syntax is consistent with the order of each linguistic term defined by the parameters of their membership functions.

1.1 Argentina I

$$ Cr_1: (1,1,(8), (4)) $$

Cr ₂: Fuzzy variable: load. Universe: (0.0, 11000.0).

$$ \begin{array}{ll} &((0.00, 0.00, 151.09, 440.62),\\&(151.09, 440.62, 441.53, 847.72),\\& (441.53, 847.72, 5100.58, 6223.19),\\&(5100.58, 6223.19, 6660.29, 6853.03), \\&(6660.29, 6853.03, 7558.87, 7725.61),\\&(7558.87, 7725.61, 8700.60, 10684.87),\\ &(8700.60, 10684.87, 10810.87, 10943.31),\\ &(10810.87, 10943.31, 11000.00, 11000.00)) \end{array} $$

Fuzzy variable: bid price. Universe: (0.0, 205.23).

$$ \begin{array}{ll} &((0.00, 0.00, 1.87, 37.24),\\&(1.87, 37.24, 119.68, 120.07),\\&(119.68, 120.07, 127.23, 196.59),\\&(127.23, 196.59, 205.23, 205.23))\end{array} $$

Cr ₃: Size of rules: 12. Number of rules: 4. Number of active rules: 1. Active rules:

$$ 11100100 \Rightarrow 1100 $$

1.2 Argentina II

$$ Cr_1: (1,1,(6),(5)) $$

Cr ₂: Fuzzy variable: load. Universe: (0.0, 11000.0).

$$ \begin{array}{ll} &((0.00, 0.00, 8.36, 51.92), \\&(8.36, 51.92, 361.73, 3767.61), \\&(361.73, 3767.61, 4988.87, 5225.23),\\&(4988.87, 5225.23, 5529.74, 6267.51),\\&(5529.74, 6267.51, 9292.04, 10954.12),\\&(9292.04, 10954.12, 11000.00, 11000.00))\end{array} $$

Fuzzy variable: bid price. Universe: (0.0, 205.23).

$$ \begin{array}{ll} &((0.00, 0.00, 14.51, 24.31),\\&(14.51, 24.31, 27.62, 83.23), \\&(27.62, 83.23, 117.88, 122.55), \\&(117.88, 122.55, 156.79, 191.14),\\&(156.79, 191.14, 205.23, 205.23))\end{array} $$

Cr ₃: Size of rules: 11. Number of rules: 12. Number of active rules: 2. Active rules:

$$ \begin{aligned} 101100 &\Rightarrow 10000\\111010 &\Rightarrow 01000 \end{aligned} $$

1.3 TermoRio

$$ Cr_1\text{:}\, (1,1((4),(6)) $$

Cr ₂: Fuzzy variable: load. Universe: (0.0, 11000.0).

$$ \begin{array}{ll} &((0.00, 0.00, 495.81, 703.40),\\&(495.81, 703.40, 1580.41, 6551.32),\\&(1580.41, 6551.32, 7236.99, 10868.00),\\&(7236.99, 10868.00, 11000.00, 11000.00)) \end{array} $$

Fuzzy variable: bid price. Universe: (0.0, 199.54).

$$ \begin{array}{ll} &((0.00, 0.00, 11.13, 43.77),\\ &(11.13, 43.77, 70.40, 76.47),\\&(70.40, 76.47, 96.22, 102.68),\\&(96.22, 102.68, 127.19, 140.53),\\&(127.19, 140.53, 145.48, 180.30),\\&(145.48, 180.30, 199.54, 199.54)) \end{array} $$

Cr ₃: Size of rules: 10. Number of rules: 14. Number of active rules: 1. Active rules:

$$ 1110 \Rightarrow 110000 $$

1.4 Ibirité

$$ Cr_1\text{:}\, (1,1,(8),(4)) $$

Cr ₂: Fuzzy variable: load. Universe: (0.0, 11000.0).

$$ \begin{array}{ll} &((0.00, 0.00, 37.88, 92.52),\\&(37.88, 92.52, 1078.46, 1245.47),\\&(1078.46, 1245.47, 4476.17, 5160.2),\\&(4476.17, 5160.2, 5256.63, 7622.05),\\&(5256.63, 7622.05, 8111.93, 8129.69),\\&(8111.93, 8129.69, 8989.09, 10335.80),\\&(8989.09, 10335.80, 10907.09, 10929.42),\\&(10907.09, 10929.42, 11000.00, 11000.00)) \end{array} $$

Fuzzy variable: bid price. Universe: (0.0, 199.27).

$$ \begin{array}{ll} &((0.00, 0.00, 0.18, 3.28),\\&(0.18, 3.28, 156.93, 164.87),\\&(156.93, 164.87, 174.6, 180.98),\\&(174.6, 180.98, 199.27, 199.27)) \end{array} $$

Cr ₃: Size of rules: 12. Number of rules: 6. Number of active rules: 2. Active rules:

$$ \begin{aligned} 01011000 &\Rightarrow 1000\\ 01000011 &\Rightarrow 0100 \end{aligned} $$