Optimal Stabilization of Social Welfare under Small Variation of Operating Condition with Bifurcation Analysis

Abstract

A social welfare optimization technique has been proposed in this paper with a developed state space based model and bifurcation analysis to offer substantial stability margin even in most inadvertent states of power system networks. The restoration of the power market dynamic price equilibrium has been negotiated in this paper, by forming Jacobian of the sensitivity matrix to regulate the state variables for the standardization of the quality of solution in worst possible contingencies of the network and even with co-option of intermittent renewable energy sources. The model has been tested in IEEE 30 bus system and illustrious particle swarm optimization has assisted the fusion of the proposed model and methodology.

Appendices

Overview of PSO

A brief overview of PSO method, which has been used in this paper for solving the problem of maximization of social welfare, is provided here.

Kennedy and Eberhart introduced the concept of function optimization by means of a particle swarm. Suppose the global optimum of an N-dimensional function is to be located. The function may be mathematically represented as:

$$f(x_{1} ,x_{2} ,x_{3} , \ldots \ldots ,x_{n} ) = f(X)$$

(4)

where X is the search vector, which actually represents the set of independent variables of the given function. The task is to find out such X, that the function value f(X) is either a minimum or a maximum denoted by f* in the search range. If the components of X assume real values then the task is to locate a particular point in the n-dimensional hyperspace which is a continum of such points.

PSO is a multi-agent parallel search technique. Particles are conceptual entities, which fly through the multidimentional search space. At any particular instant, each particle has a position and velocity. The position vector of a particle with respect to the origin of the search space represents a trial solution of the search problem. At the beginning, a population of particles is initialised with random positions marked by $\overrightarrow {{x_{i} }}$ and random velocities $\overrightarrow {{v_{i} }}$. The population of such particles is called a ‘swarm’ S. A neighbourhood relation N is defined in the swarm. N determines for any two particles $p_{i}$ whether they are neighbours or not. Thus for any particle $p$, neighbourhood can be assigned as N(p), containing all the neighbours of that particle. A popular version of PSO uses N = S for each particle. In this case, any particle has all the remaining particles in the swarm in its neighbourhood.

Each particle has two state variables, namely, its current position and velocity as stated earlier. It is also equiped with a small memory comprising its previous best position (One yielding the highest value of the fitnees fuction found so far) $\overrightarrow {p(t)}$ that is personal best experience and the best $\overrightarrow {p(t)}$ of $p \in N(p)$:$\overrightarrow {g(t)}$ that is the best position so far in the neighbourhood of the particle. When we set $N(p) = {\text{S}}$, $\overrightarrow {g(t)}$ is referred to as the globally best particle in the entire swarm.

The PSO scheme has the following algorithmic parameters:

• (i) $V_{\hbox{max} }$ or maximum velocity which restricts $\overrightarrow {{v_{i} }}$ within the interval [-$V_{\hbox{max} }$,$V_{\hbox{max} }$]

• (ii) An inertial weight factor ω

• (iii) Two uniformly distributed random numbers $\varPhi_{1}$ and $\varPhi_{2}$ that respectively determine the influence of $\overrightarrow {p(t)}$ and $\overrightarrow {g(t)}$ on the velocity update formula

• (iv) Two constant multiplier terms $c_{1}$ and $c_{2}$ known as ‘self-confidence’ and ‘swarm confidence’, respectively.

Initially the settings for $\overrightarrow {p(t)}$ and $\overrightarrow {g(t)}$ are $\overrightarrow {p(0)} = \overrightarrow {g(0)} = \overrightarrow {x(0)}$ for all particles. Once the particles are initialized, an iterative optimization process begins, where the positions and velocities of all the particles are altered by the following recursive equations. The equations are presented for the dth dimension of the position and velocity of the ith particle.

$$v_{id} (t + 1) = \omega .v_{id} (t) + c_{1} .\varPhi_{1} (p_{id} (t) - x_{id} (t)) + c_{2} .\varPhi_{2} (g_{id} (t) - x_{id} (t))$$

(5)

$$x_{id} (t + 1) = x_{id} (t) + v_{id} (t + 1)$$

(6)

The first term in the velocity updating formula represents the initial velocity of the particle. ω is the inertia factor. Venter and Sobeiski termed $c_{1}$ as ‘self confidence’ and $c_{2}$ as ‘swarm’ confidence. These terminologies provide an insight from a sociological standpoint. Since the co-efficient$c_{1}$ has a contribution towards self-exploration, we regard it as particles self-confidence. On the other hand, the particle$c_{2}$ has a contribution towards motion of the particles towards a global direction, which takes into account the motion of all the particles in the preceding program iteration, naturally its definition, as ‘Swarm Confidence’ is apparent. $\varPhi_{1}$ and $\varPhi_{2}$ stand for a uniformly distributed random number in the interval [0,1]. After having calculated the velocities and position for the next step t + 1, the first iteration of the algorithm is completed.

Description of the System with Generator and Consumer Cost Characteristics

The description of the system is given in Fig. 4.

Fig. 4

The IEEE 30 bus system

An Alternative Approach to Price Dependant Load Curtailment

In demand response scenario, the load demand is also considered to be following a predetermined schedule just like generation schedule, the ‘willingness to pay’ of individual consumers are available to the system operator. The operator under peak load condition can curtail a part of the load according to consumer’s demand response curve that but the process is entirely price dependant. This process does not concern the revenue loss suffered by the Gencos. A more rational way of curtailment should have considered not only the market price but also the generation surplus of the Gencos. The generation surplus can be defined as the difference between maximum generation available and load dispatched. The present work proposes a novel curtailment strategy, which not only adheres to consumer demand response curve, but also equally shields the Gencos against a foreseeable revenue loss.

In the proposed strategy, the ISO can set generation surplus $S_{\hbox{max} }$ for the Gencos above which the net curtailment would be zero to protect Gencos from further revenue loss, irrespective of price condition of the market. In demand response the minimum limit of demand is set by the consumer. To maintain reliability of supply and to build consumer’s confidence for the least supply this minimum dispatch has to be assured for all the time.

$${\text{S}}_{ \hbox{max} } = {\text{Maximum possible generation}} - {\text{Sum of minimum requested demand}}$$

When surplus is maximum that is S_max; the ISO should stick to zero curtailment policy conversely to maintain price equilibrium ISO must also set a maximum limit for curtailment $P_{\hbox{max} }$ when the surplus is minimum $S_{\hbox{min} }.$

The maximum curtailment limit $P_{\hbox{max} }$ can alternatively expressed in the form of

P_max = Sum maximum Limit of requested demand – Sum minimum limit of requested demand

For the rest of the cases ISO may choose the operating point from the curtailment-surplus relationship, which can be approximated by a straight line relation ship

$$P = P\hbox{max} - \frac{P{\hbox{max}} (S - S\hbox{min} )}{S\hbox{max} - S\hbox{min} }$$

(7)

where $S$ represents generation surplus and $P$ represent the corresponding upper limit of curtailment or ISO set limit of load curtailment. The proposed method utilizes this Eq. (7) to determine the maximum allowable load curtailment at any given price equilibrium point or ISO set limit of curtailment.

Objective Function of the Proposed Methodology

Maximize ($C(d) - C(glv))$ = social welfare with desired operating conditionswhere

$$C(d) = \;\sum\limits_{j = 1}^{{n_{d} }} {a_{j} P_{dj}^{2} + b_{j} P_{dj} } + \gamma_{j} Q_{dj} + \mu_{j} Q_{dj}$$

(8)

and

$$C(glv) = \;\left( {\sum\limits_{i = 1}^{{n_{g} }} {a_{i} P_{gi}^{2} + b_{i} P_{gi} } + c_{i} + P_{c} .P_{1} + T_{L} .P_{2} + V_{\hbox{min} } .P_{3} + P_{l\hbox{max} } .P_{4} + BI.P_{5} } \right)$$

(9)

$a_{i} ,b_{i} ,c_{i}$ is the cost co-efficients of the ith generator; $P_{c}$, the load curtailment, MW; $T_{L}$, the transmission loss, MW; $V_{\hbox{min} }$, the minimum bus voltage; $P_{l\hbox{max} }$, the maximum line flow; $P_{gi}$, the generation of the ith generator; $a_{j} ,\;b_{j}$, the bid coefficients of jth consumer for real power; $\gamma_{j}$,$\mu_{j}$, the bid coefficients of jth consumer for reactive power; $P_{dj}$, the power demand of the jth consumer in kW; $Q_{dj}$, the power demand of the jth consumer, kVAr, $V_{\hbox{min} }$, the minimum value of bus voltage; and$P_{1} ,P_{2} ,P_{3} ,P_{4} ,P_{5}$ are the penalties for limit violation set by ISO.

Equality or power balance constraints for the present OPF are

$$P_{Gi} - P_{Di} - V_{i} \sum\limits_{i = 1}^{n} {V_{j} } (G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij} ) = \;0$$

(10)

$$Q_{Gi} - Q_{Di} - V_{i} \sum\limits_{i = 1}^{n} {V_{j} } (G_{ij} \cos \theta_{ij} - B_{ij} \sin \theta_{ij} ) = \;0$$

(11)

where $P_{Gi}$ is the active power injected in bus i; $P_{Di}$, the active power demand on bus i; $V_{i}$, the magnitude of voltage of buses i; $V_{j}$, the magnitude of voltage of buse j; $G_{ij}$, the conductance of transmission line from bus i to j; $B_{ij}$, the susceptance of transmission line from bus i to j; and n is the number of buses.

Inequality or generator output constraints

$$P_{gi}^{\hbox{min} } \le P_{gi} \le P_{gi}^{\hbox{max} }$$

(12)

$$Q_{gi}^{\hbox{min} } \le Q_{gi} \le Q_{gi}^{\hbox{max} }$$

(13)

where, $P_{gi}$, $Q_{gi}$ are the active and reactive power of generator i respectively; $P_{gi}^{\hbox{min} }$, $Q_{gi}^{\hbox{min} }$, the lower limit of active and reactive power of the generators; $P_{gi}^{\hbox{max} }$, $Q_{gi}^{\hbox{max} }$ are the upper limit of active and reactive power of the generators.

Flow Chart of the Proposed Methodology

The flowchart of the proposed methodology is given in Fig. 5.

Fig. 5

Flowchart of the proposed methodology

Identification of Saddle Node and Hopf Bifurcations by Proposed Indices

As discussed in the earlier sections, the occurrence of small disturbances may perturb the operating point and can induce price thereby frequency instability. To insulate the system against small signal disturbances the proposed methodology introduces two bifurcation indices. In Hopf bifurcation the eigen values of the sensitivity matrix A has zero real part hence the system operating point is subjected to only oscillations within bounded limits.

Let the critical Eigen value is $\alpha + j\beta$

$$\left| {(\alpha + j\beta )I - A} \right| = 0$$

$$\left| {(\alpha I - A) + j\beta I} \right| = 0$$

(14)

A second order system $S^{2} + 2\zeta \omega_{n} S + \omega_{n}^{2}$ can be viewed as $(\omega^{2} - \omega_{n}^{2} ) + j2\omega \omega {}_{n}\zeta = 0$ where $S = j\omega$. At the point of oscillation that is Hopf bifurcation the system poles or Eigen values are placed at the imaginary axis. The magnitude of ζ can be derived from the previous equation as $- \frac{{\omega_{n}^{2} - \omega^{2} }}{{2\omega \omega_{n} }}$ that is $- \frac{{\text{Real} \,Part}}{{\sqrt {\text{Real} \,Part^{2} + \text{Imaginery} \,Part^{2} } }}$ applying the theory in Eq. (14)

one obtains

$$\zeta = - \frac{\alpha }{{\sqrt {\alpha^{2} + \beta^{2} } }}$$

(15)

for the critical Eigen values considered.

Minimum value of ζ not only ensures oscillations and low α but also optimizes the imaginary part of the equation. Thus, minimum or zero value of ζ guarantee both α = β = 0 which is the condition for saddle node bifurcation, too.

Hence against small signal oscillations we define bifurcation index (BI) corresponding to both Hopf and saddle node bifurcation as ${\rm BI} = - \frac{\alpha }{{\sqrt {\alpha^{2} + \beta^{2} } }}$ for all the critical Eigen values.

In the proposed algorithm along with generation cost and operational constraints this proposed index has been minimized. The maximum value of $\alpha_{\hbox{max} }$ can be defined as $- \frac{{\zeta_{\hbox{min} } }}{{\sqrt {\zeta_{\hbox{min} }^{2} + \beta^{2} } }}$ and the small signal stability margin at any stable operating point is defined as

$$\frac{{\alpha {}_{crit}}}{{\alpha_{\hbox{max} } }}$$

(16)

where $\alpha {}_{crit}$ is the real part of critical Eigen value at that operating point.

The Proposed Methodology and Case Studies

The base case is the state of equilibrium and the state variables obtained from this state is assumed as standard value of the state variables. The system description has been tabulated in Tables 1, 2 and 3 depicting the generator and load characteristics for the base case.

Table 1 Summary of the adopted system

Table 2 Generator cost co-efficients

Table 3 Coefficients of consumer cost benefit function

In the proposed methodology, the swarm intelligence based optimizer generates random solution particles in the workspace bounded by the maximum and minimum limit set by the generators and the consumers as described in Tables 2 and 3. In each step the optimizer generates an optimum, load schedule with an objective of maximizing social welfare. The global best solution is selected as optimal solution for given set operating constraints.

Base Case

As stated earlier, a stable operating point has to be obtained in order to perform bifurcation analysis. The state variables of this base case will be taken as state variables of this state. Now if the demand increases or contingency occurs or generation becomes intermittent, the optimal solution reconciles to a different set of state variables which can be taken as next state variables. The sensitivity matrix A between these two states is obtained by linear approximation. The eigen values of this matrix is determined. If all the Eigen values have zero real part then the solution becomes asymptotically stable against small signal disturbances To justify this equilibrium point Lyapunov’s concept has been followed that is $\dot{x} = 0$. As declared earlier, state variables are the generation of the six generators, operational status that is minimum bus voltage, maximum line flow total transmission loss, load curtailment and active and reactive power loads on 37 buses. Out of the 47 state variables, a few for the base case have been depicted in Table 4. The optimizer is trained to increase the loading up to maximum possible level. Hence the convergence of optimizer would suggest even with increase of loading the state variables are not changing or in other words $x(n + 1) = x(n)$ or $\dot{x} = 0$.

Table 4 Base case equilibrium and state variables in steady state

To monitor the convergence of rotor angle δ for the weakest bus (bus number 13), the same has been plotted against iterations as depicted in following figure (Fig. 6).

Fig. 6

Convergence of the rotor angle for the base case

Testing the Methodology with Transmission Line Contingency, Excessive Loading Stress

The examination of the feasibility and effectiveness of the algorithm proposed remains incomplete without testing its adaptability in the worst possible scenarios of the network. In each of the cases the Eigen values with the proposed methodology and small signal bifurcation index and without the index has been depicted. The BI as developed in the previous sections monitors the stability margin of each optimal solution and helps to constrict the solution within stability limit. The rotor angle or power angle convergence curve for each of the cases has also been depicted with and without the introduction of BI. In every case depicted in Table 5 the stability margin have been subjected to an improvement. The distance of the critical eigen value from the imaginary axis to the left of the plane has been referred as stability margin here.

Table 5 Testing the methodology in worst possible abnormal states of system

For the VAr loading case, the rotor angle has been plotted in Fig. 7. It is observable that the blue lines corresponding to the green has lesser deviation of rotor angle.

Fig. 7

Comparison of rotor angles for the weakest bus with and without small signal stability constraint

A comparison of final (steady state) value of rotor angles has been depicted for each and every case as an example in Fig. 8. It has been evidenced that the rotor angle for the weakest bus (no 30) is less than that with the introduction of BI. Low value of rotor angle signifies high synchronizing coefficient and stiffness of the system to maintain stability. Blue bars are for small signal stability constraint OPF (Fig. 8).

Fig. 8

Comparison of rotor angles with and without BI

The proposed methodology not only optimizes small signal stability but also improves operational condition of the power system network. To compare the performance in the first test bed active loading stress has been increased on selected number of buses and the operating conditions have been kept under observation. Though the minimum bus voltage gets least affected with active power loading stress, the Table 6 depicts a significant improvement in the line loss profile and line flow distribution compared to the traditional method. In the next step of study, reactive loading stress has been applied on the weakest bus of the system (bus no 30). As discussed earlier, excess requirement of reactive power not only degrades the operating conditions of the system especially the voltage profile but also affects the price of electricity. It is also imperative from Table 6 that the proposed algorithm working in a smart grid scenario would deliver more excelling operating conditions specially in terms of congestion management under single or multiple contingencies as compared to traditional optimization algorithms working in the environments which rely solely on forecasted demand with generation cost optimization objective and without any small signal stability constraint.

Table 6 Comparison of performance

In real time operation and control of power system, social welfare is becoming a popular objective being adopted worldwide for simultaneous optimal benefit for all the market participants. This work has depicted that the inclusion of the developed BI in social welfare objective can substantiate considerable stability margins of the solutions of the OPFs which in turn can increase the liquidity of power markets. Higher liquidity means that the price will remain same even in case of contingencies of transmission lines and generators.

Implementation of the Proposed Methodology with Intermittent Energy Sources

Promotion of distributed generation with renewable energy sources as stated earlier is one of the nonnegotiable objectives of the present and future grids. However, their intermittent generation profile requires extensive load management for power market equilibrium. With the intermittency of the generators as shown in Table 7, the performance of the proposed methodology has been observed with and without the BI for each hour. The rotor angle stability curve has been depicted as an example for one of the hours with and without the incorporation of BI.

Table 7 Forecasted percentage availability of intermittent sources

Table 8 reveals the ability of the methodology to improve stability margin even in case of intermittency of generation. As the methodology includes a small signal stability bifurcation index in the objective function, the optimal solution produced by it maximizes the distance of the operating point form the imaginary axis with standard operating conditions.

Table 8 Improvement of stability margin during intermittency of generation

With the intermittency of the generators, as shown in Table 7, the performance of the proposed algorithm has been observed to be remarkably superior in sustaining the operating conditions within safe limit (Table 9), by efficiently managing loads to cater maximum demand at minimum cost in the most optimal way maintaining all operating constraints.

Table 9 Performance comparison

Conclusion

The introduction of small signal stability criteria enriches the solution of OPF and improves the disruption resilience capability of modern power networks. The existing power markets are operated with the solution of most economical OPF in terms of generation cost. This interdependency of price of electricity and the state variables harness dynamics in price of electricity. This kind of dynamics in real time power markets creates oscillations can cause even voltage collapse and frequency instability. In pursuit of developing, a self healing small signal stability constraint OPF and to maximize social welfare, this paper introduces a state space based model and methodology to minimize the turbulences in the price of electricity in real time markets and to maximize load catering at minimum cost to improve the adoptability of modern power networks. To detect two important bifurcations namely Hopf and saddle node, this paper proposes a BI which directly estimates the margin of stability and penalizes the objective function to attain the possible degree of small signal stability. On application of this methodology in transmission line contingency, generation intermittency and excessive loading, encouraging results have been evidenced.